Elliptic curves over Q(−7)\Q(\sqrt{-7}) of conductor 26244.2

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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
26244.2-a1	26244.2-a	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{4} \cdot 3^{12}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$3$	3B.1.1	$1$	$2^{2} \cdot 3$	$0.305934883$	$3.305583379$	2.038575609	$-\frac{35937}{4}$	$\bigl[1$ , $-1$ , $0$ , $-6$ , $8\bigr]$	${y}^2+{x}{y}={x}^{3}-{x}^{2}-6{x}+8$
26244.2-a2	26244.2-a	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{12} \cdot 3^{20}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$3$	3B.1.2	$1$	$2^{2} \cdot 3$	$0.101978294$	$1.101861126$	2.038575609	$\frac{109503}{64}$	$\bigl[1$ , $-1$ , $0$ , $39$ , $-19\bigr]$	${y}^2+{x}{y}={x}^{3}-{x}^{2}+39{x}-19$
26244.2-b1	26244.2-b	$4$	$21$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{14} \cdot 3^{12}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$3, 7$	3B.1.1, 7B[2]	$1$	$3$	$10.13703660$	$0.583485219$	2.980784579	$-\frac{189613868625}{128}$	$\bigl[1$ , $-1$ , $0$ , $-1077$ , $13877\bigr]$	${y}^2+{x}{y}={x}^{3}-{x}^{2}-1077{x}+13877$
26244.2-b2	26244.2-b	$4$	$21$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{6} \cdot 3^{20}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$3, 7$	3B.1.2, 7B[2]	$1$	$3$	$0.482716028$	$1.361465512$	2.980784579	$-\frac{140625}{8}$	$\bigl[1$ , $-1$ , $0$ , $-42$ , $-100\bigr]$	${y}^2+{x}{y}={x}^{3}-{x}^{2}-42{x}-100$
26244.2-b3	26244.2-b	$4$	$21$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{42} \cdot 3^{20}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$3, 7$	3B.1.2, 7B[2]	$1$	$3$	$3.379012201$	$0.194495073$	2.980784579	$-\frac{1159088625}{2097152}$	$\bigl[1$ , $-1$ , $0$ , $-852$ , $19664\bigr]$	${y}^2+{x}{y}={x}^{3}-{x}^{2}-852{x}+19664$
26244.2-b4	26244.2-b	$4$	$21$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{2} \cdot 3^{12}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$3, 7$	3B.1.1, 7B[2]	$1$	$3$	$1.448148086$	$4.084396538$	2.980784579	$\frac{3375}{2}$	$\bigl[1$ , $-1$ , $0$ , $3$ , $-1\bigr]$	${y}^2+{x}{y}={x}^{3}-{x}^{2}+3{x}-1$
26244.2-c1	26244.2-c	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{29} \cdot 3^{8}$	$3.00916$	$(a), (-a+1), (3)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.1	$1$	$2^{3} \cdot 3$	$1$	$1.223113832$	2.465565733	$-\frac{3541149801}{16777216} a - \frac{7108840053}{8388608}$	$\bigl[1$ , $-1$ , $a$ , $4 a - 27$ , $-28 a + 95\bigr]$	${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(4a-27\right){x}-28a+95$
26244.2-c2	26244.2-c	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{23} \cdot 3^{24}$	$3.00916$	$(a), (-a+1), (3)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.2	$1$	$2^{3}$	$1$	$0.407704610$	2.465565733	$\frac{3499281}{32768} a + \frac{20975733}{32768}$	$\bigl[1$ , $-1$ , $a$ , $-41 a + 228$ , $587 a - 1598\bigr]$	${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-41a+228\right){x}+587a-1598$
26244.2-d1	26244.2-d	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{29} \cdot 3^{20}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.2	$1$	$2 \cdot 5$	$0.820812680$	$0.407704610$	5.059419044	$\frac{3541149801}{16777216} a - \frac{17758829907}{16777216}$	$\bigl[1$ , $-1$ , $a$ , $-41 a - 204$ , $-709 a - 1598\bigr]$	${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-41a-204\right){x}-709a-1598$
26244.2-d2	26244.2-d	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{23} \cdot 3^{12}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.1	$1$	$2 \cdot 3^{2} \cdot 5$	$0.273604226$	$1.223113832$	5.059419044	$-\frac{3499281}{32768} a + \frac{12237507}{16384}$	$\bigl[1$ , $-1$ , $a$ , $4 a + 21$ , $20 a + 31\bigr]$	${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(4a+21\right){x}+20a+31$
