LMFDB - Elliptic curves over $\Q$ of conductor 768

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Conductor	Discriminant	j-invariant	Rank

Bad$\ p$	Curves per isogeny class	Complex multiplication	Torsion

Isogeny class degree	Cyclic isogeny degree	Isogeny class size	Integral points

Analytic order of Ш	$p\ $div$\ $\|Ш\|	Regulator	Reduction	Faltings height

Galois image	Adelic level	Adelic index	Adelic genus

Nonmax$\ \ell$	$abc$ quality	Szpiro ratio

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Results (24 matches)

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Label	Cremona label	Class	Cremona class	Class size	Class degree	Conductor	Discriminant	Rank	Torsion	$\textrm{End}^0(E_{\overline\Q})$	Sato-Tate	Nonmax $\ell$	$\ell$-adic images	mod-$\ell$ images	Adelic level	Adelic index	Adelic genus	Regulator	$Ш_{\textrm{an}}$	Integral points	Modular degree	Faltings height	j-invariant	$abc$ quality	Szpiro ratio	Weierstrass coefficients	Weierstrass equation	mod-$m$ images	MW-generators
768.a1	768b4	768.a	768b	$4$	$10$	$ 2^{8} \cdot 3 $	$ 2^{15} \cdot 3^{5} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 5$	8.6.0.4, 5.6.0.1	2B, 5B	$240$	$288$	$5$	$5.666583769$	$1$	$3$	$320$	$0.533577$	$58591911104/243$	$1.10697$	$5.29684$	$[0, -1, 0, -2589, -49851]$	$y^2=x^3-x^2-2589x-49851$	2.3.0.a.1, 5.6.0.a.1, 8.6.0.d.1, 10.18.0.a.1, 12.6.0.f.1, $\ldots$	$[(260, 4097)]$
768.a2	768b3	768.a	768b	$4$	$10$	$ 2^{8} \cdot 3 $	$ - 2^{9} \cdot 3^{10} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 5$	8.6.0.5, 5.6.0.1	2B, 5B	$240$	$288$	$5$	$2.833291884$	$1$	$5$	$160$	$0.187004$	$-873722816/59049$	$1.08875$	$4.05447$	$[0, -1, 0, -159, -765]$	$y^2=x^3-x^2-159x-765$	2.3.0.a.1, 5.6.0.a.1, 8.6.0.a.1, 10.18.0.a.1, 12.6.0.f.1, $\ldots$	$[(17, 34)]$
768.a3	768b2	768.a	768b	$4$	$10$	$ 2^{8} \cdot 3 $	$ 2^{15} \cdot 3 $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 5$	8.6.0.4, 5.6.0.1	2B, 5B	$240$	$288$	$5$	$1.133316753$	$1$	$5$	$64$	$-0.271142$	$85184/3$	$1.24048$	$3.27370$	$[0, -1, 0, -29, 69]$	$y^2=x^3-x^2-29x+69$	2.3.0.a.1, 5.6.0.a.1, 8.6.0.d.1, 10.18.0.a.1, 12.6.0.f.1, $\ldots$	$[(4, 1)]$
768.a4	768b1	768.a	768b	$4$	$10$	$ 2^{8} \cdot 3 $	$ - 2^{9} \cdot 3^{2} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 5$	8.6.0.5, 5.6.0.1	2B, 5B	$240$	$288$	$5$	$0.566658376$	$1$	$7$	$32$	$-0.617716$	$64/9$	$1.31110$	$2.39113$	$[0, -1, 0, 1, 3]$	$y^2=x^3-x^2+x+3$	2.3.0.a.1, 5.6.0.a.1, 8.6.0.a.1, 10.18.0.a.1, 12.6.0.f.1, $\ldots$	$[(1, 2)]$
