Elliptic curves over \(\Q(\sqrt{-3}) \) of conductor 676.2

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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
676.2-a1	676.2-a	$2$	$7$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	676.2	\( 2^{2} \cdot 13^{2} \)	\( 2^{2} \cdot 13^{14} \)	$0.78920$	$(-4a+1), (4a-3), (2)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$7$	7B.1.3	$1$	\( 1 \)	$1$	$0.560128502$	0.646780683	\( -\frac{1064019559329}{125497034} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -213\) , \( -1257\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-213{x}-1257$
676.2-a2	676.2-a	$2$	$7$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	676.2	\( 2^{2} \cdot 13^{2} \)	\( 2^{14} \cdot 13^{2} \)	$0.78920$	$(-4a+1), (4a-3), (2)$	0	$\Z/7\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$7$	7B.1.1	$1$	\( 7 \)	$1$	$3.920899519$	0.646780683	\( -\frac{2146689}{1664} \)	\( \bigl[1\) , \( -1\) , \( 1\) , \( -3\) , \( 3\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-3{x}+3$
676.2-b1	676.2-b	$5$	$9$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	676.2	\( 2^{2} \cdot 13^{2} \)	\( 2^{18} \cdot 13^{2} \)	$0.78920$	$(-4a+1), (4a-3), (2)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3B.1.1[2]	$1$	\( 3^{2} \)	$1$	$0.896934130$	1.035690323	\( -\frac{10730978619193}{6656} \)	\( \bigl[1\) , \( 0\) , \( 1\) , \( -460\) , \( -3830\bigr] \)	${y}^2+{x}{y}+{y}={x}^{3}-460{x}-3830$
676.2-b2	676.2-b	$5$	$9$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	676.2	\( 2^{2} \cdot 13^{2} \)	\( 2^{6} \cdot 13^{6} \)	$0.78920$	$(-4a+1), (4a-3), (2)$	0	$\Z/3\Z\oplus\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3Cs.1.1[2]	$1$	\( 3^{3} \)	$1$	$2.690802392$	1.035690323	\( -\frac{10218313}{17576} \)	\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( -5 a + 4\) , \( -8\bigr] \)	${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-5a+4\right){x}-8$
676.2-b3	676.2-b	$5$	$9$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	676.2	\( 2^{2} \cdot 13^{2} \)	\( 2^{2} \cdot 13^{2} \)	$0.78920$	$(-4a+1), (4a-3), (2)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3B.1.1[2]	$1$	\( 1 \)	$1$	$8.072407178$	1.035690323	\( \frac{12167}{26} \)	\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( -1\) , \( 0\bigr] \)	${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}-{x}$
676.2-b4	676.2-b	$5$	$9$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	676.2	\( 2^{2} \cdot 13^{2} \)	\( 2^{2} \cdot 13^{10} \)	$0.78920$	$(-4a+1), (4a-3), (2)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$3$	3B.1.1[2]	$1$	\( 3^{2} \)	$1$	$0.896934130$	1.035690323	\( -\frac{1824558573109097}{21208998746} a + \frac{26221448217163305}{21208998746} \)	\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( 135 a + 94\) , \( 724 a - 1240\bigr] \)	${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(135a+94\right){x}+724a-1240$
676.2-b5	676.2-b	$5$	$9$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	676.2	\( 2^{2} \cdot 13^{2} \)	\( 2^{2} \cdot 13^{10} \)	$0.78920$	$(-4a+1), (4a-3), (2)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$3$	3B.1.1[2]	$1$	\( 3^{2} \)	$1$	$0.896934130$	1.035690323	\( \frac{1824558573109097}{21208998746} a + \frac{12198444822027104}{10604499373} \)	\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( -95 a - 136\) , \( -724 a - 516\bigr] \)	${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-95a-136\right){x}-724a-516$

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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.