Elliptic curves over Q(−3)\Q(\sqrt{-3}) of conductor 9984.1

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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
9984.1-a1	9984.1-a	$4$	$4$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	9984.1	$2^{8} \cdot 3 \cdot 13$	$2^{16} \cdot 3^{2} \cdot 13$	$1.54712$	$(-2a+1), (-4a+1), (2)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{2}$	$1.753693398$	$1.893750525$	1.917413612	$\frac{178274368}{39} a - \frac{114526480}{13}$	$\bigl[0$ , $a$ , $0$ , $-16 a - 48$ , $-92 a - 116\bigr]$	${y}^2={x}^{3}+a{x}^{2}+\left(-16a-48\right){x}-92a-116$
9984.1-a2	9984.1-a	$4$	$4$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	9984.1	$2^{8} \cdot 3 \cdot 13$	$2^{16} \cdot 3^{8} \cdot 13$	$1.54712$	$(-2a+1), (-4a+1), (2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{2}$	$0.438423349$	$1.893750525$	1.917413612	$\frac{12519872}{1053} a - \frac{4457840}{351}$	$\bigl[0$ , $a$ , $0$ , $-21 a + 2$ , $35 a - 21\bigr]$	${y}^2={x}^{3}+a{x}^{2}+\left(-21a+2\right){x}+35a-21$
9984.1-a3	9984.1-a	$4$	$4$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	9984.1	$2^{8} \cdot 3 \cdot 13$	$2^{16} \cdot 3^{2} \cdot 13^{4}$	$1.54712$	$(-2a+1), (-4a+1), (2)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{4}$	$0.438423349$	$1.893750525$	1.917413612	$\frac{165562496}{85683} a - \frac{32807504}{28561}$	$\bigl[0$ , $a$ , $0$ , $14 a - 3$ , $-11 a - 14\bigr]$	${y}^2={x}^{3}+a{x}^{2}+\left(14a-3\right){x}-11a-14$
9984.1-a4	9984.1-a	$4$	$4$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	9984.1	$2^{8} \cdot 3 \cdot 13$	$2^{8} \cdot 3^{4} \cdot 13^{2}$	$1.54712$	$(-2a+1), (-4a+1), (2)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	$2^{2}$	$0.876846699$	$3.787501051$	1.917413612	$-\frac{4554752}{1521} a + \frac{2244608}{1521}$	$\bigl[0$ , $a$ , $0$ , $-a - 3$ , $-2 a - 2\bigr]$	${y}^2={x}^{3}+a{x}^{2}+\left(-a-3\right){x}-2a-2$
9984.1-b1	9984.1-b	$6$	$8$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	9984.1	$2^{8} \cdot 3 \cdot 13$	$2^{22} \cdot 3 \cdot 13^{2}$	$1.54712$	$(-2a+1), (-4a+1), (2)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{3}$	$2.896857298$	$0.628385753$	2.101952031	$-\frac{1028251680722}{507} a - \frac{94596173570}{507}$	$\bigl[0$ , $a$ , $0$ , $-42 a + 981$ , $13529 a - 5938\bigr]$	${y}^2={x}^{3}+a{x}^{2}+\left(-42a+981\right){x}+13529a-5938$
9984.1-b2	9984.1-b	$6$	$8$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	9984.1	$2^{8} \cdot 3 \cdot 13$	$2^{16} \cdot 3^{4} \cdot 13^{2}$	$1.54712$	$(-2a+1), (-4a+1), (2)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	$2^{3}$	$0.724214324$	$2.513543014$	2.101952031	$-\frac{52528}{1521} a + \frac{5072}{507}$	$\bigl[0$ , $a$ , $0$ , $-2 a + 1$ , $9 a - 6\bigr]$	${y}^2={x}^{3}+a{x}^{2}+\left(-2a+1\right){x}+9a-6$
9984.1-b3	9984.1-b	$6$	$8$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	9984.1	$2^{8} \cdot 3 \cdot 13$	$2^{22} \cdot 3 \cdot 13^{8}$	$1.54712$	$(-2a+1), (-4a+1), (2)$	$1$	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{5}$	$0.724214324$	$0.628385753$	2.101952031	$\frac{2457991737026}{2447192163} a + \frac{3468181544930}{2447192163}$	$\bigl[0$ , $a$ , $0$ , $38 a + 101$ , $-215 a + 382\bigr]$	${y}^2={x}^{3}+a{x}^{2}+\left(38a+101\right){x}-215a+382$