768.b1	768a1	768.b	768a	$2$	$2$	$ 2^{8} \cdot 3 $	$ 2^{9} \cdot 3^{2} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	16.48.0.247	2B	$48$	$96$	$1$	$0.439295861$	$1$	$7$	$64$	$-0.431715$	$2744000/9$	$1.10672$	$3.17037$	$[0, -1, 0, -23, 51]$	$y^2=x^3-x^2-23x+51$	2.3.0.a.1, 4.6.0.e.1, 8.24.0.s.1, 12.12.0.n.1, 16.48.0-8.s.1.3, $\ldots$	$[(2, 3)]$
768.b2	768a2	768.b	768a	$2$	$2$	$ 2^{8} \cdot 3 $	$ - 2^{15} \cdot 3^{4} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.48.0.206	2B	$24$	$96$	$1$	$0.878591722$	$1$	$7$	$128$	$-0.085141$	$-8000/81$	$1.24244$	$3.35681$	$[0, -1, 0, -13, 85]$	$y^2=x^3-x^2-13x+85$	2.3.0.a.1, 4.12.0.f.1, 8.48.0-8.n.1.4, 24.96.1-24.cm.1.3	$[(3, 8)]$
768.c1	768e2	768.c	768e	$2$	$2$	$ 2^{8} \cdot 3 $	$ 2^{15} \cdot 3^{2} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	16.48.0.247	2B	$48$	$96$	$1$	$1$	$1$	$1$	$128$	$-0.085141$	$2744000/9$	$1.10672$	$3.79635$	$[0, -1, 0, -93, -315]$	$y^2=x^3-x^2-93x-315$	2.3.0.a.1, 4.6.0.e.1, 8.24.0.s.1, 12.12.0.n.1, 16.48.0-8.s.1.3, $\ldots$	$[]$
768.c2	768e1	768.c	768e	$2$	$2$	$ 2^{8} \cdot 3 $	$ - 2^{9} \cdot 3^{4} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.48.0.206	2B	$24$	$96$	$1$	$1$	$1$	$1$	$64$	$-0.431715$	$-8000/81$	$1.24244$	$2.73083$	$[0, -1, 0, -3, -9]$	$y^2=x^3-x^2-3x-9$	2.3.0.a.1, 4.12.0.f.1, 8.48.0-8.n.1.4, 24.96.1-24.cm.1.3	$[]$
768.d1	768f3	768.d	768f	$4$	$10$	$ 2^{8} \cdot 3 $	$ 2^{9} \cdot 3^{5} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 5$	8.6.0.4, 5.6.0.1	2B, 5B	$240$	$288$	$5$	$1$	$1$	$1$	$160$	$0.187004$	$58591911104/243$	$1.10697$	$4.67086$	$[0, -1, 0, -647, 6555]$	$y^2=x^3-x^2-647x+6555$	2.3.0.a.1, 5.6.0.a.1, 8.6.0.d.1, 10.18.0.a.1, 12.6.0.f.1, $\ldots$	$[]$
768.d2	768f4	768.d	768f	$4$	$10$	$ 2^{8} \cdot 3 $	$ - 2^{15} \cdot 3^{10} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 5$	8.6.0.5, 5.6.0.1	2B, 5B	$240$	$288$	$5$	$1$	$1$	$1$	$320$	$0.533577$	$-873722816/59049$	$1.08875$	$4.68045$	$[0, -1, 0, -637, 6757]$	$y^2=x^3-x^2-637x+6757$	2.3.0.a.1, 5.6.0.a.1, 8.6.0.a.1, 10.18.0.a.1, 12.6.0.f.1, $\ldots$	$[]$
768.d3	768f1	768.d	768f	$4$	$10$	$ 2^{8} \cdot 3 $	$ 2^{9} \cdot 3 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 5$	8.6.0.4, 5.6.0.1	2B, 5B	$240$	$288$	$5$	$1$	$1$	$1$	$32$	$-0.617716$	$85184/3$	$1.24048$	$2.64772$	$[0, -1, 0, -7, -5]$	$y^2=x^3-x^2-7x-5$	2.3.0.a.1, 5.6.0.a.1, 8.6.0.d.1, 10.18.0.a.1, 12.6.0.f.1, $\ldots$	$[]$
