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J invariant 1728

elliptic curves - Why 1728 in $j$-invariant? - Mathematics

• The j -invariant for elliptic curves has a 1728 in it. According to Hartshorne, this is supposedly for characteristic- 2 and 3 reasons, despite appearances to the contrary. Indeed, it is unfathomable why it would help in char 2 and 3 when it would vanish. For that matter, the functions g 2, g 3 and ќФ too have these constants
• The j-invariant is given by j(E) = 1728 4a3 4a3 + 27b2: Theorem Let E;E0be elliptic curves over Q. Then EЋШ=E0over C if and only if j(E) = j(E0). In general, given a eld Kand elliptic curves E;E0over Kthen EЋШ=E0over Kif and only if j(E) = j(E0). Dylan Pentland The j-invariant of an Elliptic Curve 20 May 2018 7 / 1
• Over the complex numbers, the j -invariant is a map j: н†µні• вЯґ вДВ from the upper half plane to the complex numbers. This is a branched cover, with two branching points being 0, 1728 вИИ вДВ
• The j invariant DeпђБnition The j invariant7 is deпђБned as j(ѕД) = 1728 g3 2 (ѕД) вИЖ(ѕД), ѕД вИИ H. Note j istheratiooftwomodularforms of weight 12, hence it is a modular func-tion of weight 0. Since вИЖ has a simple zero at inпђБnity but vanishes nowhere else, j has a simple pole at inпђБnity and is holo-morphic on H. Let ќУd\H denote the closure of the modu
• The j-invariant of E is defined by j (E) = 1728¬Ј4A3/D.The aim of this note is to establish some results concerning the cardinality of the group of points on elliptic curves over Fp with j-invariants equals to 0 or 1728, and the connection between these cardinalities and some expressions of sum of squares.}, author = {Munuera G√≥mez, Carlos}
• The j-invariant of an elliptic curve The j-invariant of an elliptic curve with a-ne equation y2 = x3 +px+q is deпђВned to be j = 1728 4p3 4p3 +27q2: p and q are not uniquely deпђВned, if we change co-ordinates so that y0 = пђБ3y and x0 = пђБ2x for some пђБ 6= 0, then the new co-e-cients are p0 = пђБ4p and q0 = пђБ6q, so we still get the same value of j (1.1) DeпђБnition (j-Funktion/absolute Invariante (modular invariant)) Die j-Funktion/absolute Invariante wird deпђБniert als j := 1728 g3 2 D = 123 g3 2 D!. (1.2) Lemma a) Die j-Funktion ist eine Modulfunktion vom Gewicht 0. b) Sie ist holomorph auf H und hat einen einfachen Pol in ¬•. c) Sie deпђБniert eine Bijektion von H/G auf C. Beweis a) Die j-Funktion ist ein Quotient zweier Modulfunktionen vom Gewicht 12. Da j-invariant of Eis j(E) = 1728 4a3=(4a3 + 27b2). One can easily check that j(E) = 0 if and only if a= 0, and j(E) = 1728 if and only if b= 0. For any extension Kof k, the set of K-rational points on Eis E(K) = f(x;y) 2K K : y2 = x3 + ax+ bg[f1g, where 1is the point at in nity; we write E= E(k). The chord-and-tangent addition la Modular j-invariant. Table of contents: Definitions - Illustrations - Modular transformations - Special values - Connection formulas - Derivatives - Analytic properties - Hilbert class polynomials. Definition If the endomorphism ring is equivalent to the maximal order of the field, why is the j invariant equal to 0 or 1728? I know that given E: y 2 = x 3 + A x + B, if A = 0 then j = 0 and if B = 0 then j = 1728. The endomorphism ring will be isomorphic to an imaginary quadratic field if E is ordinary and to a quaternion algebra if E is supersingular

j-invariant of an elliptic curve. Given an elliptic curve ( E / K) where c h a r ( K) вЙ† 2, 3 defined by the Weierstrass equation y 2 = x 3 + a x + b. The j -invariant is j = 1728 4 a 3 4 a 3 + 27 b 2. I want to understand very clearly how this j-invariant is constructed and especially from where does the 1728 come In addition if p вЙ°3 mod 4 there is a supersingular elliptic curve (with j -invariant 1728) whose automorphism group is cyclic or order 4 unless p =3 in which case it has order 12, and if p вЙ°2 mod 3 there is a supersingular elliptic curve (with j -invariant 0) whose automorphism group is cyclic of order 6 unless p =2 in which case it has order 24 1 DIE J-INVARIANTE Satz 1.6. Es sei j6= 0 ;1728. Dann ist j die j-Invariante von y2 = x3 + 3j 1728 j x+ 2j 1728 j (1) Beweis. Nachrechnen. De nition 1.7. Wir de nieren die Standardrepr