Elliptic curves over \(\Q(\sqrt{13}) \) of conductor 36.3

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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
36.3-a1	36.3-a	$6$	$45$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	36.3	\( 2^{2} \cdot 3^{2} \)	\( 2^{10} \cdot 3^{6} \)	$0.78920$	$(-a+1), (2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3, 5$	3B.1.2, 5B.4.2	$1$	\( 5 \)	$1$	$0.759892509$	1.053781309	\( -\frac{1250637664527933}{32} a - \frac{1629300280935823}{32} \)	\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -85 a + 94\) , \( -352 a + 209\bigr] \)	${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-85a+94\right){x}-352a+209$
36.3-a2	36.3-a	$6$	$45$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	36.3	\( 2^{2} \cdot 3^{2} \)	\( 2^{2} \cdot 3^{6} \)	$0.78920$	$(-a+1), (2)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3, 5$	3B.1.1, 5B.4.1	$1$	\( 1 \)	$1$	$34.19516291$	1.053781309	\( -\frac{461373}{2} a - \frac{601423}{2} \)	\( \bigl[a\) , \( a + 1\) , \( 1\) , \( -a - 1\) , \( 0\bigr] \)	${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-a-1\right){x}$
36.3-a3	36.3-a	$6$	$45$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	36.3	\( 2^{2} \cdot 3^{2} \)	\( 2^{30} \cdot 3^{6} \)	$0.78920$	$(-a+1), (2)$	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	\( -\frac{1680914269}{32768} \)	\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -124 a - 173\) , \( -1087 a - 1405\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-124a-173\right){x}-1087a-1405$
36.3-a4	36.3-a	$6$	$45$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	36.3	\( 2^{2} \cdot 3^{2} \)	\( 2^{2} \cdot 3^{6} \)	$0.78920$	$(-a+1), (2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3, 5$	3B.1.2, 5B.4.1	$1$	\( 1 \)	$1$	$3.799462546$	1.053781309	\( \frac{461373}{2} a - 531398 \)	\( \bigl[1\) , \( -a\) , \( a\) , \( -4 a - 4\) , \( -15 a - 20\bigr] \)	${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-4a-4\right){x}-15a-20$
36.3-a5	36.3-a	$6$	$45$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	36.3	\( 2^{2} \cdot 3^{2} \)	\( 2^{6} \cdot 3^{6} \)	$0.78920$	$(-a+1), (2)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3, 5$	3Cs.1.1, 5B.4.1	$1$	\( 3 \)	$1$	$11.39838763$	1.053781309	\( \frac{1331}{8} \)	\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( a + 2\) , \( 3 a + 4\bigr] \)	${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(a+2\right){x}+3a+4$
36.3-a6	36.3-a	$6$	$45$	\(\Q(\sqrt{13}) \)	$2$	$[2, 0]$	36.3	\( 2^{2} \cdot 3^{2} \)	\( 2^{10} \cdot 3^{6} \)	$0.78920$	$(-a+1), (2)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$3, 5$	3B.1.1, 5B.4.2	$1$	\( 5 \)	$1$	$6.839032582$	1.053781309	\( \frac{1250637664527933}{32} a - \frac{719984486365939}{8} \)	\( \bigl[a\) , \( a + 1\) , \( 1\) , \( 114 a - 191\) , \( -670 a + 1528\bigr] \)	${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(114a-191\right){x}-670a+1528$

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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.