Elliptic curves over \(\Q(\sqrt{-3}) \) of conductor 6561.1

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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
6561.1-CMf1	6561.1-CMf	$1$	$1$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	6561.1	\( 3^{8} \)	\( 3^{14} \)	$1.39297$	$(-2a+1)$	0	$\Z/3\Z$	$\textsf{yes}$	$-3$	$\mathrm{U}(1)$	✓			✓	$3$	3B.1.1[2]	$1$	\( 3 \)	$1$	$3.898221876$	1.500426299	\( 0 \)	\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 0\) , \( -3 a\bigr] \)	${y}^2+\left(a+1\right){y}={x}^{3}-3a$
6561.1-CMe1	6561.1-CMe	$1$	$1$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	6561.1	\( 3^{8} \)	\( 3^{14} \)	$1.39297$	$(-2a+1)$	0	$\Z/3\Z$	$\textsf{yes}$	$-3$	$\mathrm{U}(1)$	✓			✓	$3$	3B.1.1[2]	$1$	\( 3 \)	$1$	$3.898221876$	1.500426299	\( 0 \)	\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( 2 a - 2\bigr] \)	${y}^2+a{y}={x}^{3}+2a-2$
6561.1-CMd1	6561.1-CMd	$1$	$1$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	6561.1	\( 3^{8} \)	\( 3^{14} \)	$1.39297$	$(-2a+1)$	0	$\Z/3\Z$	$\textsf{yes}$	$-3$	$\mathrm{U}(1)$	✓	✓		✓	$3$	3B.1.1[2]	$1$	\( 3 \)	$1$	$3.898221876$	1.500426299	\( 0 \)	\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 2\bigr] \)	${y}^2+{y}={x}^{3}+2$
6561.1-CMc1	6561.1-CMc	$1$	$1$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	6561.1	\( 3^{8} \)	\( 3^{10} \)	$1.39297$	$(-2a+1)$	0	$\Z/3\Z$	$\textsf{yes}$	$-3$	$\mathrm{U}(1)$	✓			✓	$3$	3B.1.1[2]	$1$	\( 3 \)	$1$	$5.622208826$	2.163989186	\( 0 \)	\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 0\) , \( 0\bigr] \)	${y}^2+\left(a+1\right){y}={x}^{3}$
6561.1-CMb1	6561.1-CMb	$1$	$1$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	6561.1	\( 3^{8} \)	\( 3^{10} \)	$1.39297$	$(-2a+1)$	0	$\Z/3\Z$	$\textsf{yes}$	$-3$	$\mathrm{U}(1)$	✓			✓	$3$	3B.1.1[2]	$1$	\( 3 \)	$1$	$5.622208826$	2.163989186	\( 0 \)	\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( -a + 1\bigr] \)	${y}^2+a{y}={x}^{3}-a+1$
6561.1-CMa1	6561.1-CMa	$1$	$1$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	6561.1	\( 3^{8} \)	\( 3^{10} \)	$1.39297$	$(-2a+1)$	$2$	$\Z/3\Z$	$\textsf{yes}$	$-3$	$\mathrm{U}(1)$	✓	✓		✓	$3$	3B.1.1[2]	$1$	\( 3 \)	$0.192305923$	$5.622208826$	1.664591759	\( 0 \)	\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( -1\bigr] \)	${y}^2+{y}={x}^{3}-1$
6561.1-a1	6561.1-a	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	6561.1	\( 3^{8} \)	\( 3^{14} \)	$1.39297$	$(-2a+1)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2, 3$	2Cn, 3B.1.1[2]	$1$	\( 3 \)	$1$	$3.153971350$	1.213964138	\( -33507 a - 11637 \)	\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 4 a - 11\) , \( 6 a - 10\bigr] \)	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(4a-11\right){x}+6a-10$
6561.1-a2	6561.1-a	$2$	$3$	\(\Q(\sqrt{-3}) \)	$2$	$[0, 1]$	6561.1	\( 3^{8} \)	\( 3^{14} \)	$1.39297$	$(-2a+1)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$2, 3$	2Cn, 3B.1.1[2]	$1$	\( 3 \)	$1$	$3.153971350$	1.213964138	\( 33507 a - 45144 \)	\( \bigl[1\) , \( -1\) , \( a\) , \( -5 a - 6\) , \( -7 a - 3\bigr] \)	${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-5a-6\right){x}-7a-3$

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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.