asentante f ur j6= 0 ;1728 durch (1). F ur j= 0 sei die Standardrepr asentante durch y2 = x3 + 1 und f ur j= 1728 durch y2 = x3 + xgegeben. Folgerung 1.8. Es gibt eine Bijektion K$Curves with j-invariant 1728 The elliptic curves with j-inarianvt 1728 are curves which have the form of E: y2 = x3 + Ax, and it is de ned over F p. romF Washington (2008), this curve is ordinary only when p 1 (mod 4). This curve corresponds to unique discriminant of the complex quadratic eld, D= 4 that is belong to class number one eld, see Cohen (1996) where K= Q (i). olloFw Proposition 2.1. • 1728 (number): is the number of cubic inches in a cubic foot. 1728 occurs in the algebraic formula for the j-invariant of an elliptic curve. As a consequence, it is sometimes Glossary of classical algebraic geometry: to the other pair. 2. A harmonic cubic is an elliptic curve with j-invariant 1728, given by a double cover of the projective. • жХ∞е≠¶гБІгБѓгАБи§Ззі†е§ЙжХ∞ ѕД гБЃеЗљжХ∞гБІгБВгВЛгГХгВІгГ™гГГгВѓгВєгГїгВѓгГ©гВ§гГ≥гБЃ j-дЄНе§ЙйЗП пЉИгВВгБЧгБПгБѓj-еЗљжХ∞пЉЙгБ®гБѓгАБи§Ззі†жХ∞гБЃдЄКеНКеє≥йЭҐдЄКгБЂеЃЪзЊ©гБХгВМгБЯ SL гБЃгВ¶гВІгВ§гГИ 0 гБЃгГҐгВЄгГ•гГ©гГЉеЗљжХ∞гБІгБВгВЛгАВj-дЄНе§ЙйЗПгБ®гБЧгБ¶гАБе∞ЦзВєгБІдЄАдљНгБЃж•µгВТжМБгБ§дї•е§ЦгБѓж≠£еЙЗгБ™йЦҐжХ∞гБІгБВгВКгАБжђ°гВТжЇАгБЯгБЩгВВгБЃгБМдЄАжДПгБЂеЃЪгБЊгВЛгАВ j = 0, j = 1728 j\left=0,\quad j=1728} jгБЃжЬЙзРЖеЗљжХ∞гБѓгГҐгВЄгГ•гГ©гГЉгБІгБВгВКгАБеЃЯйЪЫгБЂ. • 2 Answers2. This is not true. For instance j ( ѕД) = j ( ѕД + 1) = j ( вИТ 1 / ѕД) for every ѕД вИИ H. More generally, ( 2, Z). It is true, however, that j: H / ќУ ( 1) вЖТ C is a bijection. See Chapter 1, Section 3, of Silverman's Advanced Topics in the Theory of Elliptic Curves • Case of j-invariant 1728 Consider an elliptic F q2-curve E a: y2 = x3 +ax of j-invariant 1728, where p 1 (mod 4 ), that is i := p 1 2F p. The latter condition is necessary and su cient for the ordinariness of E a. Our technique also remains to be valid for compressing F q-points of E2 a (if a 2F q) and F q2-points of E a, because there is on E a the F q-automorphism [i]: (x;y) 7!( x;iy) of. • 1.2 The j-function De nition 1.2.1. The j-function is a complex-valued function de ned on all ЋЭ 2C such that j(ЋЭ) = 1728 g 2(ЋЭ)3 g 2(ЋЭ)3 27g 3(ЋЭ)2; where g 2(ЋЭ) and g 3(ЋЭ) are certain in nite sums over all the points in a particular lattice, as de ned in Section 2.4. Jay has many guises, though! Setting q= e2ЋЗiЋЭ, we can write j(ЋЭ) = 1 • j(ѕД) 1728: 0: 0: вИЮ : вИТ15 3: вИТ32 3: вИТ96 3 нГАмЫР л™®лУИлЭЉ j-нХ®мИШ (elliptic modular function, j-invariant) . гАКмИШнХЩлЕЄнКЄгАЛ. мЭі лђЄмДЬлКФ 2020лЕД 5мЫФ 21мЭЉ (л™©) 12:27мЧР лІИмІАлІЙмЬЉл°Ь нОЄмІСлРШмЧИмКµлЛИлЛ§. л™®лУ† лђЄмДЬлКФ нБђл¶ђмЧРмЭінЛ∞лЄМ мї§л®Љм¶И м†АмЮСмЮРнСЬмЛЬ-лПЩмЭЉм°∞к±іл≥Ак≤љнЧИлЭљ 3.0мЧР лФ∞лЭЉ мВђмЪ©нХ† мИШ мЮИмЬЉл©∞, мґФк∞Ам†БмЭЄ м°∞к±імЭі м†БмЪ©лР†. In mathematics, Klein's j-invariant or j function, regarded as a function of a complex variable ѕД, is a modular function of weight zero for SL(2, Z) defined on the upper half-plane of complex numbers. It is the unique such function which is holomorphic away from a simple pole at the cusp such tha j -invariants of elliptic curves over finite fields. Let K be a finite field, and K ¬ѓ its algebraic closure. It is well known that two curves are isomorphic over K ¬ѓ if and only if they have the same j -invariant. If two such curves are also K -isogenous, I believe we can conclude that they are K -isomorphic, but I cannot find any reference. DOI: 10.1007/s00145-010-9065-y) is answered, that is, studying the performance of 4-dimensional GLV method for faster point multiplication on some GLS curves over Fp2 with j-invariant 1728. Finally some results and examples are presented, showing that the 4-dimensional GLV method runs in between 70% and 73% the time of the 2-dimensional GLV method which Galbraith et al. did in their work.EI0225-28 Have a look at the MathWorld article for the Klein invariant, for instance.