LMFDB - Elliptic curves over $\Q(\sqrt{5}) $ of conductor 2401.1

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Base field	Conductor norm	Rank*	Torsion order	CM discriminant

Base field degree/signature	Bad $p$	$\Q$-curves	Torsion structure	CM

Analytic order of Ш*	Cyclic isogeny degree	Isogeny class size	Reduction	Galois image

j-invariant	Regulator*	Curves per isogeny class	Isogeny class degree	Nonmax$\ \ell$

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Label	Class	Class size	Class degree	Base field	Field degree	Field signature	Conductor	Conductor norm	Discriminant norm	Root analytic conductor	Bad primes	Torsion	CM	CM	Sato-Tate	$\Q$-curve	Base change	Potentially good	Nonmax $\ell$	mod-$\ell$ images	$Ш_{\textrm{an}}$	Tamagawa	Regulator	Period	Leading coeff	j-invariant	Weierstrass coefficients	Weierstrass equation
2401.1-a1	2401.1-a	$2$	$37$	$\Q(\sqrt{5}) $	$2$	$[2, 0]$	2401.1	$ 7^{4} $	$ 7^{4} $	$1.39869$	$(7)$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓	$37$	37B.11.2	$1$	$ 1 $	$1$	$4.444942730$	1.987838820	$ -162677523113838677 $	$ \bigl[\phi + 1$ , $ -\phi - 1$ , $ 1$ , $ -41617 \phi - 41617$ , $ 5859391 \phi + 4394543\bigr] $	${y}^2+\left(\phi+1\right){x}{y}+{y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(-41617\phi-41617\right){x}+5859391\phi+4394543$
2401.1-a2	2401.1-a	$2$	$37$	$\Q(\sqrt{5}) $	$2$	$[2, 0]$	2401.1	$ 7^{4} $	$ 7^{4} $	$1.39869$	$(7)$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓	$37$	37B.11.1	$1$	$ 1 $	$1$	$4.444942730$	1.987838820	$ -9317 $	$ \bigl[\phi + 1$ , $ -\phi - 1$ , $ 1$ , $ -2 \phi - 2$ , $ -\phi - 1\bigr] $	${y}^2+\left(\phi+1\right){x}{y}+{y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(-2\phi-2\right){x}-\phi-1$
2401.1-b1	2401.1-b	$4$	$35$	$\Q(\sqrt{5}) $	$2$	$[2, 0]$	2401.1	$ 7^{4} $	$ 7^{6} $	$1.39869$	$(7)$	$\mathsf{trivial}$	$\textsf{potential}$	$-35$	$N(\mathrm{U}(1))$	✓		✓	$7$	7B.1.5	$1$	$ 2 $	$1$	$2.437755509$	2.180394812	$ -52756480 a - 32604160 $	$ \bigl[0$ , $ -1$ , $ 1$ , $ -14 \phi + 19$ , $ 21 \phi - 36\bigr] $	${y}^2+{y}={x}^{3}-{x}^{2}+\left(-14\phi+19\right){x}+21\phi-36$
2401.1-b2	2401.1-b	$4$	$35$	$\Q(\sqrt{5}) $	$2$	$[2, 0]$	2401.1	$ 7^{4} $	$ 7^{18} $	$1.39869$	$(7)$	$\mathsf{trivial}$	$\textsf{potential}$	$-35$	$N(\mathrm{U}(1))$	✓		✓	$7$	7B.1.2	$1$	$ 2 $	$1$	$2.437755509$	2.180394812	$ -52756480 a - 32604160 $	$ \bigl[0$ , $ 1$ , $ 1$ , $ -686 \phi + 915$ , $ -5831 \phi + 10420\bigr] $	${y}^2+{y}={x}^{3}+{x}^{2}+\left(-686\phi+915\right){x}-5831\phi+10420$
2401.1-b3	2401.1-b	$4$	$35$	$\Q(\sqrt{5}) $	$2$	$[2, 0]$	2401.1	$ 7^{4} $	$ 7^{6} $	$1.39869$	$(7)$	$\mathsf{trivial}$	$\textsf{potential}$	$-35$	$N(\mathrm{U}(1))$	✓		✓	$7$	7B.1.5	$1$	$ 2 $	$1$	$2.437755509$	2.180394812	$ 52756480 a - 85360640 $	$ \bigl[0$ , $ -1$ , $ 1$ , $ 14 \phi + 5$ , $ -21 \phi - 15\bigr] $	${y}^2+{y}={x}^{3}-{x}^{2}+\left(14\phi+5\right){x}-21\phi-15$
2401.1-b4	2401.1-b	$4$	$35$	$\Q(\sqrt{5}) $	$2$	$[2, 0]$	2401.1	$ 7^{4} $	$ 7^{18} $	$1.39869$	$(7)$	$\mathsf{trivial}$	$\textsf{potential}$	$-35$	$N(\mathrm{U}(1))$	✓		✓	$7$	7B.1.2	$1$	$ 2 $	$1$	$2.437755509$	2.180394812	$ 52756480 a - 85360640 $	$ \bigl[0$ , $ 1$ , $ 1$ , $ 686 \phi + 229$ , $ 5831 \phi + 4589\bigr] $	${y}^2+{y}={x}^{3}+{x}^{2}+\left(686\phi+229\right){x}+5831\phi+4589$
