Elliptic curve search results

Results (12 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
24.1-a7	24.1-a	$8$	$16$	$\Q(\sqrt{3})$	$2$	$[2, 0]$	24.1	$2^{3} \cdot 3$	$2^{10} \cdot 3^{4}$	$0.68514$	$(a+1), (a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{2}$	$1$	$2.325279868$	0.671250479	$\frac{3065617154}{9}$	$\bigl[a + 1$ , $1$ , $0$ , $385 a - 671$ , $5582 a - 9681\bigr]$	${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(385a-671\right){x}+5582a-9681$
24.1-b7	24.1-b	$8$	$16$	$\Q(\sqrt{3})$	$2$	$[2, 0]$	24.1	$2^{3} \cdot 3$	$2^{10} \cdot 3^{4}$	$0.68514$	$(a+1), (a)$	0	$\Z/8\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{3}$	$1$	$22.73403407$	0.820343793	$\frac{3065617154}{9}$	$\bigl[a + 1$ , $-a$ , $a + 1$ , $383 a - 674$ , $-5198 a + 9008\bigr]$	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(383a-674\right){x}-5198a+9008$
144.1-a7	144.1-a	$8$	$16$	$\Q(\sqrt{3})$	$2$	$[2, 0]$	144.1	$2^{4} \cdot 3^{2}$	$2^{10} \cdot 3^{10}$	$1.07231$	$(a+1), (a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{2}$	$1$	$4.197737158$	1.211782339	$\frac{3065617154}{9}$	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $-a - 290$ , $-1040 a - 1\bigr]$	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-a-290\right){x}-1040a-1$
144.1-c7	144.1-c	$8$	$16$	$\Q(\sqrt{3})$	$2$	$[2, 0]$	144.1	$2^{4} \cdot 3^{2}$	$2^{10} \cdot 3^{10}$	$1.07231$	$(a+1), (a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{2}$	$1$	$4.197737158$	1.211782339	$\frac{3065617154}{9}$	$\bigl[a + 1$ , $-1$ , $0$ , $16142 a - 27960$ , $1464590 a - 2536744\bigr]$	${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(16142a-27960\right){x}+1464590a-2536744$
768.1-e7	768.1-e	$8$	$16$	$\Q(\sqrt{3})$	$2$	$[2, 0]$	768.1	$2^{8} \cdot 3$	$2^{22} \cdot 3^{4}$	$1.62956$	$(a+1), (a)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{3}$	$1$	$11.36701703$	1.640687586	$\frac{3065617154}{9}$	$\bigl[0$ , $a - 1$ , $0$ , $-1538 a - 2689$ , $43117 a + 74761\bigr]$	${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-1538a-2689\right){x}+43117a+74761$
768.1-l7	768.1-l	$8$	$16$	$\Q(\sqrt{3})$	$2$	$[2, 0]$	768.1	$2^{8} \cdot 3$	$2^{22} \cdot 3^{4}$	$1.62956$	$(a+1), (a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓			$2$	2B	$1$	$2^{4}$	$1.079273864$	$1.162639934$	2.897852395	$\frac{3065617154}{9}$	$\bigl[0$ , $-a + 1$ , $0$ , $-1538 a - 2689$ , $-43117 a - 74761\bigr]$	${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-1538a-2689\right){x}-43117a-74761$
2304.1-q7	2304.1-q	$8$	$16$	$\Q(\sqrt{3})$	$2$	$[2, 0]$	2304.1	$2^{8} \cdot 3^{2}$	$2^{22} \cdot 3^{10}$	$2.14462$	$(a+1), (a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{2}$	$2.786181258$	$2.098868579$	3.376245242	$\frac{3065617154}{9}$	$\bigl[0$ , $-a$ , $0$ , $-1152$ , $-8316 a\bigr]$	${y}^2={x}^{3}-a{x}^{2}-1152{x}-8316a$
2304.1-bb7	2304.1-bb	$8$	$16$	$\Q(\sqrt{3})$	$2$	$[2, 0]$	2304.1	$2^{8} \cdot 3^{2}$	$2^{22} \cdot 3^{10}$	$2.14462$	$(a+1), (a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{2}$	$2.786181258$	$2.098868579$	3.376245242	$\frac{3065617154}{9}$	$\bigl[0$ , $a$ , $0$ , $-1152$ , $8316 a\bigr]$	${y}^2={x}^{3}+a{x}^{2}-1152{x}+8316a$
3072.1-e7	3072.1-e	$8$	$16$	$\Q(\sqrt{3})$	$2$	$[2, 0]$	3072.1	$2^{10} \cdot 3$	$2^{28} \cdot 3^{4}$	$2.30454$	$(a+1), (a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{4}$	$1$	$2.570578528$	2.968248410	$\frac{3065617154}{9}$	$\bigl[0$ , $-a + 1$ , $0$ , $768 a - 1536$ , $-16632 a + 27720\bigr]$	${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(768a-1536\right){x}-16632a+27720$
3072.1-f7	3072.1-f	$8$	$16$	$\Q(\sqrt{3})$	$2$	$[2, 0]$	3072.1	$2^{10} \cdot 3$	$2^{28} \cdot 3^{4}$	$2.30454$	$(a+1), (a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{3}$	$0.929871924$	$2.570578528$	2.760090861	$\frac{3065617154}{9}$	$\bigl[0$ , $-a - 1$ , $0$ , $-768 a - 1536$ , $-16632 a - 27720\bigr]$	${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-768a-1536\right){x}-16632a-27720$
3072.1-bt7	3072.1-bt	$8$	$16$	$\Q(\sqrt{3})$	$2$	$[2, 0]$	3072.1	$2^{10} \cdot 3$	$2^{28} \cdot 3^{4}$	$2.30454$	$(a+1), (a)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{4}$	$1$	$2.570578528$	2.968248410	$\frac{3065617154}{9}$	$\bigl[0$ , $a + 1$ , $0$ , $-768 a - 1536$ , $16632 a + 27720\bigr]$	${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-768a-1536\right){x}+16632a+27720$
3072.1-cc7	3072.1-cc	$8$	$16$	$\Q(\sqrt{3})$	$2$	$[2, 0]$	3072.1	$2^{10} \cdot 3$	$2^{28} \cdot 3^{4}$	$2.30454$	$(a+1), (a)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓				$2$	2B	$1$	$2^{3}$	$0.929871924$	$2.570578528$	2.760090861	$\frac{3065617154}{9}$	$\bigl[0$ , $a - 1$ , $0$ , $768 a - 1536$ , $16632 a - 27720\bigr]$	${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(768a-1536\right){x}+16632a-27720$

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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.