$\endgroup$- J. M.'s torpor вЩ¶ Dec 19 '16 at 4:43$\begingroup$@J.M. - The series expansion Series[1728*KleinInvariantJ[ѕД], {ѕД, 0, 4}] has the ѕД in the denominator of the exponent of E$\endgroup$- Bob Hanlon Dec 19 '16 at 4:5 Klein's absolute invariant J=j/1728 is Gamma-modular. (n+1)*A000521(n)/24 yields integral values - see A161395. - Alexander R. Povolotsky, Jun 09 2009. The Mathematica implementation of KleinInvariantJ[] (versions 6 to 8) had bugs giving wrong value for a, a, a and other values. - Michael Somos, Mar 07 2012. It is an open question if there are infinitely many k such that a(k) is. j-invariant in nLa The j-invariant vanishes at the corner of the fundamental domain at. Here are a few more special values (only the first four of which are well known; in what follows, j means J/1728 throughout): Several special values were calculated in 2014: and let, All preceding values are real sage: E = EllipticCurve (j = 1728); E; E. j_invariant (); E. label Elliptic Curve defined by y^2 = x^3 - x over Rational Field 1728 '32a2' sage: E = EllipticCurve (j = GF (5)(2)); E; E. j_invariant Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 5 2. See trac ticket #6657. sage: EllipticCurve (GF (144169), j = 1728) Elliptic Curve defined by y^2 = x^3 + x over Finite. SUPERSINGULAR j-INVARIANTS, HYPERGEOMETRIC SERIES, AND ATKIN'S ORTHOGONAL POLYNOMIALS M. Kaneko and D. Zagier x1. Introduction. An elliptic curve E over a eld K of characteristic p>0 is called supersingular if the group E(K ) has no p-torsion. This condition depends only on the j-invariant of E and it is well known (cf. x2 for a review) that there are only nitely many supersingular j. EUDML Elliptic curves with j-invariant equals 0 or 1728 1. istic finite field mapping h: Fq вЖТ Ea(Fq) F F. to the case of any elliptic Fq. F. -curve Ea: y2 = x3 вИТ ax of j -invariant 1728 2. ant of the cubic x3 +Ax +B ((r1 вИТr2)(r1 вИТr3)(r2 вИТr3))2 = вИТ(4A3 +27B2) change of variables (2.8) leaves j unchanged. The converse is true, too! Rong-Jaye Chen 2.6 The j-invariant ECC 2008 4 / 8 Cryptanalysis Lab. Theorem 2.18 Theorem 2.18 Given y2 1 = x 3 1 +A1x1 +B1 with j1 y2 2= x 3 2 +A x +B with j If j1 = j2. 3. Elliptic curves over a nite eld Fq with j-invariant 0 or 1728, both supersingular and ordinary, whose embedding degree k is low are studied. In the ordinary case we give conditions characterizing such elliptic curves with xed embedding degree with respect to a subgroup of prime order вДУ. For k = 1;2, these conditions give parameterizations of q in terms of вДУ and two integers m, n. We show. Modular j-invariant - Fungrim: The Mathematical Functions Displaying similar documents to Elliptic curves with j-invariant equals 0 or 1728 over a finite prime field. K 2 of elliptic curves with sufficient torsion over Q. Raymond Ross (1992) Compositio Mathematica. Similarity: q-series, elliptic curves, and odd values of the partition function. Eriksson, Nicholas (1999) International Journal of Mathematics and Mathematical Sciences. Similarity. Title: j-invariant: Canonical name: Jinvariant: Date of creation: 2013-03-22 13:49:54: Last modified on: 2013-03-22 13:49:54: Owner: alozano (2414) Last modified by. A curve defined over a given field K the j -invariant of an elliptic curve is an element of that field. Therefore for a finite field of prime order the j -invariant can be represented by a number less than p. As for which j -invariants are possible, they all are! The curve y2 + xy = x3 вИТ 36 j0 вИТ 1728x вИТ 1 j0 вИТ 1728 Medientyp: Buch Titel: Congruences for the coefficients of 3j(t) and j(t)-1728 where j(t) is the modular invariant Beteiligte: Reidar Erevik, Olaf [VerfasserIn] Erschienen: Bergen [u.a.]: Norwegian Univ. Press, 1967 [erschienen] 1968 Erschienen als: ¬∞Arbok for Universitetet i Bergen; 1966, Le j-invariant, parfois appel√© fonction j, est une fonction introduite par Felix Klein pour l'√©tude des courbes elliptiques, qui a depuis trouv√© des applications au-del√† de la seule g√©om√©trie alg√©brique, par exemple dans l'√©tude des fonctions modulaires, de la th√©orie des corps de classes et du monstrous moonshine.. Motivation : birapport et j-invariant La j- invariante desaparece en la esquina del dominio fundamental en ( + ) = Aqu√≠ hay algunos valores especiales m√°s dados en t√©rminos de la notaci√≥n alternativa J ( ѕД ) вЙ° 1 / 1728 j ( ѕД ) (solo los primeros cuatro son bien conocidos) ж•ХеЖЖжЫ≤зЈЪгБЃj дЄНе§ЙйЗПгБЂйЦҐгБЩгВЛи©±й°М