2401.1-c1	2401.1-c	$2$	$37$	$\Q(\sqrt{5}) $	$2$	$[2, 0]$	2401.1	$ 7^{4} $	$ 7^{16} $	$1.39869$	$(7)$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓	$37$	37B.10.2	$1369$	$ 1 $	$1$	$0.003990320$	2.443015397	$ -162677523113838677 $	$ \bigl[\phi + 1$ , $ \phi$ , $ \phi$ , $ -2039213 \phi - 2039213$ , $ -2007731903 \phi - 1505289124\bigr] $	${y}^2+\left(\phi+1\right){x}{y}+\phi{y}={x}^{3}+\phi{x}^{2}+\left(-2039213\phi-2039213\right){x}-2007731903\phi-1505289124$
2401.1-c2	2401.1-c	$2$	$37$	$\Q(\sqrt{5}) $	$2$	$[2, 0]$	2401.1	$ 7^{4} $	$ 7^{16} $	$1.39869$	$(7)$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓	$37$	37B.10.1	$1$	$ 1 $	$1$	$5.462748497$	2.443015397	$ -9317 $	$ \bigl[\phi$ , $ \phi - 1$ , $ \phi$ , $ 78 \phi - 157$ , $ -575 \phi + 986\bigr] $	${y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(78\phi-157\right){x}-575\phi+986$
2401.1-d1	2401.1-d	$2$	$5$	$\Q(\sqrt{5}) $	$2$	$[2, 0]$	2401.1	$ 7^{4} $	$ 7^{22} $	$1.39869$	$(7)$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			$5$	5B.4.1[2]	$1$	$ 2 $	$1$	$2.200325409$	1.968030875	$ -\frac{2887553024}{16807} $	$ \bigl[0$ , $ \phi - 1$ , $ 1$ , $ -1454 \phi - 1453$ , $ 37868 \phi + 28764\bigr] $	${y}^2+{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(-1454\phi-1453\right){x}+37868\phi+28764$
2401.1-d2	2401.1-d	$2$	$5$	$\Q(\sqrt{5}) $	$2$	$[2, 0]$	2401.1	$ 7^{4} $	$ 7^{14} $	$1.39869$	$(7)$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			$5$	5B.4.1[2]	$1$	$ 2 $	$1$	$2.200325409$	1.968030875	$ \frac{4096}{7} $	$ \bigl[0$ , $ -\phi$ , $ 1$ , $ -16 \phi + 33$ , $ 58 \phi - 106\bigr] $	${y}^2+{y}={x}^{3}-\phi{x}^{2}+\left(-16\phi+33\right){x}+58\phi-106$
2401.1-e1	2401.1-e	$4$	$14$	$\Q(\sqrt{5}) $	$2$	$[2, 0]$	2401.1	$ 7^{4} $	$ 7^{6} $	$1.39869$	$(7)$	$\Z/2\Z$	$\textsf{potential}$	$-7$	$N(\mathrm{U}(1))$	✓	✓	✓	$7$	7B.1.5	$1$	$ 2 $	$1$	$3.737694151$	0.835773820	$ -3375 $	$ \bigl[1$ , $ -1$ , $ 0$ , $ -2$ , $ -1\bigr] $	${y}^2+{x}{y}={x}^{3}-{x}^{2}-2{x}-1$
2401.1-e2	2401.1-e	$4$	$14$	$\Q(\sqrt{5}) $	$2$	$[2, 0]$	2401.1	$ 7^{4} $	$ 7^{18} $	$1.39869$	$(7)$	$\Z/2\Z$	$\textsf{potential}$	$-7$	$N(\mathrm{U}(1))$	✓	✓	✓	$7$	7B.1.2	$1$	$ 2 $	$1$	$3.737694151$	0.835773820	$ -3375 $	$ \bigl[1$ , $ -1$ , $ 0$ , $ -107$ , $ 552\bigr] $	${y}^2+{x}{y}={x}^{3}-{x}^{2}-107{x}+552$
2401.1-e3	2401.1-e	$4$	$14$	$\Q(\sqrt{5}) $	$2$	$[2, 0]$	2401.1	$ 7^{4} $	$ 7^{6} $	$1.39869$	$(7)$	$\Z/2\Z$	$\textsf{potential}$	$-28$	$N(\mathrm{U}(1))$	✓	✓	✓	$7$	7B.1.5	$1$	$ 2 $	$1$	$3.737694151$	0.835773820	$ 16581375 $	$ \bigl[1$ , $ -1$ , $ 0$ , $ -37$ , $ -78\bigr] $	${y}^2+{x}{y}={x}^{3}-{x}^{2}-37{x}-78$
2401.1-e4	2401.1-e	$4$	$14$	$\Q(\sqrt{5}) $	$2$	$[2, 0]$	2401.1	$ 7^{4} $	$ 7^{18} $	$1.39869$	$(7)$	$\Z/2\Z$	$\textsf{potential}$	$-28$	$N(\mathrm{U}(1))$	✓	✓	✓	$7$	7B.1.2	$1$	$ 2 $	$1$	$3.737694151$	0.835773820	$ 16581375 $	$ \bigl[1$ , $ -1$ , $ 0$ , $ -1822$ , $ 30393\bigr] $	${y}^2+{x}{y}={x}^{3}-{x}^{2}-1822{x}+30393$

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*The rank, regulator and analytic order of Ш are not known for all curves in the database; curves for which these are unknown will not appear in searches specifying one of these quantities.

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