26244.2-e1	26244.2-e	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{7} \cdot 3^{12}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.1	$1$	$2 \cdot 3^{2}$	$0.607627350$	$2.973335577$	5.462886890	$\frac{20817}{16} a + \frac{26271}{16}$	$\bigl[1$ , $-1$ , $a$ , $-5 a + 3$ , $2 a - 5\bigr]$	${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-5a+3\right){x}+2a-5$
26244.2-e2	26244.2-e	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{13} \cdot 3^{20}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.2	$1$	$2$	$1.822882051$	$0.991111859$	5.462886890	$-\frac{11308041}{4096} a + \frac{12488499}{4096}$	$\bigl[1$ , $-1$ , $a$ , $40 a - 42$ , $101 a + 22\bigr]$	${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(40a-42\right){x}+101a+22$
26244.2-f1	26244.2-f	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{13} \cdot 3^{8}$	$3.00916$	$(a), (-a+1), (3)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.1	$1$	$2^{2} \cdot 3$	$1$	$2.973335577$	2.996840572	$\frac{11308041}{4096} a + \frac{590229}{2048}$	$\bigl[1$ , $-1$ , $a$ , $-5 a$ , $5 a - 4\bigr]$	${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}-5a{x}+5a-4$
26244.2-f2	26244.2-f	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{7} \cdot 3^{24}$	$3.00916$	$(a), (-a+1), (3)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.2	$1$	$2^{2}$	$1$	$0.991111859$	2.996840572	$-\frac{20817}{16} a + 2943$	$\bigl[1$ , $-1$ , $a$ , $40 a - 15$ , $20 a + 103\bigr]$	${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(40a-15\right){x}+20a+103$
26244.2-g1	26244.2-g	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{29} \cdot 3^{8}$	$3.00916$	$(a), (-a+1), (3)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.1	$1$	$2^{3} \cdot 3$	$1$	$1.223113832$	2.465565733	$\frac{3541149801}{16777216} a - \frac{17758829907}{16777216}$	$\bigl[1$ , $-1$ , $a + 1$ , $-5 a - 23$ , $27 a + 67\bigr]$	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-5a-23\right){x}+27a+67$
26244.2-g2	26244.2-g	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{23} \cdot 3^{24}$	$3.00916$	$(a), (-a+1), (3)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.2	$1$	$2^{3}$	$1$	$0.407704610$	2.465565733	$-\frac{3499281}{32768} a + \frac{12237507}{16384}$	$\bigl[1$ , $-1$ , $a + 1$ , $40 a + 187$ , $-588 a - 1011\bigr]$	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(40a+187\right){x}-588a-1011$
26244.2-h1	26244.2-h	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{29} \cdot 3^{20}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.2	$1$	$2 \cdot 5$	$0.820812680$	$0.407704610$	5.059419044	$-\frac{3541149801}{16777216} a - \frac{7108840053}{8388608}$	$\bigl[1$ , $-1$ , $a + 1$ , $40 a - 245$ , $708 a - 2307\bigr]$	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(40a-245\right){x}+708a-2307$
26244.2-h2	26244.2-h	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{23} \cdot 3^{12}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.1	$1$	$2 \cdot 3^{2} \cdot 5$	$0.273604226$	$1.223113832$	5.059419044	$\frac{3499281}{32768} a + \frac{20975733}{32768}$	$\bigl[1$ , $-1$ , $a + 1$ , $-5 a + 25$ , $-21 a + 51\bigr]$	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-5a+25\right){x}-21a+51$
26244.2-i1	26244.2-i	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{13} \cdot 3^{20}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.2	$1$	$2$	$1.822882051$	$0.991111859$	5.462886890	$\frac{11308041}{4096} a + \frac{590229}{2048}$	$\bigl[1$ , $-1$ , $a + 1$ , $-41 a - 2$ , $-102 a + 123\bigr]$	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-41a-2\right){x}-102a+123$