дєЭеЈЮе§Іе≠¶ жХ∞зРЖе≠¶з†Фз©ґзІС йЗСе≠РжШМдњ° гБѓгБШгВБгБЂеАЛдЇЇзЪДгБ™гБУгБ®гВТе∞СгБЧгАВиЗ™еИЖгБМгБѓгБШгВБгБ¶j invariant гБ®гБДгБЖгВВгБЃгБЂеЗЇдЉЪгБ£гБЯгБЃгБМгБДгБ§гБ†гБ£ гБЯгБЛгАБгВВгБЖж≠£зҐЇгБЂгБѓжАЭгБДеЗЇгБЫгБЊгБЫгВУгБМгАБoпђГcial гБЂгБѓгАБе≠¶йГ®4еєігБЃгВЉгГЯгБІдЉКеОЯеЇЈйЪЖеЕИзФЯгБЃгВВгБ®гАБ Lang гБЃElliptic Functions гВТи™≠гВУгБ†гБ®гБН. A direct calculation shows that each of these curves has j-invariant equal to 1728. We will show that the second curve cannot occur; the proof of the пђБrst is the similar. If this curve had good reduction, we could use a transformations of the form Y0 = 8Y+sX+t, X0 = 4X+r, and we пђБnd (8Y+sX+t)2 = (4X+r)3 +6(4X+r)2 +8(4X+r) GROUP SCHEMES 5 which is 64Y2 +16sXY+16tY = 64X3 +(48r+96вИТs2)X2. DeпђБnition14.11. Thej-invariant oftheellipticcurveE: y2 = x3 + Ax+ Bis j(E) = j(A;B) = 1728 4A3 4A3 + 27B2: Note that the denominator of j(E) is nonzero, since it is the discriminant of the cubic x3 + Ax+ B, which has no repeated roots. There are two special cases worth noting: if A= 0 thenj(A;B) = 0,andifB= 0 thenj(A;B) = 1728. The j-invariant of E, which determines the isomorphism class of E over F вАЊ p, is j (E) = 1728 вЛЕ 4 a 3 4 a 3 + 27 b 2. The endomorphism ring End (E) of E is the set of all isogenies between E and itself$1728$Discriminant$-1728$j-invariant $$0$$ CM: yes ($$D=-3$$) Rank:$1$Torsion structure: trivial: Related objects. Isogeny class 1728.o; Minimal quadratic twist (by -8) 27.a4; All twists ; L-function; Modular form 1728.2.a.o; Downloads. q-expansion to text; All stored data to text; Code to Magma; Code to SageMath; Code to GP ; Learn more. Source and acknowledgments; Completeness of the. The curve with j-invariant 1728 has a unique discriminant of the imaginary quadratic field. This unique discriminant of quadratic field yields a unique efficiently computable endomorphism, which later able to speed up the computations on this curve. However, the ISD method needs three endomorphisms to be accomplished. Hence, we choose all three endomorphisms to be from the same imaginary. The exceptions for j-invariants 0 = j(ѕБ) and 1728 = j(i) arise from the fact . that the corresponding elliptic curves y 2 3 = x + B and y 2 3 = x + Ax have automorphisms ѕБ: (x, y) 7вЖТ(ѕБx, y) and i: (x, y) 7вЖТ( x, iy), respectively, where ѕБ and i denote third and fourth roots of unity, respectively, in both End(E) and k ¬ѓ . The automorphism 1 does not pose a problem because it пђБxes. thej-invariant is one of173, 2573, 153, 2553, or 203. The DofTheorem2(b) is 65, 65,-7, 7, 2 respectively. As65, 65, -7,-7, 2, respectively, mustbearational normfromk, onlythefirst twovaluesofA cangive curves over complexquadratic fields. This is consistent with . Section 1 gives the proofs of the Theorems and Section 2 gives some examples. Section 1 Thefollowing lemmacollects all the facts. 1.2 Characterization by j-invariant. The j-invariant of an elliptic curve E(K) : y. 2 = x. 3 +Ax+Bis an invariant of the isomorphism class of Ein an algebraic closure of Kde ned as j= 1728 4A. 3. 4A. 3 + 27B. 2: (5) Note that all curves of j-invariant 0 take the form y. 2 = x. 3 +B, and all curves of j-invariant 1728 are of the form y. 2 = x. 3. The j-invariant of E is defined by j(E) = 1728¬Ј4A3/D. The aim of this note is to establish some results concerning the cardinality of the group of points on elliptic curves over Fp with j-invariants equals to 0 or 1728, and the connection between these cardinalities and some expressions of sum of squares j= c3 4 = the last two quantities, and j (we call this the 'j-invariant') are particularly important. If we have characteristic not 2 or 3 and our equation takes the form y2 = x3 + Ax+ B then it turns out = 16(4A3 + 27B2) and j= 1728 (4A)3 Theorem 2.7. A Curve described by a Weierstrass equation is singular if and only if = 0 Proof. Recall. This condition depends only on the j-invariant of E and it is well known that there are only finitely many supersingular j-invariants in \bbfFp. \par The authors of the paper under review describe several different ways of constructing canonical polynomials in \bbfQ [j] whose reduction modulo p gives the supersingular polynomial ssp(j):= \prod\Sb