768.d4	768f2	768.d	768f	$4$	$10$	$ 2^{8} \cdot 3 $	$ - 2^{15} \cdot 3^{2} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 5$	8.6.0.5, 5.6.0.1	2B, 5B	$240$	$288$	$5$	$1$	$1$	$1$	$64$	$-0.271142$	$64/9$	$1.31110$	$3.01711$	$[0, -1, 0, 3, -27]$	$y^2=x^3-x^2+3x-27$	2.3.0.a.1, 5.6.0.a.1, 8.6.0.a.1, 10.18.0.a.1, 12.6.0.f.1, $\ldots$	$[]$
768.e1	768h4	768.e	768h	$4$	$10$	$ 2^{8} \cdot 3 $	$ 2^{15} \cdot 3^{5} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 5$	8.6.0.4, 5.6.0.1	2B, 5B	$240$	$288$	$5$	$0.448964398$	$1$	$9$	$320$	$0.533577$	$58591911104/243$	$1.10697$	$5.29684$	$[0, 1, 0, -2589, 49851]$	$y^2=x^3+x^2-2589x+49851$	2.3.0.a.1, 5.6.0.a.1, 8.6.0.d.1, 10.18.0.a.1, 12.6.0.f.1, $\ldots$	$[(30, 9)]$
768.e2	768h3	768.e	768h	$4$	$10$	$ 2^{8} \cdot 3 $	$ - 2^{9} \cdot 3^{10} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 5$	8.6.0.5, 5.6.0.1	2B, 5B	$240$	$288$	$5$	$0.224482199$	$1$	$11$	$160$	$0.187004$	$-873722816/59049$	$1.08875$	$4.05447$	$[0, 1, 0, -159, 765]$	$y^2=x^3+x^2-159x+765$	2.3.0.a.1, 5.6.0.a.1, 8.6.0.a.1, 10.18.0.a.1, 12.6.0.f.1, $\ldots$	$[(3, 18)]$
768.e3	768h2	768.e	768h	$4$	$10$	$ 2^{8} \cdot 3 $	$ 2^{15} \cdot 3 $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 5$	8.6.0.4, 5.6.0.1	2B, 5B	$240$	$288$	$5$	$2.244821992$	$1$	$3$	$64$	$-0.271142$	$85184/3$	$1.24048$	$3.27370$	$[0, 1, 0, -29, -69]$	$y^2=x^3+x^2-29x-69$	2.3.0.a.1, 5.6.0.a.1, 8.6.0.d.1, 10.18.0.a.1, 12.6.0.f.1, $\ldots$	$[(6, 3)]$
768.e4	768h1	768.e	768h	$4$	$10$	$ 2^{8} \cdot 3 $	$ - 2^{9} \cdot 3^{2} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 5$	8.6.0.5, 5.6.0.1	2B, 5B	$240$	$288$	$5$	$1.122410996$	$1$	$7$	$32$	$-0.617716$	$64/9$	$1.31110$	$2.39113$	$[0, 1, 0, 1, -3]$	$y^2=x^3+x^2+x-3$	2.3.0.a.1, 5.6.0.a.1, 8.6.0.a.1, 10.18.0.a.1, 12.6.0.f.1, $\ldots$	$[(3, 6)]$
768.f1	768g2	768.f	768g	$2$	$2$	$ 2^{8} \cdot 3 $	$ 2^{15} \cdot 3^{2} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	16.48.0.244	2B	$48$	$96$	$1$	$0.771945058$	$1$	$5$	$128$	$-0.085141$	$2744000/9$	$1.10672$	$3.79635$	$[0, 1, 0, -93, 315]$	$y^2=x^3+x^2-93x+315$	2.3.0.a.1, 4.6.0.e.1, 8.24.0.s.1, 12.12.0.n.1, 16.48.0-8.s.1.1, $\ldots$	$[(6, 3)]$
