Elliptic curves over \(\Q(\sqrt{13}) \) of conductor 23.2

The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 5000 over real quadratic fields with discriminant 497

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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
23.2-a1	23.2-a	$2$	$2$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	23.2	\( 23 \)	\( - 23^{2} \)	$0.70557$	$(3a-4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	\( 2 \)	$1$	$6.496490929$	0.900901198	\( -\frac{50955500525}{529} a + \frac{117339251834}{529} \)	\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 8 a - 7\) , \( 10 a - 14\bigr] \)	${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(8a-7\right){x}+10a-14$
23.2-a2	23.2-a	$2$	$2$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	23.2	\( 23 \)	\( -23 \)	$0.70557$	$(3a-4)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	\( 1 \)	$1$	$12.99298185$	0.900901198	\( \frac{156803}{23} a - \frac{207479}{23} \)	\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 3 a + 3\) , \( 2 a + 2\bigr] \)	${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a+3\right){x}+2a+2$

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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.