LMFDB - Elliptic curves over $\Q$ of conductor 67081

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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	Manin constant

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Label	Cremona label	Class	Cremona class	Class size	Class degree	Conductor	Discriminant	Rank	Torsion	$\textrm{End}^0(E_{\overline\Q})$	CM	Sato-Tate	Potentially good	Nonmax $\ell$	$\ell$-adic images	mod-$\ell$ images	Adelic level	Adelic index	Adelic genus	Regulator	$Ш_{\textrm{an}}$	Ш primes	Integral points	Modular degree	Faltings height	j-invariant	$abc$ quality	Szpiro ratio	Weierstrass coefficients	Weierstrass equation	mod-$m$ images	MW-generators
67081.a1	67081j2	67081.a	67081j	$2$	$5$	$ 7^{2} \cdot 37^{2} $	$ 7^{6} \cdot 37^{9} $	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓	$3, 5$	3.3.0.1, 5.6.0.1	3Nn, 5B	$7770$	$288$	$9$	$38.00375988$	$1$		$0$	$8791200$	$3.196571$	$38477541376$	$1.06078$	$6.16782$	$[0, -1, 1, -174567122, -887694063726]$	$y^2+y=x^3-x^2-174567122x-887694063726$	3.3.0.a.1, 5.6.0.a.1, 6.6.0.c.1, 10.12.0.a.2, 15.36.0.a.2, $\ldots$	$[(-265611461692260410141/186599703, 821246889025903246092923446/186599703)]$
67081.a2	67081j1	67081.a	67081j	$2$	$5$	$ 7^{2} \cdot 37^{2} $	$ 7^{6} \cdot 37^{9} $	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓	$3, 5$	3.3.0.1, 5.6.0.1	3Nn, 5B	$7770$	$288$	$9$	$7.600751977$	$1$		$0$	$1758240$	$2.391853$	$4096$	$0.76978$	$4.72315$	$[0, -1, 1, -827332, 220507048]$	$y^2+y=x^3-x^2-827332x+220507048$	3.3.0.a.1, 5.6.0.a.1, 6.6.0.c.1, 10.12.0.a.1, 15.36.0.a.1, $\ldots$	$[(-183782/19, 154817465/19)]$
67081.b1	67081c4	67081.b	67081c	$4$	$14$	$ 7^{2} \cdot 37^{2} $	$ 7^{9} \cdot 37^{6} $	$0$	$\Z/2\Z$	$\Q(\sqrt{-7})$	$-28$	$N(\mathrm{U}(1))$	✓							$1$	$1$		$0$	$709632$	$2.325848$	$16581375$	$1.19804$	$5.02107$	$[1, -1, 1, -2494575, 1517046248]$	$y^2+xy+y=x^3-x^2-2494575x+1517046248$		$[ ]$
67081.b2	67081c3	67081.b	67081c	$4$	$14$	$ 7^{2} \cdot 37^{2} $	$ - 7^{9} \cdot 37^{6} $	$0$	$\Z/2\Z$	$\Q(\sqrt{-7})$	$-7$	$N(\mathrm{U}(1))$	✓							$1$	$1$		$1$	$354816$	$1.979275$	$-3375$	$0.98030$	$4.29348$	$[1, -1, 1, -146740, 26640590]$	$y^2+xy+y=x^3-x^2-146740x+26640590$		$[ ]$
67081.b3	67081c2	67081.b	67081c	$4$	$14$	$ 7^{2} \cdot 37^{2} $	$ 7^{3} \cdot 37^{6} $	$0$	$\Z/2\Z$	$\Q(\sqrt{-7})$	$-28$	$N(\mathrm{U}(1))$	✓							$1$	$1$		$0$	$101376$	$1.352894$	$16581375$	$1.19804$	$3.97052$	$[1, -1, 1, -50910, -4408330]$	$y^2+xy+y=x^3-x^2-50910x-4408330$		$[ ]$
67081.b4	67081c1	67081.b	67081c	$4$	$14$	$ 7^{2} \cdot 37^{2} $	$ - 7^{3} \cdot 37^{6} $	$0$	$\Z/2\Z$	$\Q(\sqrt{-7})$	$-7$	$N(\mathrm{U}(1))$	✓							$1$	$1$		$1$	$50688$	$1.006321$	$-3375$	$0.98030$	$3.24293$	$[1, -1, 1, -2995, -76814]$	$y^2+xy+y=x^3-x^2-2995x-76814$		$[ ]$
