LMFDB - Elliptic curves over $\Q$ of conductor 1296

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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	Potentially good	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
1296.a1	1296i1	1296.a	1296i	$2$	$3$	$ 2^{4} \cdot 3^{4} $	$ - 2^{8} \cdot 3^{4} $	$0$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	✓	$2, 3, 5$	8.4.0.2, 3.4.0.1, 5.5.0.1	3B, 5S4	$120$	$160$	$4$	$1$	$1$	$0$	$144$	$-0.304559$	$-316368$	$0.93828$	$3.15468$	$[0, 0, 0, -39, -94]$	$y^2=x^3-39x-94$	3.4.0.a.1, 4.2.0.a.1, 5.5.0.a.1, 6.8.0-3.a.1.1, 8.4.0-4.a.1.1, $\ldots$	$[]$
1296.a2	1296i2	1296.a	1296i	$2$	$3$	$ 2^{4} \cdot 3^{4} $	$ - 2^{8} \cdot 3^{12} $	$0$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	✓	$2, 3, 5$	8.4.0.2, 3.4.0.1, 5.5.0.1	3B, 5S4	$120$	$160$	$4$	$1$	$1$	$0$	$432$	$0.244747$	$432$	$0.77371$	$3.61315$	$[0, 0, 0, 81, -486]$	$y^2=x^3+81x-486$	3.4.0.a.1, 4.2.0.a.1, 5.5.0.a.1, 6.8.0-3.a.1.2, 8.4.0-4.a.1.1, $\ldots$	$[]$
1296.b1	1296l2	1296.b	1296l	$2$	$3$	$ 2^{4} \cdot 3^{4} $	$ 2^{4} \cdot 3^{10} $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	✓	$2, 3$	2.2.0.1, 3.4.0.1	2Cn, 3B	$36$	$96$	$2$	$0.288119981$	$1$	$4$	$216$	$0.054953$	$790272$	$0.98305$	$3.81452$	$[0, 0, 0, -189, 999]$	$y^2=x^3-189x+999$	2.2.0.a.1, 3.4.0.a.1, 6.8.0.a.1, 12.32.0-12.a.2.3, 18.24.0.a.1, $\ldots$	$[(6, 9)]$
1296.b2	1296l1	1296.b	1296l	$2$	$3$	$ 2^{4} \cdot 3^{4} $	$ 2^{4} \cdot 3^{6} $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	✓	$2, 3$	2.2.0.1, 3.4.0.1	2Cn, 3B	$36$	$96$	$2$	$0.864359944$	$1$	$2$	$72$	$-0.494353$	$6912$	$0.77371$	$2.54014$	$[0, 0, 0, -9, -9]$	$y^2=x^3-9x-9$	2.2.0.a.1, 3.4.0.a.1, 6.8.0.a.1, 12.32.0-12.a.1.2, 18.24.0.a.1, $\ldots$	$[(-2, 1)]$
1296.c1	1296j1	1296.c	1296j	$2$	$3$	$ 2^{4} \cdot 3^{4} $	$ - 2^{14} \cdot 3^{6} $	$0$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$		$2, 3$	4.8.0.2, 3.4.0.1	3B	$12$	$128$	$1$	$1$	$1$	$0$	$288$	$0.095341$	$-35937/4$	$1.00607$	$3.56840$	$[0, 0, 0, -99, -414]$	$y^2=x^3-99x-414$	3.4.0.a.1, 4.8.0.b.1, 6.8.0-3.a.1.1, 12.128.1-12.b.2.1	$[]$
1296.c2	1296j2	1296.c	1296j	$2$	$3$	$ 2^{4} \cdot 3^{4} $	$ - 2^{18} \cdot 3^{10} $	$0$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$		$2, 3$	4.8.0.2, 3.4.0.1	3B	$12$	$128$	$1$	$1$	$1$	$0$	$864$	$0.644647$	$109503/64$	$1.28549$	$4.31246$	$[0, 0, 0, 621, 594]$	$y^2=x^3+621x+594$	3.4.0.a.1, 4.8.0.b.1, 6.8.0-3.a.1.2, 12.128.1-12.b.1.1	$[]$