E/\overline\bbfFp\\ E\text supersingular \endSb. We have the following q j for supersingular j-invariant j in F p: q 57 = 11, q 59 = 59, q 66 = 67, q 64 = 83, q 2 = 139, q 21 = 163. Thus q j can be bigger than p. 3. Neighborhood of supersingular elliptic curves. In this section we assume that j вИИ F p \ {0, 1728} is a supersingular j-invariant, E = E j, End (E) = O and q = q j Endomorphism ring of a Elliptic Curve and$j$invarian En matem√°ticas, j-invariante de Klein o funci√≥n j, considerada como una funci√≥n de una variable compleja ѕД, es una forma modular de peso cero para SL(2, Z) definida sobre el semiplano positivo de n√Їmeros complejos.Es la √Їnica funci√≥n tal que es holomorfa lejos de un polo simple de la c√Їspide tal que =, =Las funciones racionales de j son modulares, y de hecho proporcionan todas las. j-Funktion. j. -Funktion. einer elliptischen Kurve auf folgende Weise zugeordnete Invariante. Es sei k der Grundk√ґrper. Wenn seine Charakteristik вЙ† 2 und вЙ† 3 ist, so l√§√Яt sich die Kurve so in \ ( { {\mathbb {P}}}^ {2}\) einbetten, da√Я sie in affinen Koordinaten durch eine Gleichung \begin {eqnarray} {y}^ {2}= {x}^ {3}+ab+b\end {eqnarray. (2.1) De nition (j-Funktion, absolute Invariante) Die Funktion j : H!C,z 7! (12g 4 (z)) 3 D(z) heiЋЗt j-Funktion oder absolute Invariante. Die ersten Eigenschaften der j-Funktion werden zusammengefasst in dem folgenden (2.2) Lemma 1. Die j-Funktion ist eine modulare Funktion vom Gewicht 0. 2. Die absolute Invariante ist holomorph auf H und hat einen einfachen Pol in ¬• mit Res¬• (j(z)) = 1. 3. We present an algorithm to compute the number ofF q -rational points on elliptic curves defined over a finite fieldF q , withj-invariant 0 or 1728. This algorithm takesO(log3 p) bit operations, werep is the characteristic ofF q 1728 is the cube of 12 and, as such, is important in the duodecimal number system, in which it is represented as 1000. It is the number of cubic inches in a cubic foot. 1728 occurs in the algebraic formula for the j-invariant of an elliptic curve. As a consequence, it is sometimes called a Zagier as a pun on the Gross-Zagier theorem j-invariant of an elliptic curve - Cryptography Stack Exchang 1. Liftingthe j-invariant:QuestionsofMazurandTate Lu√≠sR.A.Finotti DepartmentofMathematics,UniversityofTennessee,Knoxville,TN37996,UnitedStates article info abstract Articlehistory: Received11 December 2008 Availableonline31 December2009 Communicatedby FernandoRodriguez Villegas Keywords: Ellipticcurves Canonical lifting Pseudo-canonical lifting Modularpolynomial In this paper we analyze the j. 2. Introduction Complex multiplication theory Real quadratic number elds unctionF elds in positive characteristic Haeys theory The quantum j invariant Proof Quantum j invariant and Real Multiplication program for global function elds Luca Demangos (Joint work with T. M. Gendron - UNAM, Mexico) University of Stellenbosc 3. En matem√†tiques, el j-invariant o funcio j de Felix Klein, considerada com a funci√≥ d'una variable complexa ѕД, √©s una funci√≥ modular de pes zero per a SL(2, Z) definida al semipl√† superior dels nombres complexos. √Йs l'√Їnica funci√≥ que √©s holomorfa allunyada d'un simple pol a la c√Їspide de manera que (/) =, = =.Les funcions racionals de j s√≥n modulars i, de fet, ofereixen totes les. 4. Abstract: We study the j-invariant of the canonical lifting of an elliptic curve as a Witt vector. We prove that its Witt coordinates lie in an open affine subset of the j-line and deduce the existence of a universal formula for the j-invariant of the canonical lifting. The canonical lifting of the elliptic curves with j-invariant 0 and 1728 over any characteristic are also explicitly found. 