67081.c1	67081d2	67081.c	67081d	$2$	$7$	$ 7^{2} \cdot 37^{2} $	$ - 7^{6} \cdot 37^{2} $	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓	$2, 7$	4.2.0.1, 7.8.0.1	7B	$2072$	$192$	$6$	$2.780787133$	$1$		$4$	$90720$	$1.234652$	$-371323264041$	$1.11663$	$4.09745$	$[1, -1, 1, -81472, 8971092]$	$y^2+xy+y=x^3-x^2-81472x+8971092$	4.2.0.a.1, 7.8.0.a.1, 28.16.0.a.1, 56.32.0.b.1, 259.48.0.?, $\ldots$	$[(165, -84), (1474/3, -1465/3)]$
67081.c2	67081d1	67081.c	67081d	$2$	$7$	$ 7^{2} \cdot 37^{2} $	$ - 7^{6} \cdot 37^{2} $	$2$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓	$2, 7$	4.2.0.1, 7.8.0.1	7B	$2072$	$192$	$6$	$2.780787133$	$1$		$6$	$12960$	$0.261696$	$999$	$0.76978$	$2.32183$	$[1, -1, 1, 113, 368]$	$y^2+xy+y=x^3-x^2+113x+368$	4.2.0.a.1, 7.8.0.a.1, 28.16.0.a.1, 56.32.0.b.2, 259.48.0.?, $\ldots$	$[(2, 23), (47/2, 403/2)]$
67081.d1	67081e2	67081.d	67081e	$2$	$2$	$ 7^{2} \cdot 37^{2} $	$ 7^{12} \cdot 37^{7} $	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$		$2$	2.3.0.1	2B	$1036$	$12$	$0$	$1$	$9$	$3$	$0$	$4727808$	$2.893696$	$760798453689/4353013$	$0.97619$	$5.46162$	$[1, -1, 1, -12757968, 17455894334]$	$y^2+xy+y=x^3-x^2-12757968x+17455894334$	2.3.0.a.1, 28.6.0.c.1, 74.6.0.?, 1036.12.0.?	$[ ]$
67081.d2	67081e1	67081.d	67081e	$2$	$2$	$ 7^{2} \cdot 37^{2} $	$ - 7^{9} \cdot 37^{8} $	$0$	$\Z/2\Z$	$\Q$		$\mathrm{SU}(2)$		$2$	2.3.0.1	2B	$1036$	$12$	$0$	$1$	$9$	$3$	$1$	$2363904$	$2.547123$	$-15438249/469567$	$0.97720$	$4.84756$	$[1, -1, 1, -347983, 578314734]$	$y^2+xy+y=x^3-x^2-347983x+578314734$	2.3.0.a.1, 14.6.0.b.1, 148.6.0.?, 1036.12.0.?	$[ ]$
67081.e1	67081h1	67081.e	67081h	$1$	$1$	$ 7^{2} \cdot 37^{2} $	$ - 7^{9} \cdot 37^{3} $	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$		$3$	3.6.0.1	3Ns	$1554$	$24$	$1$	$0.722412656$	$1$		$4$	$103680$	$1.055111$	$-262144/343$	$0.92294$	$3.25428$	$[0, 1, 1, -2417, -83415]$	$y^2+y=x^3+x^2-2417x-83415$	3.6.0.b.1, 42.12.0.a.1, 111.12.0.?, 518.2.0.?, 1554.24.1.?	$[(345, 6345)]$
67081.f1	67081g1	67081.f	67081g	$1$	$1$	$ 7^{2} \cdot 37^{2} $	$ - 7^{9} \cdot 37^{9} $	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$		$3$	3.6.0.1	3Ns	$1554$	$24$	$1$	$11.94724437$	$1$		$0$	$3836160$	$2.860569$	$-262144/343$	$0.92294$	$5.20372$	$[0, 1, 1, -3309329, -4185497256]$	$y^2+y=x^3+x^2-3309329x-4185497256$	3.6.0.b.1, 42.12.0.a.1, 111.12.0.?, 518.2.0.?, 1554.24.1.?	$[(682998864/547, 2068802898669/547)]$
67081.g1	67081a3	67081.g	67081a	$3$	$9$	$ 7^{2} \cdot 37^{2} $	$ 7^{6} \cdot 37^{7} $	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$		$3$	27.36.0.1	3B	$13986$	$1296$	$43$	$1$	$1$		$0$	$3102624$	$3.000496$	$727057727488000/37$	$1.08598$	$6.07910$	$[0, -1, 1, -125665073, 542254977186]$	$y^2+y=x^3-x^2-125665073x+542254977186$	3.4.0.a.1, 9.12.0.a.1, 27.36.0.a.1, 42.8.0-3.a.1.1, 74.2.0.?, $\ldots$	$[ ]$