1296.d1	1296b1	1296.d	1296b	$1$	$1$	$ 2^{4} \cdot 3^{4} $	$ - 2^{10} \cdot 3^{4} $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	✓	$2, 5$	4.16.0.2, 5.5.0.1	5S4	$20$	$80$	$2$	$0.863309381$	$1$	$4$	$96$	$-0.375659$	$-36$	$1.04124$	$2.62330$	$[0, 0, 0, -3, -14]$	$y^2=x^3-3x-14$	4.16.0-4.b.1.1, 5.5.0.a.1, 20.80.2-20.b.1.2	$[(3, 2)]$
1296.e1	1296c1	1296.e	1296c	$1$	$1$	$ 2^{4} \cdot 3^{4} $	$ 2^{4} \cdot 3^{4} $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	✓	$2$	4.4.0.2	2Cn	$36$	$12$	$0$	$0.600867403$	$1$	$2$	$48$	$-0.706588$	$2304$	$0.77371$	$2.08028$	$[0, 0, 0, -3, 1]$	$y^2=x^3-3x+1$	2.2.0.a.1, 4.4.0-2.a.1.1, 18.6.0.a.1, 36.12.0-18.a.1.4	$[(0, 1)]$
1296.f1	1296k3	1296.f	1296k	$4$	$21$	$ 2^{4} \cdot 3^{4} $	$ - 2^{19} \cdot 3^{6} $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$		$2, 3, 7$	8.2.0.1, 3.4.0.1, 7.8.0.1	3B, 7B	$504$	$768$	$21$	$3.080173468$	$1$	$2$	$1008$	$0.962515$	$-189613868625/128$	$1.12596$	$5.70357$	$[0, 0, 0, -17235, -870894]$	$y^2=x^3-17235x-870894$	3.4.0.a.1, 7.8.0.a.1, 8.2.0.a.1, 12.8.0-3.a.1.1, 21.64.1.a.4, $\ldots$	$[(217, 2368)]$
1296.f2	1296k4	1296.f	1296k	$4$	$21$	$ 2^{4} \cdot 3^{4} $	$ - 2^{33} \cdot 3^{10} $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$		$2, 3, 7$	8.2.0.1, 3.4.0.1, 7.8.0.1	3B, 7B	$504$	$768$	$21$	$1.026724489$	$1$	$4$	$3024$	$1.511822$	$-1159088625/2097152$	$1.11235$	$5.80326$	$[0, 0, 0, -13635, -1244862]$	$y^2=x^3-13635x-1244862$	3.4.0.a.1, 7.8.0.a.1, 8.2.0.a.1, 12.8.0-3.a.1.2, 21.64.1.a.1, $\ldots$	$[(1761, 73728)]$
1296.f3	1296k2	1296.f	1296k	$4$	$21$	$ 2^{4} \cdot 3^{4} $	$ - 2^{15} \cdot 3^{10} $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$		$2, 3, 7$	8.2.0.1, 3.4.0.1, 7.8.0.1	3B, 7B	$504$	$768$	$21$	$0.146674927$	$1$	$8$	$432$	$0.538866$	$-140625/8$	$1.17810$	$4.36045$	$[0, 0, 0, -675, 7074]$	$y^2=x^3-675x+7074$	3.4.0.a.1, 7.8.0.a.1, 8.2.0.a.1, 12.8.0-3.a.1.2, 21.64.1.a.3, $\ldots$	$[(33, 144)]$
1296.f4	1296k1	1296.f	1296k	$4$	$21$	$ 2^{4} \cdot 3^{4} $	$ - 2^{13} \cdot 3^{6} $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$		$2, 3, 7$	8.2.0.1, 3.4.0.1, 7.8.0.1	3B, 7B	$504$	$768$	$21$	$0.440024781$	$1$	$4$	$144$	$-0.010440$	$3375/2$	$1.42657$	$3.21382$	$[0, 0, 0, 45, 18]$	$y^2=x^3+45x+18$	3.4.0.a.1, 7.8.0.a.1, 8.2.0.a.1, 12.8.0-3.a.1.1, 21.64.1.a.2, $\ldots$	$[(1, 8)]$