9984.1-b4	9984.1-b	$6$	$8$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	9984.1	$2^{8} \cdot 3 \cdot 13$	$2^{8} \cdot 3^{2} \cdot 13$	$1.54712$	$(-2a+1), (-4a+1), (2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2$	$0.362107162$	$5.027086028$	2.101952031	$\frac{539392}{39} a + \frac{478208}{39}$	$\bigl[0$ , $a$ , $0$ , $3 a - 4$ , $2 a - 3\bigr]$	${y}^2={x}^{3}+a{x}^{2}+\left(3a-4\right){x}+2a-3$
9984.1-b5	9984.1-b	$6$	$8$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	9984.1	$2^{8} \cdot 3 \cdot 13$	$2^{20} \cdot 3^{2} \cdot 13^{4}$	$1.54712$	$(-2a+1), (-4a+1), (2)$	$1$	$\Z/2\Z\oplus\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	$2^{5}$	$1.448428649$	$1.256771507$	2.101952031	$-\frac{4035638236}{85683} a + \frac{3926320864}{85683}$	$\bigl[0$ , $a$ , $0$ , $-2 a + 61$ , $225 a - 90\bigr]$	${y}^2={x}^{3}+a{x}^{2}+\left(-2a+61\right){x}+225a-90$
9984.1-b6	9984.1-b	$6$	$8$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	9984.1	$2^{8} \cdot 3 \cdot 13$	$2^{20} \cdot 3^{8} \cdot 13$	$1.54712$	$(-2a+1), (-4a+1), (2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{3}$	$0.362107162$	$1.256771507$	2.101952031	$\frac{89672548}{1053} a + \frac{92596592}{1053}$	$\bigl[0$ , $a$ , $0$ , $-82 a + 21$ , $241 a - 194\bigr]$	${y}^2={x}^{3}+a{x}^{2}+\left(-82a+21\right){x}+241a-194$
9984.1-c1	9984.1-c	$4$	$4$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	9984.1	$2^{8} \cdot 3 \cdot 13$	$2^{16} \cdot 3^{2} \cdot 13$	$1.54712$	$(-2a+1), (-4a+1), (2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{2}$	$1$	$1.893750525$	2.186714751	$\frac{178274368}{39} a - \frac{114526480}{13}$	$\bigl[0$ , $-a$ , $0$ , $-16 a - 48$ , $92 a + 116\bigr]$	${y}^2={x}^{3}-a{x}^{2}+\left(-16a-48\right){x}+92a+116$
9984.1-c2	9984.1-c	$4$	$4$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	9984.1	$2^{8} \cdot 3 \cdot 13$	$2^{16} \cdot 3^{8} \cdot 13$	$1.54712$	$(-2a+1), (-4a+1), (2)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{4}$	$1$	$1.893750525$	2.186714751	$\frac{12519872}{1053} a - \frac{4457840}{351}$	$\bigl[0$ , $-a$ , $0$ , $-21 a + 2$ , $-35 a + 21\bigr]$	${y}^2={x}^{3}-a{x}^{2}+\left(-21a+2\right){x}-35a+21$
9984.1-c3	9984.1-c	$4$	$4$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	9984.1	$2^{8} \cdot 3 \cdot 13$	$2^{16} \cdot 3^{2} \cdot 13^{4}$	$1.54712$	$(-2a+1), (-4a+1), (2)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2B	$1$	$2^{4}$	$1$	$1.893750525$	2.186714751	$\frac{165562496}{85683} a - \frac{32807504}{28561}$	$\bigl[0$ , $-a$ , $0$ , $14 a - 3$ , $11 a + 14\bigr]$	${y}^2={x}^{3}-a{x}^{2}+\left(14a-3\right){x}+11a+14$
9984.1-c4	9984.1-c	$4$	$4$	$\Q(\sqrt{-3})$	$2$	$[0, 1]$	9984.1	$2^{8} \cdot 3 \cdot 13$	$2^{8} \cdot 3^{4} \cdot 13^{2}$	$1.54712$	$(-2a+1), (-4a+1), (2)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2$	2Cs	$1$	$2^{3}$	$1$	$3.787501051$	2.186714751	$-\frac{4554752}{1521} a + \frac{2244608}{1521}$	$\bigl[0$ , $-a$ , $0$ , $-a - 3$ , $2 a + 2\bigr]$	${y}^2={x}^{3}-a{x}^{2}+\left(-a-3\right){x}+2a+2$

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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.