67081.g2	67081a2	67081.g	67081a	$3$	$9$	$ 7^{2} \cdot 37^{2} $	$ 7^{6} \cdot 37^{9} $	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$		$3$	9.36.0.2	3Cs	$13986$	$1296$	$43$	$1$	$1$		$0$	$1034208$	$2.451191$	$1404928000/50653$	$0.97274$	$4.89526$	$[0, -1, 1, -1565223, 730389729]$	$y^2+y=x^3-x^2-1565223x+730389729$	3.12.0.a.1, 9.36.0.b.1, 42.24.0-3.a.1.1, 74.2.0.?, 126.72.0.?, $\ldots$	$[ ]$
67081.g3	67081a1	67081.g	67081a	$3$	$9$	$ 7^{2} \cdot 37^{2} $	$ 7^{6} \cdot 37^{7} $	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$		$3$	27.36.0.1	3B	$13986$	$1296$	$43$	$1$	$1$		$0$	$344736$	$1.901884$	$4096000/37$	$0.88268$	$4.36998$	$[0, -1, 1, -223603, -40303880]$	$y^2+y=x^3-x^2-223603x-40303880$	3.4.0.a.1, 9.12.0.a.1, 27.36.0.a.1, 42.8.0-3.a.1.2, 74.2.0.?, $\ldots$	$[ ]$
67081.h1	67081b2	67081.h	67081b	$2$	$7$	$ 7^{2} \cdot 37^{2} $	$ - 7^{6} \cdot 37^{8} $	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓	$2, 7$	4.2.0.1, 7.16.0.1	7B.2.1	$2072$	$192$	$6$	$1$	$4$	$2$	$0$	$3356640$	$3.040112$	$-371323264041$	$1.11663$	$6.04690$	$[1, -1, 0, -111534740, 453408923869]$	$y^2+xy=x^3-x^2-111534740x+453408923869$	4.2.0.a.1, 7.16.0-7.a.1.2, 28.32.0-28.a.1.2, 56.64.0-56.b.1.1, 259.48.0.?, $\ldots$	$[ ]$
67081.h2	67081b1	67081.h	67081b	$2$	$7$	$ 7^{2} \cdot 37^{2} $	$ - 7^{6} \cdot 37^{8} $	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓	$2, 7$	4.2.0.1, 7.16.0.2	7B.2.3	$2072$	$192$	$6$	$1$	$4$	$2$	$0$	$479520$	$2.067154$	$999$	$0.76978$	$4.27128$	$[1, -1, 0, 155125, 20049882]$	$y^2+xy=x^3-x^2+155125x+20049882$	4.2.0.a.1, 7.16.0-7.a.1.1, 28.32.0-28.a.1.4, 56.64.0-56.b.2.1, 259.48.0.?, $\ldots$	$[ ]$
67081.i1	67081i2	67081.i	67081i	$2$	$5$	$ 7^{2} \cdot 37^{2} $	$ 7^{6} \cdot 37^{3} $	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓	$3, 5$	3.3.0.1, 5.6.0.1	3Nn, 5B	$7770$	$288$	$9$	$14.48443622$	$1$		$0$	$237600$	$1.391113$	$38477541376$	$1.06078$	$4.21837$	$[0, -1, 1, -127514, -17483649]$	$y^2+y=x^3-x^2-127514x-17483649$	3.3.0.a.1, 5.6.0.a.1, 6.6.0.c.1, 10.12.0.a.2, 15.36.0.a.2, $\ldots$	$[(-437542907/1458, 356183641/1458)]$
67081.i2	67081i1	67081.i	67081i	$2$	$5$	$ 7^{2} \cdot 37^{2} $	$ 7^{6} \cdot 37^{3} $	$1$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$	✓	$3, 5$	3.3.0.1, 5.6.0.1	3Nn, 5B	$7770$	$288$	$9$	$2.896887245$	$1$		$0$	$47520$	$0.586393$	$4096$	$0.76978$	$2.77370$	$[0, -1, 1, -604, 4549]$	$y^2+y=x^3-x^2-604x+4549$	3.3.0.a.1, 5.6.0.a.1, 6.6.0.c.1, 10.12.0.a.1, 15.36.0.a.1, $\ldots$	$[(-11/2, 625/2)]$
67081.j1	67081f1	67081.j	67081f	$1$	$1$	$ 7^{2} \cdot 37^{2} $	$ 7^{6} \cdot 37^{7} $	$0$	$\mathsf{trivial}$	$\Q$		$\mathrm{SU}(2)$					$74$	$2$	$0$	$1$	$9$	$3$	$0$	$1034208$	$1.781872$	$110592/37$	$0.76978$	$4.04498$	$[0, 0, 1, -67081, -4343495]$	$y^2+y=x^3-67081x-4343495$	74.2.0.?	$[ ]$

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Elliptic curves over $\Q$
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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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