768.f2	768g1	768.f	768g	$2$	$2$	$ 2^{8} \cdot 3 $	$ - 2^{9} \cdot 3^{4} $	$1$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.48.0.205	2B	$24$	$96$	$1$	$0.385972529$	$1$	$9$	$64$	$-0.431715$	$-8000/81$	$1.24244$	$2.73083$	$[0, 1, 0, -3, 9]$	$y^2=x^3+x^2-3x+9$	2.3.0.a.1, 4.12.0.f.1, 8.48.0-8.n.1.1, 24.96.1-24.cm.1.1	$[(0, 3)]$
768.g1	768c1	768.g	768c	$2$	$2$	$ 2^{8} \cdot 3 $	$ 2^{9} \cdot 3^{2} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	16.48.0.244	2B	$48$	$96$	$1$	$1$	$1$	$1$	$64$	$-0.431715$	$2744000/9$	$1.10672$	$3.17037$	$[0, 1, 0, -23, -51]$	$y^2=x^3+x^2-23x-51$	2.3.0.a.1, 4.6.0.e.1, 8.24.0.s.1, 12.12.0.n.1, 16.48.0-8.s.1.1, $\ldots$	$[]$
768.g2	768c2	768.g	768c	$2$	$2$	$ 2^{8} \cdot 3 $	$ - 2^{15} \cdot 3^{4} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2$	8.48.0.205	2B	$24$	$96$	$1$	$1$	$1$	$1$	$128$	$-0.085141$	$-8000/81$	$1.24244$	$3.35681$	$[0, 1, 0, -13, -85]$	$y^2=x^3+x^2-13x-85$	2.3.0.a.1, 4.12.0.f.1, 8.48.0-8.n.1.1, 24.96.1-24.cm.1.1	$[]$
768.h1	768d3	768.h	768d	$4$	$10$	$ 2^{8} \cdot 3 $	$ 2^{9} \cdot 3^{5} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 5$	8.6.0.4, 5.6.0.1	2B, 5B	$240$	$288$	$5$	$1$	$1$	$1$	$160$	$0.187004$	$58591911104/243$	$1.10697$	$4.67086$	$[0, 1, 0, -647, -6555]$	$y^2=x^3+x^2-647x-6555$	2.3.0.a.1, 5.6.0.a.1, 8.6.0.d.1, 10.18.0.a.1, 12.6.0.f.1, $\ldots$	$[]$
768.h2	768d4	768.h	768d	$4$	$10$	$ 2^{8} \cdot 3 $	$ - 2^{15} \cdot 3^{10} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 5$	8.6.0.5, 5.6.0.1	2B, 5B	$240$	$288$	$5$	$1$	$1$	$1$	$320$	$0.533577$	$-873722816/59049$	$1.08875$	$4.68045$	$[0, 1, 0, -637, -6757]$	$y^2=x^3+x^2-637x-6757$	2.3.0.a.1, 5.6.0.a.1, 8.6.0.a.1, 10.18.0.a.1, 12.6.0.f.1, $\ldots$	$[]$
768.h3	768d1	768.h	768d	$4$	$10$	$ 2^{8} \cdot 3 $	$ 2^{9} \cdot 3 $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 5$	8.6.0.4, 5.6.0.1	2B, 5B	$240$	$288$	$5$	$1$	$1$	$1$	$32$	$-0.617716$	$85184/3$	$1.24048$	$2.64772$	$[0, 1, 0, -7, 5]$	$y^2=x^3+x^2-7x+5$	2.3.0.a.1, 5.6.0.a.1, 8.6.0.d.1, 10.18.0.a.1, 12.6.0.f.1, $\ldots$	$[]$
768.h4	768d2	768.h	768d	$4$	$10$	$ 2^{8} \cdot 3 $	$ - 2^{15} \cdot 3^{2} $	$0$	$\Z/2\Z$	$\Q$	$\mathrm{SU}(2)$	$2, 5$	8.6.0.5, 5.6.0.1	2B, 5B	$240$	$288$	$5$	$1$	$1$	$1$	$64$	$-0.271142$	$64/9$	$1.31110$	$3.01711$	$[0, 1, 0, 3, 27]$	$y^2=x^3+x^2+3x+27$	2.3.0.a.1, 5.6.0.a.1, 8.6.0.a.1, 10.18.0.a.1, 12.6.0.f.1, $\ldots$	$[]$

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

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