Elliptic curves over \(\Q(\sqrt{157}) \) of conductor 39.2

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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	Rank	Torsion	CM	CM	Sato-Tate	$\Q$-curve	Base change	Semistable	Potentially good	Nonmax $\ell$	mod-$\ell$ images	$Ш_{\textrm{an}}$	Tamagawa	Regulator	Period	Leading coeff	j-invariant	Weierstrass coefficients	Weierstrass equation
39.2-a1	39.2-a	$1$	$1$	\(\Q(\sqrt{157}) \)	$2$	$[2, 0]$	39.2	\( 3 \cdot 13 \)	\( 3^{5} \cdot 13 \)	$2.79804$	$(-a-6), (-2a-11)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 5 \)	$2.475533218$	$3.347309730$	6.613248349	\( \frac{26222359015}{3159} a - \frac{13645384489}{243} \)	\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 15357 a - 103761\) , \( 2589646 a - 17518538\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(15357a-103761\right){x}+2589646a-17518538$
39.2-b1	39.2-b	$1$	$1$	\(\Q(\sqrt{157}) \)	$2$	$[2, 0]$	39.2	\( 3 \cdot 13 \)	\( 3^{29} \cdot 13^{3} \)	$2.79804$	$(-a-6), (-2a-11)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓				$1$	\( 29 \)	$0.555699910$	$2.193832103$	5.643153734	\( \frac{30994677524215436107}{150780939070647951} a - \frac{16201418628140718394}{11598533774665227} \)	\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -382 a - 2189\) , \( 170330 a + 981950\bigr] \)	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-382a-2189\right){x}+170330a+981950$
39.2-c1	39.2-c	$1$	$1$	\(\Q(\sqrt{157}) \)	$2$	$[2, 0]$	39.2	\( 3 \cdot 13 \)	\( - 3^{3} \cdot 13^{3} \)	$2.79804$	$(-a-6), (-2a-11)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$3$	3Ns	$1$	\( 3 \)	$0.350253304$	$19.97727795$	3.350579895	\( -\frac{16236544}{59319} a + \frac{7573504}{4563} \)	\( \bigl[0\) , \( 1\) , \( 1\) , \( 139 a - 940\) , \( 1571 a - 10628\bigr] \)	${y}^2+{y}={x}^{3}+{x}^{2}+\left(139a-940\right){x}+1571a-10628$
39.2-d1	39.2-d	$1$	$1$	\(\Q(\sqrt{157}) \)	$2$	$[2, 0]$	39.2	\( 3 \cdot 13 \)	\( 3^{9} \cdot 13^{3} \)	$2.79804$	$(-a-6), (-2a-11)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$3$	3Ns	$1$	\( 3^{2} \)	$0.103787514$	$10.51848882$	1.568271099	\( -\frac{1059488906}{43243551} a + \frac{466674431}{3326427} \)	\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 4 a + 34\) , \( 18 a + 111\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(4a+34\right){x}+18a+111$

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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.