Elliptic curve search results

Results (4 matches)

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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
686.2-c1	686.2-c	$2$	$3$	\(\Q(\sqrt{2}) \)	$2$	$[2, 0]$	686.2	\( 2 \cdot 7^{3} \)	\( 2^{2} \cdot 7^{5} \)	$1.29349$	$(a), (-2a+1), (2a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B	$1$	\( 2 \)	$0.125216197$	$8.219192181$	1.455474647	\( \frac{341511377481251}{686} a - \frac{482970021761709}{686} \)	\( \bigl[1\) , \( -a\) , \( a + 1\) , \( 30 a - 50\) , \( -122 a + 124\bigr] \)	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(30a-50\right){x}-122a+124$
686.2-e1	686.2-e	$2$	$3$	\(\Q(\sqrt{2}) \)	$2$	$[2, 0]$	686.2	\( 2 \cdot 7^{3} \)	\( 2^{2} \cdot 7^{11} \)	$1.29349$	$(a), (-2a+1), (2a+1)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.2	$1$	\( 2 \cdot 3^{2} \)	$1$	$0.348888885$	2.220315274	\( \frac{341511377481251}{686} a - \frac{482970021761709}{686} \)	\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 478 a - 695\) , \( 6733 a - 9853\bigr] \)	${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(478a-695\right){x}+6733a-9853$
4802.1-w1	4802.1-w	$2$	$3$	\(\Q(\sqrt{2}) \)	$2$	$[2, 0]$	4802.1	\( 2 \cdot 7^{4} \)	\( 2^{2} \cdot 7^{17} \)	$2.10397$	$(a), (-2a+1), (2a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B	$1$	\( 2^{2} \)	$3.414994725$	$0.253427211$	2.447869585	\( \frac{341511377481251}{686} a - \frac{482970021761709}{686} \)	\( \bigl[1\) , \( a + 1\) , \( a\) , \( 1503 a - 2458\) , \( 48068 a - 51240\bigr] \)	${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(1503a-2458\right){x}+48068a-51240$
4802.1-bd1	4802.1-bd	$2$	$3$	\(\Q(\sqrt{2}) \)	$2$	$[2, 0]$	4802.1	\( 2 \cdot 7^{4} \)	\( 2^{2} \cdot 7^{11} \)	$2.10397$	$(a), (-2a+1), (2a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B	$1$	\( 2^{3} \)	$0.145513853$	$5.970287511$	4.914446106	\( \frac{341511377481251}{686} a - \frac{482970021761709}{686} \)	\( \bigl[a + 1\) , \( 0\) , \( a\) , \( 75 a - 207\) , \( 217 a + 2431\bigr] \)	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(75a-207\right){x}+217a+2431$

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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.