5. j-дЄНе§ЙйЗПгБѓгБВгВЛзД°йЩРеТМпЉИдЄЛи®ШгБЃ g 2, g 3 гВТеПВзЕІпЉЙгБІзіФз≤ЛгБЂеЃЪзЊ©гБЩгВЛгБУгБ®гБМгБІгБНгВЛгБМгАБгБУгВМгВЙгБѓж•ХеЖЖжЫ≤зЈЪгБЃеРМеЮЛй°ЮгВТиАГгБИгВЛгБУгБ®гБМеЛХж©ЯгБ®гБ™гВЛгАВ C дЄКгБЃгБЩгБєгБ¶гБЃж•ХеЖЖжЫ≤зЈЪ E гБѓи§Ззі†гГИгГЉгГ©гВєгБІгБВгВЛгБЃгБІгАБгГ©гГ≥гВѓ 2 гБЃж†Ље≠РгАБгБ§гБЊгВК C гБЃ 2 жђ°еЕГж†Ље≠РгБ®еРМдЄАи¶ЦгБІгБНгВЛгАВ ж†Ље≠РгБЃдЇТгБДгБЂеє≥и°МгБ™еПНеѓЊеБігБЃиЊЇгВТеРМдЄАи¶Ц. International Journal of Computer Mathematics. Periodical Home; Latest Issue; Archive; Authors; Affiliations; Home Browse by Title Periodicals International Journal of Computer Mathematics Vol. 93, No. 12 Elliptic curves with j = 0,1728 and low embedding degree Browse by Title Periodicals International Journal of Computer Mathematics Vol. 93, No. 12 Elliptic curves wit If you have a disability and are having trouble accessing information on this website or need materials in an alternate format, contact web-accessibility@cornell.edu for assistance.web-accessibility@cornell.edu for assistance For instance, if j = 1728, this would return a different curve (of conductor 32 instead of 64). The library syntax is GEN ellfromj(GEN j). ellgenerators(E) If E is an elliptic curve over the rationals, return a ℤ-basis of the free part of the Mordell-Weil group attached to E. This relies on the elldata database being installed and referencing the curve, and so is only available for curves. Supersingular elliptic curve - Wikipedi j-invariants, we will describe the possible images of the '-adic representation arising from elliptic curves over Q. Denote by ЋЖ E(Gal Q) the group generated by Iand ЋЖ E(Gal Q). The following theorem describes the possibilities for ЋЖ E(Gal Q), up to conjugacy, when j E is not in the ( nite!) set J '. Theorem 1.4. (i) The set J ' is nite 101774 2021 69 Finite Fields Their Appl. https://doi.org/10.1016/j.ffa.2020.101774 db/journals/ffa/ffa69.html#Koshelev21 Dmitrii Koshele May 03, 2021 ¬Ј 1728 is the cube of 12 and, as such, is important in the duodecimal number system, in which it is represented as 1000. It is the number of cubic inches in a cubic foot. 1728 occurs in the algebraic formula for the j-invariant of an elliptic curve жЦЗзМЃгАМ4-dimensional GLV method on GLS elliptic curves with j-invariant 1728гАНгБЃи©≥зі∞жГЕе†±гБІгБЩгАВJ-GLOBAL зІСе≠¶жКАи°УзЈПеРИгГ™гГ≥гВѓгВїгГ≥гВњгГЉгБѓз†Фз©ґиАЕгАБжЦЗзМЃгАБзЙєи®±гБ™гБ©гБЃжГЕе†±гВТгБ§гБ™гБРгБУгБ®гБІгАБзХ∞еИЖйЗОгБЃзЯ•гВДжДПе§ЦгБ™зЩЇи¶ЛгБ™гБ©гВТжФѓжПігБЩгВЛжЦ∞гБЧгБДгВµгГЉгГУгВєгБІгБЩгАВгБЊгБЯJSTеЖЕе§ЦгБЃиЙѓи≥™гБ™гВ≥гГ≥гГЖгГ≥гГДгБЄж°ИеЖЕгБДгБЯгБЧгБЊгБЩгА %i %s -1728,-233,0,90,539,1539,3375,6511,11663,19897,32768,52512,82306, %t 126616,191658,286008,421407,613808,884736,1263051,1787217,2508208, %u 3493226,4830420. The same was done by Najman for cyclic $$\ell$$-isogenies of elliptic curves with rational j-invariant . We aim to give a more precise characterization of the degrees of number fields over which elliptic curves exhibit certain properties deducible from the images of their Galois representations. Recently, Sutherland developed an algorithm to efficiently compute Galois images of elliptic. 1728 is the natural number following 1727 and preceding 1729.1728 is a dozen gross, one great gross (or grand gross, or, in Germanic, Mass [citation needed]).In mathematics [ edit ] 1728 is the cube of 12 and, as such, is important in the duodecimal number system, in which it is represented as 1000 1728 is the natural number following 1727 and preceding 1729. 1728 is a dozen gross, one great gross (or grand gross, or, in Germanic, Mass). (en) 1728пЉИеНГдЄГзЩЊдЇМеНБеЕЂгАБгБЫгВУгБ™гБ™гБ≤гВГгБПгБЂгБШгВЕгБЖгБѓгБ°пЉЙгБѓиЗ™зДґжХ∞гАБгБЊгБЯжХіжХ∞гБЂгБКгБДгБ¶гАБ1727гБЃжђ°гБІ1729гБЃеЙНгБЃжХ∞гБІгБВгВЛгАВ (ja Imaginary quadratic field data (2013 updates) Numbers of newforms (rational, cuspidal, weight 2) These counts are expected to equal the number of isogeny classes of. There are various forms of the j-invariant, but this wiki article shows the constant 1728 j-invariant - Wikipedia It mentions that the modular discriminant is defined in terms of two functions $g_2$ and $g_3$ . The discrimina.. 1728 ¬™-2√§pz+744+√Ґ k=1 ¬• Transcendentality of Klein invariant J for algebraic argument The value JHaL is transcendental for any algebraic a where ImHaL>0 and a not being a quadratic irrational. Moonshine conjecture The dimensions of the representations of the monster group (the largest simple sporadic group) of order 246¬Ј320¬Ј59¬Ј112¬Ј133¬Ј17¬Ј19¬Ј23¬Ј29¬Ј31¬Ј41¬Ј47¬Ј59¬Ј71 are simple.  