Elliptic curves over \(\Q(\sqrt{13}) \) of conductor 36.2

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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
36.2-a1	36.2-a	$6$	$45$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	36.2	\( 2^{2} \cdot 3^{2} \)	\( 2^{10} \cdot 3^{6} \)	$0.78920$	$(-a), (2)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3, 5$	3B.1.1, 5B.4.2	$1$	\( 5 \)	$1$	$6.839032582$	1.053781309	\( -\frac{1250637664527933}{32} a - \frac{1629300280935823}{32} \)	\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -113 a - 80\) , \( 590 a + 599\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-113a-80\right){x}+590a+599$
36.2-a2	36.2-a	$6$	$45$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	36.2	\( 2^{2} \cdot 3^{2} \)	\( 2^{2} \cdot 3^{6} \)	$0.78920$	$(-a), (2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3, 5$	3B.1.2, 5B.4.1	$1$	\( 1 \)	$1$	$3.799462546$	1.053781309	\( -\frac{461373}{2} a - \frac{601423}{2} \)	\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 3 a - 8\) , \( 14 a - 35\bigr] \)	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(3a-8\right){x}+14a-35$
36.2-a3	36.2-a	$6$	$45$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	36.2	\( 2^{2} \cdot 3^{2} \)	\( 2^{30} \cdot 3^{6} \)	$0.78920$	$(-a), (2)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3, 5$	3Cs.1.1, 5B.4.2	$1$	\( 3 \cdot 5 \)	$1$	$2.279677527$	1.053781309	\( -\frac{1680914269}{32768} \)	\( \bigl[a\) , \( 0\) , \( 0\) , \( 124 a - 297\) , \( 1087 a - 2492\bigr] \)	${y}^2+a{x}{y}={x}^{3}+\left(124a-297\right){x}+1087a-2492$
36.2-a4	36.2-a	$6$	$45$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	36.2	\( 2^{2} \cdot 3^{2} \)	\( 2^{2} \cdot 3^{6} \)	$0.78920$	$(-a), (2)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3, 5$	3B.1.1, 5B.4.1	$1$	\( 1 \)	$1$	$34.19516291$	1.053781309	\( \frac{461373}{2} a - 531398 \)	\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 2 a - 5\) , \( -5 a + 11\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(2a-5\right){x}-5a+11$
36.2-a5	36.2-a	$6$	$45$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	36.2	\( 2^{2} \cdot 3^{2} \)	\( 2^{6} \cdot 3^{6} \)	$0.78920$	$(-a), (2)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3, 5$	3Cs.1.1, 5B.4.1	$1$	\( 3 \)	$1$	$11.39838763$	1.053781309	\( \frac{1331}{8} \)	\( \bigl[a\) , \( 0\) , \( 0\) , \( -a + 3\) , \( -3 a + 7\bigr] \)	${y}^2+a{x}{y}={x}^{3}+\left(-a+3\right){x}-3a+7$
36.2-a6	36.2-a	$6$	$45$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	36.2	\( 2^{2} \cdot 3^{2} \)	\( 2^{10} \cdot 3^{6} \)	$0.78920$	$(-a), (2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3, 5$	3B.1.2, 5B.4.2	$1$	\( 5 \)	$1$	$0.759892509$	1.053781309	\( \frac{1250637664527933}{32} a - \frac{719984486365939}{8} \)	\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 87 a + 6\) , \( 358 a + 111\bigr] \)	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(87a+6\right){x}+358a+111$

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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.