1296.g1	1296e4	1296.g	1296e	$4$	$21$	$ 2^{4} \cdot 3^{4} $	$ - 2^{19} \cdot 3^{12} $	$0$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$		$2, 3, 7$	8.2.0.1, 3.4.0.1, 7.8.0.1	3B, 7B	$504$	$768$	$21$	$1$	$1$	$0$	$3024$	$1.511822$	$-189613868625/128$	$1.12596$	$6.62329$	$[0, 0, 0, -155115, 23514138]$	$y^2=x^3-155115x+23514138$	3.4.0.a.1, 7.8.0.a.1, 8.2.0.a.1, 12.8.0-3.a.1.2, 21.64.1.a.4, $\ldots$	$[]$
1296.g2	1296e3	1296.g	1296e	$4$	$21$	$ 2^{4} \cdot 3^{4} $	$ - 2^{33} \cdot 3^{4} $	$0$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$		$2, 3, 7$	8.2.0.1, 3.4.0.1, 7.8.0.1	3B, 7B	$504$	$768$	$21$	$1$	$1$	$0$	$1008$	$0.962515$	$-1159088625/2097152$	$1.11235$	$4.88354$	$[0, 0, 0, -1515, 46106]$	$y^2=x^3-1515x+46106$	3.4.0.a.1, 7.8.0.a.1, 8.2.0.a.1, 12.8.0-3.a.1.1, 21.64.1.a.1, $\ldots$	$[]$
1296.g3	1296e1	1296.g	1296e	$4$	$21$	$ 2^{4} \cdot 3^{4} $	$ - 2^{15} \cdot 3^{4} $	$0$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$		$2, 3, 7$	8.2.0.1, 3.4.0.1, 7.8.0.1	3B, 7B	$504$	$768$	$21$	$1$	$1$	$0$	$144$	$-0.010440$	$-140625/8$	$1.17810$	$3.44073$	$[0, 0, 0, -75, -262]$	$y^2=x^3-75x-262$	3.4.0.a.1, 7.8.0.a.1, 8.2.0.a.1, 12.8.0-3.a.1.1, 21.64.1.a.3, $\ldots$	$[]$
1296.g4	1296e2	1296.g	1296e	$4$	$21$	$ 2^{4} \cdot 3^{4} $	$ - 2^{13} \cdot 3^{12} $	$0$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$		$2, 3, 7$	8.2.0.1, 3.4.0.1, 7.8.0.1	3B, 7B	$504$	$768$	$21$	$1$	$1$	$0$	$432$	$0.538866$	$3375/2$	$1.42657$	$4.13354$	$[0, 0, 0, 405, -486]$	$y^2=x^3+405x-486$	3.4.0.a.1, 7.8.0.a.1, 8.2.0.a.1, 12.8.0-3.a.1.2, 21.64.1.a.2, $\ldots$	$[]$
1296.h1	1296a1	1296.h	1296a	$1$	$1$	$ 2^{4} \cdot 3^{4} $	$ - 2^{10} \cdot 3^{10} $	$1$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	✓	$2, 5$	4.8.0.2, 5.5.0.1	5S4	$60$	$80$	$2$	$0.274047555$	$1$	$6$	$288$	$0.173646$	$-36$	$1.04124$	$3.54302$	$[0, 0, 0, -27, 378]$	$y^2=x^3-27x+378$	4.8.0.b.1, 5.5.0.a.1, 12.16.0-4.b.1.1, 20.40.2.b.1, 60.80.2-20.b.1.1	$[(3, 18)]$
1296.i1	1296d1	1296.i	1296d	$1$	$1$	$ 2^{4} \cdot 3^{4} $	$ 2^{4} \cdot 3^{10} $	$0$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	✓	$2$	2.2.0.1	2Cn	$36$	$12$	$0$	$1$	$1$	$0$	$144$	$-0.157282$	$2304$	$0.77371$	$3.00000$	$[0, 0, 0, -27, -27]$	$y^2=x^3-27x-27$	2.2.0.a.1, 12.4.0-2.a.1.1, 18.6.0.a.1, 36.12.0-18.a.1.2	$[]$
