Elliptic curves over \(\Q(\sqrt{-11}) \) of conductor 1089.3

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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
1089.3-a1	1089.3-a	$2$	$11$	\(\Q(\sqrt{-11}) \)	$2$	$[0, 1]$	1089.3	\( 3^{2} \cdot 11^{2} \)	\( 3^{6} \cdot 11^{10} \)	$1.70252$	$(a-1), (-2a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$11$	11B.10.4[2]	$1$	\( 2 \)	$1.143742936$	$0.490498967$	1.353194323	\( -24729001 \)	\( \bigl[a\) , \( a + 1\) , \( 1\) , \( -330 a - 662\) , \( -6012 a - 4518\bigr] \)	${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-330a-662\right){x}-6012a-4518$
1089.3-a2	1089.3-a	$2$	$11$	\(\Q(\sqrt{-11}) \)	$2$	$[0, 1]$	1089.3	\( 3^{2} \cdot 11^{2} \)	\( 3^{6} \cdot 11^{2} \)	$1.70252$	$(a-1), (-2a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓			✓	$11$	11B.10.2[2]	$1$	\( 2 \)	$0.103976630$	$5.395488645$	1.353194323	\( -121 \)	\( \bigl[a\) , \( a + 1\) , \( 1\) , \( -2\) , \( 0\bigr] \)	${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}-2{x}$
1089.3-b1	1089.3-b	$2$	$2$	\(\Q(\sqrt{-11}) \)	$2$	$[0, 1]$	1089.3	\( 3^{2} \cdot 11^{2} \)	\( 3^{3} \cdot 11^{8} \)	$1.70252$	$(a-1), (-2a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{3} \)	$1$	$1.280829557$	1.544738568	\( \frac{393194}{11} a - \frac{16965365}{11} \)	\( \bigl[a + 1\) , \( -a\) , \( a\) , \( 58 a - 56\) , \( 212 a + 55\bigr] \)	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(58a-56\right){x}+212a+55$
1089.3-b2	1089.3-b	$2$	$2$	\(\Q(\sqrt{-11}) \)	$2$	$[0, 1]$	1089.3	\( 3^{2} \cdot 11^{2} \)	\( 3^{3} \cdot 11^{7} \)	$1.70252$	$(a-1), (-2a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	\( 2^{2} \)	$1$	$2.561659115$	1.544738568	\( \frac{7136}{11} a + \frac{4759}{11} \)	\( \bigl[a + 1\) , \( -a\) , \( a\) , \( 3 a - 1\) , \( 3 a\bigr] \)	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(3a-1\right){x}+3a$
1089.3-c1	1089.3-c	$3$	$25$	\(\Q(\sqrt{-11}) \)	$2$	$[0, 1]$	1089.3	\( 3^{2} \cdot 11^{2} \)	\( 3^{6} \cdot 11^{8} \)	$1.70252$	$(a-1), (-2a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 5$	2Cn, 5B.4.2	$1$	\( 2^{2} \)	$9.765222448$	$0.064462474$	3.036775978	\( -\frac{52893159101157376}{11} \)	\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -86024 a - 172048\) , \( -23367062 a - 20639733\bigr] \)	${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-86024a-172048\right){x}-23367062a-20639733$
1089.3-c2	1089.3-c	$3$	$25$	\(\Q(\sqrt{-11}) \)	$2$	$[0, 1]$	1089.3	\( 3^{2} \cdot 11^{2} \)	\( 3^{6} \cdot 11^{16} \)	$1.70252$	$(a-1), (-2a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 5$	2Cn, 5Cs.4.1	$1$	\( 2^{2} \)	$1.953044489$	$0.322312373$	3.036775978	\( -\frac{122023936}{161051} \)	\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -114 a - 228\) , \( -1962 a - 1973\bigr] \)	${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-114a-228\right){x}-1962a-1973$
1089.3-c3	1089.3-c	$3$	$25$	\(\Q(\sqrt{-11}) \)	$2$	$[0, 1]$	1089.3	\( 3^{2} \cdot 11^{2} \)	\( 3^{6} \cdot 11^{8} \)	$1.70252$	$(a-1), (-2a+1)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2, 5$	2Cn, 5B.4.1	$1$	\( 2^{2} \)	$0.390608897$	$1.611561869$	3.036775978	\( -\frac{4096}{11} \)	\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -4 a - 8\) , \( 18 a + 7\bigr] \)	${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4a-8\right){x}+18a+7$

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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.