Elliptic curve search results

Results (14 matches)

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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
288.2-b3	288.2-b	$8$	$16$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	288.2	$2^{5} \cdot 3^{2}$	$2^{12} \cdot 3^{8}$	$1.04119$	$(a), (-a-1), (a-1)$	0	$\Z/2\Z\oplus\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{5}$	$1$	$3.353492831$	1.185638761	$-\frac{8000}{81}$	$\bigl[0$ , $a$ , $0$ , $1$ , $3 a\bigr]$	${y}^2={x}^{3}+a{x}^{2}+{x}+3a$
288.2-c3	288.2-c	$8$	$16$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	288.2	$2^{5} \cdot 3^{2}$	$2^{12} \cdot 3^{8}$	$1.04119$	$(a), (-a-1), (a-1)$	0	$\Z/2\Z\oplus\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{5}$	$1$	$3.353492831$	1.185638761	$-\frac{8000}{81}$	$\bigl[0$ , $-a$ , $0$ , $1$ , $-3 a\bigr]$	${y}^2={x}^{3}-a{x}^{2}+{x}-3a$
2592.3-b3	2592.3-b	$8$	$16$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	2592.3	$2^{5} \cdot 3^{4}$	$2^{12} \cdot 3^{20}$	$1.80340$	$(a), (-a-1), (a-1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{5}$	$1$	$1.117830943$	1.580851681	$-\frac{8000}{81}$	$\bigl[0$ , $0$ , $0$ , $15$ , $-68 a\bigr]$	${y}^2={x}^{3}+15{x}-68a$
2592.3-f3	2592.3-f	$8$	$16$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	2592.3	$2^{5} \cdot 3^{4}$	$2^{12} \cdot 3^{20}$	$1.80340$	$(a), (-a-1), (a-1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{5}$	$1$	$1.117830943$	1.580851681	$-\frac{8000}{81}$	$\bigl[0$ , $0$ , $0$ , $15$ , $68 a\bigr]$	${y}^2={x}^{3}+15{x}+68a$
6912.2-e3	6912.2-e	$8$	$16$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	6912.2	$2^{8} \cdot 3^{3}$	$2^{12} \cdot 3^{14}$	$2.30454$	$(a), (-a-1), (a-1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{4}$	$1$	$1.936139989$	1.369057715	$-\frac{8000}{81}$	$\bigl[0$ , $a + 1$ , $0$ , $4 a - 2$ , $-12 a - 8\bigr]$	${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(4a-2\right){x}-12a-8$
6912.2-j3	6912.2-j	$8$	$16$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	6912.2	$2^{8} \cdot 3^{3}$	$2^{12} \cdot 3^{14}$	$2.30454$	$(a), (-a-1), (a-1)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{5}$	$0.788156716$	$1.936139989$	4.316128133	$-\frac{8000}{81}$	$\bigl[0$ , $-a - 1$ , $0$ , $4 a - 2$ , $12 a + 8\bigr]$	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(4a-2\right){x}+12a+8$
6912.3-b3	6912.3-b	$8$	$16$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	6912.3	$2^{8} \cdot 3^{3}$	$2^{12} \cdot 3^{14}$	$2.30454$	$(a), (-a-1), (a-1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{4}$	$1$	$1.936139989$	1.369057715	$-\frac{8000}{81}$	$\bigl[0$ , $-a + 1$ , $0$ , $-4 a - 2$ , $12 a - 8\bigr]$	${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4a-2\right){x}+12a-8$
6912.3-m3	6912.3-m	$8$	$16$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	6912.3	$2^{8} \cdot 3^{3}$	$2^{12} \cdot 3^{14}$	$2.30454$	$(a), (-a-1), (a-1)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{5}$	$0.788156716$	$1.936139989$	4.316128133	$-\frac{8000}{81}$	$\bigl[0$ , $a - 1$ , $0$ , $-4 a - 2$ , $-12 a + 8\bigr]$	${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-4a-2\right){x}-12a+8$
9216.2-h3	9216.2-h	$8$	$16$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	9216.2	$2^{10} \cdot 3^{2}$	$2^{18} \cdot 3^{8}$	$2.47639$	$(a), (-a-1), (a-1)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$1$	$2^{4}$	$0.878591722$	$2.371277522$	2.946351043	$-\frac{8000}{81}$	$\bigl[0$ , $-1$ , $0$ , $-3$ , $-9\bigr]$	${y}^2={x}^{3}-{x}^{2}-3{x}-9$
9216.2-u3	9216.2-u	$8$	$16$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	9216.2	$2^{10} \cdot 3^{2}$	$2^{18} \cdot 3^{8}$	$2.47639$	$(a), (-a-1), (a-1)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2Cs	$1$	$2^{6}$	$0.385972529$	$2.371277522$	5.177424439	$-\frac{8000}{81}$	$\bigl[0$ , $1$ , $0$ , $-3$ , $9\bigr]$	${y}^2={x}^{3}+{x}^{2}-3{x}+9$
27648.2-c3	27648.2-c	$8$	$16$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	27648.2	$2^{10} \cdot 3^{3}$	$2^{18} \cdot 3^{14}$	$3.25911$	$(a), (-a-1), (a-1)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{5}$	$0.880355775$	$1.369057715$	3.408984042	$-\frac{8000}{81}$	$\bigl[0$ , $a + 1$ , $0$ , $-6 a + 3$ , $9 a - 45\bigr]$	${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-6a+3\right){x}+9a-45$
27648.2-bu3	27648.2-bu	$8$	$16$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	27648.2	$2^{10} \cdot 3^{3}$	$2^{18} \cdot 3^{14}$	$3.25911$	$(a), (-a-1), (a-1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{6}$	$1$	$1.369057715$	3.872279978	$-\frac{8000}{81}$	$\bigl[0$ , $-a - 1$ , $0$ , $-6 a + 3$ , $-9 a + 45\bigr]$	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6a+3\right){x}-9a+45$
27648.3-b3	27648.3-b	$8$	$16$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	27648.3	$2^{10} \cdot 3^{3}$	$2^{18} \cdot 3^{14}$	$3.25911$	$(a), (-a-1), (a-1)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{5}$	$0.880355775$	$1.369057715$	3.408984042	$-\frac{8000}{81}$	$\bigl[0$ , $-a + 1$ , $0$ , $6 a + 3$ , $-9 a - 45\bigr]$	${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(6a+3\right){x}-9a-45$
27648.3-bt3	27648.3-bt	$8$	$16$	$\Q(\sqrt{-2})$	$2$	$[0, 1]$	27648.3	$2^{10} \cdot 3^{3}$	$2^{18} \cdot 3^{14}$	$3.25911$	$(a), (-a-1), (a-1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2Cs	$1$	$2^{6}$	$1$	$1.369057715$	3.872279978	$-\frac{8000}{81}$	$\bigl[0$ , $a - 1$ , $0$ , $6 a + 3$ , $9 a + 45\bigr]$	${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(6a+3\right){x}+9a+45$

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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.