1296.j1	1296g2	1296.j	1296g	$2$	$3$	$ 2^{4} \cdot 3^{4} $	$ - 2^{8} \cdot 3^{10} $	$0$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	✓	$2, 3, 5$	4.2.0.1, 3.4.0.1, 5.5.0.1	3B, 5S4	$120$	$160$	$4$	$1$	$1$	$0$	$432$	$0.244747$	$-316368$	$0.93828$	$4.07440$	$[0, 0, 0, -351, 2538]$	$y^2=x^3-351x+2538$	3.4.0.a.1, 4.2.0.a.1, 5.5.0.a.1, 6.8.0-3.a.1.2, 12.16.0-12.a.1.2, $\ldots$	$[]$
1296.j2	1296g1	1296.j	1296g	$2$	$3$	$ 2^{4} \cdot 3^{4} $	$ - 2^{8} \cdot 3^{6} $	$0$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	✓	$2, 3, 5$	4.2.0.1, 3.4.0.1, 5.5.0.1	3B, 5S4	$120$	$160$	$4$	$1$	$1$	$0$	$144$	$-0.304559$	$432$	$0.77371$	$2.69343$	$[0, 0, 0, 9, 18]$	$y^2=x^3+9x+18$	3.4.0.a.1, 4.2.0.a.1, 5.5.0.a.1, 6.8.0-3.a.1.1, 12.16.0-12.a.1.4, $\ldots$	$[]$
1296.k1	1296f2	1296.k	1296f	$2$	$3$	$ 2^{4} \cdot 3^{4} $	$ 2^{4} \cdot 3^{12} $	$0$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	✓	$2, 3$	4.4.0.2, 3.4.0.1	2Cn, 3B	$36$	$96$	$2$	$1$	$1$	$0$	$216$	$0.054953$	$6912$	$0.77371$	$3.45986$	$[0, 0, 0, -81, 243]$	$y^2=x^3-81x+243$	2.2.0.a.1, 3.4.0.a.1, 4.4.0-2.a.1.1, 6.8.0.a.1, 12.32.0-12.a.1.3, $\ldots$	$[]$
1296.k2	1296f1	1296.k	1296f	$2$	$3$	$ 2^{4} \cdot 3^{4} $	$ 2^{4} \cdot 3^{4} $	$0$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$	✓	$2, 3$	4.4.0.2, 3.4.0.1	2Cn, 3B	$36$	$96$	$2$	$1$	$1$	$0$	$72$	$-0.494353$	$790272$	$0.98305$	$2.89480$	$[0, 0, 0, -21, -37]$	$y^2=x^3-21x-37$	2.2.0.a.1, 3.4.0.a.1, 4.4.0-2.a.1.1, 6.8.0.a.1, 12.32.0-12.a.2.2, $\ldots$	$[]$
1296.l1	1296h2	1296.l	1296h	$2$	$3$	$ 2^{4} \cdot 3^{4} $	$ - 2^{14} \cdot 3^{12} $	$0$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$		$2, 3$	4.16.0.2, 3.4.0.1	3B	$12$	$128$	$1$	$1$	$1$	$0$	$864$	$0.644647$	$-35937/4$	$1.00607$	$4.48812$	$[0, 0, 0, -891, 11178]$	$y^2=x^3-891x+11178$	3.4.0.a.1, 4.16.0-4.b.1.1, 6.8.0-3.a.1.2, 12.128.1-12.b.2.2	$[]$
1296.l2	1296h1	1296.l	1296h	$2$	$3$	$ 2^{4} \cdot 3^{4} $	$ - 2^{18} \cdot 3^{4} $	$0$	$\mathsf{trivial}$	$\Q$	$\mathrm{SU}(2)$		$2, 3$	4.16.0.2, 3.4.0.1	3B	$12$	$128$	$1$	$1$	$1$	$0$	$288$	$0.095341$	$109503/64$	$1.28549$	$3.39274$	$[0, 0, 0, 69, -22]$	$y^2=x^3+69x-22$	3.4.0.a.1, 4.16.0-4.b.1.1, 6.8.0-3.a.1.1, 12.128.1-12.b.1.2	$[]$

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
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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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