1728 and 1729 - This pair just didn't have quite enough going for them to make it. 1728 is an important j-invariant of elliptic curves and modular forms, and is a perfect cube. 1729 happens to be the third Carmichael number, but the primary motivation for including 1729 is because of the mathematical folklore associated it to being the first. modular j¬°invariant and Ramanujan's cubic theory of elliptic functions to alternative bases. We also show that for certain integers n, tn gen-erates the Hilbert class пђБeld of Q(p ¬°n). This shows that tn is a new class invariant according to H. Weber's deпђБnition of class invariants. 1. Introduction Except for four entries, the last two pages in Ramanujan's third notebook, pages 392. J-invariante. Origem: Wikip√©dia, a enciclop√©dia livre. Em matem√°tica, Klein j-invariante, tida como uma fun√І√£o de uma vari√°vel complexa ѕД, √© uma fun√І√£o modular definida sobre o semiplano superior de n√Їmeros complexos. N√≥s temos: Klein j-invariante em um plano complexo. j ( ѕД ) = 1728 g 2 3 ќФ . j (\tau )=1728 {g_ {2. Inom matematiken √§r Kleins j-invariant, sedd som en funktion av komplexa variabeln ѕД, en modul√§r funktion av vikt noll f√ґr SL(2, Z) definierad i √ґvre planhalvan av komplexa planet. Den √§r den unika funktionen med dessa egenskaper som √§r analytisk f√ґrutom vid en spets d√§r den har en enkel pol s√• att Kleins j-invariant i komplexa planet =, = Rationella funktioner av j √§r modul√§ra. J 3 Figure 3: Euler's resolvent cubic ( ) with three real roots (вДО2 > 2 , i.e. 4 3 > 2) which are all positive (н†µнЉА 2>0, 2 > ). The conditions н†µнЉА 2 >0, 2 < are associated with two negative roots (dashed curve). Note that 2, , , are constant multiples respectively of the resolvent's geometric parameters , , 2 , ( 1 = 64 6, 2 = 48 2, 3 = 1728 3, 4 = 12 ). The invariants ,. кЈЄлЯђл©і мЬ†л™ЕнХЬ j-invariantлКФ лЛ§мЭМк≥Љ к∞ЩмЭА нШХнГЬл°Ь мУЄ мИШ мЮИлЛ§. $j(\tau)=\frac{E_4^3}{\Delta(z)}=1728\frac{E_4^3}{E_4^3-E_6^2}=12^3\frac{E_4^3}{E_4^3-E_6^2}.$ лђЉл°† мЭілЯђнХЬ к≤ГлУ§мЧР лМАнХімДЬ лЭЉлІИлИДмЮФмЭі лІ§мЪ∞ мЮШ мХМк≥† мЮИмЧИмЭМмЭА мЭШмЛђмЭШ мЧђмІАк∞А мЧЖлЛ§. лЭЉлІИлИДмЮФмЭі лґДнХ†мИШмЧР лМАнХШмЧђ мЭіл£ђ л∞Ьк≤ђлПД нХЬл≤И мВінОіл≥Љ нХДмЪФк∞А мЮИлЛ§. \[ \begin. ISSN: 1029-0265. Compartir. Estad√≠sticas Ver Estad√≠stica J Invariant 1728 - The 1 Best Images, Videos & Discussions 1728 Google Scholar. Crossref. Search ADS. PubMed 3 by epidermal Langerhans cells correlates with the level of biosynthesis of MHC class II molecules and expression of invariant chain. J Exp Med. 1990. 172. 1459. 1469 Google Scholar. Crossref. Search ADS. PubMed 8. Schuler. G, Steinman. RM. Murine epidermal Langerhans cells mature into potent immunostimulatory dendritic cells in vitro. J. Un invariant j √©tant donn√©, comment retrouver un point ѕД qui s'envoie sur j. On souhaite √©galement que ѕД varie contin√їment en fonction de j. L'√©quation de Picard Fuchs permet d'apporter une r√©ponse √† ce probl√®me. Pour un invariant j distinct de 0 et 1 728, on peut obtenir une courbe elliptique d'invariant j dans la famille de Legendr with xed j-invariant j passing through 3dвИТ 1 general points P2. E d;j is de ned for d 3and16=j2M1;1. In this note, the following relations are established: 8j6=0;1728;1;E d;j = dвИТ1 2 N d; j=0;E d;0 = 1 3 dвИТ1 2 N d; j= 1728;E d;1728 = 1 2 dвИТ1 2 N d: If d 0 mod3, then 3 - dвИТ1 2.SinceE3l;0 is an integer, N3l 0 mod3forl 1. In fact, a check of values in [DF-I] shows N d 0 mod3 if and only. j-invariant Le j-invariant, parfois appel√© fonction j, est une fonction introduite par Felix Klein pour l'√©tude des courbes elliptiques, qui a depuis trouv√© des applications au-del√† de la seule g√©om√©trie alg√©brique, par exemple dans l'√©tude des fonctions modulaires, de la th√©orie des corps de classes et du monstrous moonshine j-INVARIANTS AND j-ZEROS OF CERTAIN EISENSTEIN SERIES P. GUERZHOY AND Z. KENT Abstract. Let p>3 be a prime. We consider j-zeros of Eisen-stein series E k of weights k= p 1+Mpa(p2 1) with M;a 0 as elements of Q p. If M= 0, the j-zeros of E 1belong to Q ( 2) by Hensel's Lemma. Call these j-zeros p-adic liftings of supersin-gular j-invariants. We show that for every such lifting uthere is a j. Article 4-dimensional GLV method on GLS elliptic curves with j-invariant 1728 Detailed information of the J-GLOBAL is a service based on the concept of Linking, Expanding, and Sparking, linking science and technology information which hitherto stood alone to support the generation of ideas. By linking the information entered, we provide opportunities to make unexpected discoveries and. j-дЄНе§ЙйЗП - Wikipedi Es bezeichne j die j-Invariante. Wir deпђБnieren weiter die Mengen M1:ЋШ ' z 2Cjz ЋШвА∞¬ѓit,t 2RвАЪ0 , M2:ЋШ{z 2Cjz ЋШit,t 2RвАЪ1}, M3:ЋШ вА∞ z 2Cjz ЋШe2i¬µ, 1 4 вАҐ¬µвАҐ 1 3 ¬Њ. wobei wie √Љblich вА∞:ЋШe2i/3. Zeigen Sie: (i) j(M1) ЋШ(¬°1,0], (ii) j(M2) ЋШ[1728,1), (iii) j(M3) ЋШ[0,1728]. Abgabe: Montag, 07.05, bissp√§testens11Uhrct. imTutorenbriefkastenNr. 