26244.2-i2	26244.2-i	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{7} \cdot 3^{12}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.1	$1$	$2 \cdot 3^{2}$	$0.607627350$	$2.973335577$	5.462886890	$-\frac{20817}{16} a + 2943$	$\bigl[1$ , $-1$ , $a + 1$ , $4 a - 2$ , $-3 a - 3\bigr]$	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(4a-2\right){x}-3a-3$
26244.2-j1	26244.2-j	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{7} \cdot 3^{24}$	$3.00916$	$(a), (-a+1), (3)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.2	$1$	$2^{2}$	$1$	$0.991111859$	2.996840572	$\frac{20817}{16} a + \frac{26271}{16}$	$\bigl[1$ , $-1$ , $a + 1$ , $-41 a + 25$ , $-21 a + 123\bigr]$	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-41a+25\right){x}-21a+123$
26244.2-j2	26244.2-j	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{13} \cdot 3^{8}$	$3.00916$	$(a), (-a+1), (3)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$3$	3B.1.1	$1$	$2^{2} \cdot 3$	$1$	$2.973335577$	2.996840572	$-\frac{11308041}{4096} a + \frac{12488499}{4096}$	$\bigl[1$ , $-1$ , $a + 1$ , $4 a - 5$ , $-6 a + 1\bigr]$	${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(4a-5\right){x}-6a+1$
26244.2-k1	26244.2-k	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{4} \cdot 3^{24}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$3$	3B.1.2	$1$	$2^{2}$	$1.373693536$	$1.101861126$	9.153510392	$-\frac{35937}{4}$	$\bigl[1$ , $-1$ , $1$ , $-56$ , $-161\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-56{x}-161$
26244.2-k2	26244.2-k	$2$	$3$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{12} \cdot 3^{8}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$3$	3B.1.1	$1$	$2^{2} \cdot 3^{2}$	$0.457897845$	$3.305583379$	9.153510392	$\frac{109503}{64}$	$\bigl[1$ , $-1$ , $1$ , $4$ , $-1\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+4{x}-1$
26244.2-l1	26244.2-l	$4$	$21$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{14} \cdot 3^{24}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$3, 7$	3B.1.2, 7B.2.1[2]	$1$	$7^{2}$	$0.719702593$	$0.194495073$	10.36976045	$-\frac{189613868625}{128}$	$\bigl[1$ , $-1$ , $1$ , $-9695$ , $-364985\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-9695{x}-364985$
26244.2-l2	26244.2-l	$4$	$21$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{6} \cdot 3^{8}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$3, 7$	3B.1.1, 7B.2.1[2]	$1$	$3^{2}$	$1.679306052$	$4.084396538$	10.36976045	$-\frac{140625}{8}$	$\bigl[1$ , $-1$ , $1$ , $-5$ , $5\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-5{x}+5$
26244.2-l3	26244.2-l	$4$	$21$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{42} \cdot 3^{8}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$3, 7$	3B.1.1, 7B.2.1[2]	$1$	$3^{2} \cdot 7^{2}$	$0.239900864$	$0.583485219$	10.36976045	$-\frac{1159088625}{2097152}$	$\bigl[1$ , $-1$ , $1$ , $-95$ , $-697\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-95{x}-697$
26244.2-l4	26244.2-l	$4$	$21$	$\Q(\sqrt{-7})$	$2$	$[0, 1]$	26244.2	$2^{2} \cdot 3^{8}$	$2^{2} \cdot 3^{24}$	$3.00916$	$(a), (-a+1), (3)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$3, 7$	3B.1.2, 7B.2.1[2]	$1$	$1$	$5.037918157$	$1.361465512$	10.36976045	$\frac{3375}{2}$	$\bigl[1$ , $-1$ , $1$ , $25$ , $1\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+25{x}+1$

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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.