53inINF 205im erstenStock. Let K be a field, E/K and E'/K two elliptic curves over K with the same j invariant: j(E/K) = j(E'/K) in K. Suppose further that j(E/K)(j (E/K) - 1728) is invertible in K. Then E'/K is a quadratic twist of E/K. Our main interest will be in the case when 6 is invertible in K. But first let us review the cases of characteristics 2 and 3. (11.1.2) If K has characteristic 2, then E is ordinary. A special modular function:$ j \$-invariant

Question: Consider The J-invariants Of Elliptic Curve: E : —Г–≥-x3 +ax+ B, Such That J(E) 1728 (E1)j(E2), There Exists U E C So That A2a And B 6bi 4a3 8276 Prove: For Elliptic Curves E1, E2 Over Q, Such Tha We show that this number is independent of the position of the points and the value of the j-invariant and that it coincides with the number of complex elliptic curves (with j-invariant j / вИИ {0, 1728}). The result can be used to simplify G. Mikhalkin's algorithm to count curves via lattice paths (see ) in the case of rational plane. Word definitions in dictionaries Wikipedia. In mathematics, Klein's j-invariant or j function, regarded as a function of a complex variable ѕД, is a modular function of weight zero for defined on the upper half-plane of complex numbers.It is the unique such function which is holomorphic away from a simple pole at the cusp such tha Counting tropical elliptic plane curves with fixed j-invariant. Commentarii Mathematici Helvetici, 2000. Hannah Markwig. Michael Kerber. Hannah Markwig. Michael Kerber. Download PDF. Download Full PDF Package. This paper. A short summary of this paper. 37 Full PDFs related to this paper. READ PAPER. Counting tropical elliptic plane curves with fixed j-invariant. We show that this number is independent of the position of the points and the value of the j-invariant and that it coincides with the number of complex elliptic curves (with j-invariant j / вИИ {0, 1728}). The result can be used to simplify Mikhalkin's algorithm to count curves via lattice paths (see ) in the case of rational plane curves. 1

See Page 1. , in particular the embedding in which j maps to the j -invariant of C / O , which is an embedding into R . We put K ( f ) := K Q ( f ) , the f -ring class field of K . For all f вЙ• 2 we have [Co89, Cor. 7.24] (5) [ K ( f ) : K (1) ] = 2 w K f Y p | f 1 - ќФ K p 1 p , TORSION POINTS AND ISOGENIES ON CM ELLIPTIC CURVES 7 where K (1. j-лґИл≥АлЯЙ нБілЭЉмЭЄмЭШ absolute j-invariant лЭЉлКФ мЭіл¶ДмЬЉл°Ь лґИл¶ђкЄ∞лПД нХ®; нГАмЫР л™®лУИлЭЉ нХ®мИШ(elliptic modular function) л°Ь лґИл¶ђкЄ∞лПД нХ®; л≥µмЖМ мЭім∞® мИШм≤імЭШ class field мЭіл°†мЧРмДЬ м§СмЪФнХЬ мЧ≠нХ†; л™ђмК§нД∞ кµ∞мЭШ monstrous moonshineмЧР лУ±мЮ•; м†ХмЭШ $$q=e^{2\pi i\tau},\tau\in \mathbb{H}$$лЭЉ лСРмЮ In the report we will generalize the simplified Shallue-van de Woestijne-Ulas (SWU) method of a deterministic finite field mapping F_q->E(F_q) to the case of any elliptic (F_q)-curve E of j-invariant 1728. More precisely, we will obtain a rational F_q-curve (and its explicit quite simple proper F_q-parametrization) on the Kummer surface K' associated with the direct product ExE', where. Forum Mathematicum (FORUM) is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact. Topics en:j-invariant гВТжЧ•жЬђи™ЮеМЦгБЧгБЊгБЧгБЯгБМгАБжЧ•жЬђи™ЮзЙИгБІгБѓи®ШдЇЛгБЃгВњгВ§гГИгГЂгБЃеЕИй†≠гБЂе§ІжЦЗе≠ЧгБЧгБЛдљњгБЖгБУгБ®гБМгБІгБНгБЪгАБе§ІжЦЗе≠ЧгБ®гБ™гБ£гБ¶гБЧгБЊгБДгБЊгБЧгБЯпЉОи®ИзЃЧгБЃйГљеРИдЄКгАБе∞ПжЦЗе≠Ч j-invariant гВВгБЧгБПгБѓ j-function гБМе§ЪгБДгБ®жАЭгБДгБЊгБЩпЉОдїЦгБЃи®ШдЇЛгБІгВВе∞ПжЦЗе≠ЧгБІеЉХзФ®гБЩгВЛзЃЗжЙАгБМе§ЪгБДгБ®жАЭгБД.

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