Elliptic curves over Q(2)\Q(\sqrt{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
9.1-a1	9.1-a	$4$	$10$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	9.1	$3^{2}$	$3^{20}$	$0.43777$	$(3)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$2, 5$	2B, 5B.1.2	$1$	$2$	$1$	$1.266923342$	0.223962521	$-\frac{873722816}{59049}$	$\bigl[a$ , $-a - 1$ , $a + 1$ , $-40 a - 60$ , $-153 a - 220\bigr]$	${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-40a-60\right){x}-153a-220$
9.1-a2	9.1-a	$4$	$10$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	9.1	$3^{2}$	$3^{4}$	$0.43777$	$(3)$	0	$\Z/10\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$2, 5$	2B, 5B.1.1	$1$	$2$	$1$	$31.67308356$	0.223962521	$\frac{64}{9}$	$\bigl[a$ , $-a - 1$ , $a + 1$ , $0$ , $0\bigr]$	${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}$
9.1-a3	9.1-a	$4$	$10$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	9.1	$3^{2}$	$3^{2}$	$0.43777$	$(3)$	0	$\Z/10\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$2, 5$	2B, 5B.1.1	$1$	$1$	$1$	$63.34616712$	0.223962521	$\frac{85184}{3}$	$\bigl[a$ , $a$ , $a + 1$ , $-2 a - 3$ , $0\bigr]$	${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-2a-3\right){x}$
9.1-a4	9.1-a	$4$	$10$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	9.1	$3^{2}$	$3^{10}$	$0.43777$	$(3)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓		✓		$2, 5$	2B, 5B.1.2	$1$	$1$	$1$	$2.533846685$	0.223962521	$\frac{58591911104}{243}$	$\bigl[a$ , $a$ , $a + 1$ , $-162 a - 243$ , $-1495 a - 2130\bigr]$	${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-162a-243\right){x}-1495a-2130$
17.1-a1	17.1-a	$4$	$6$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	17.1	$17$	$17^{2}$	$0.51321$	$(-3a-1)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.1	$1$	$2$	$1$	$22.23161552$	0.436670169	$-\frac{94464}{289} a + \frac{58688}{289}$	$\bigl[a$ , $-a + 1$ , $a + 1$ , $-2 a + 1$ , $-a\bigr]$	${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2a+1\right){x}-a$
17.1-a2	17.1-a	$4$	$6$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	17.1	$17$	$17^{6}$	$0.51321$	$(-3a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.2	$1$	$2$	$1$	$2.470179502$	0.436670169	$\frac{1522678220544}{24137569} a - \frac{2147745195712}{24137569}$	$\bigl[a$ , $-a + 1$ , $a + 1$ , $13 a - 19$ , $47 a - 69\bigr]$	${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(13a-19\right){x}+47a-69$
17.1-a3	17.1-a	$4$	$6$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	17.1	$17$	$17^{3}$	$0.51321$	$(-3a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.2	$1$	$1$	$1$	$4.940359004$	0.436670169	$-\frac{55615383938816}{4913} a + \frac{78652069441856}{4913}$	$\bigl[a$ , $-1$ , $a + 1$ , $-34 a - 53$ , $-113 a - 163\bigr]$	${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-34a-53\right){x}-113a-163$
17.1-a4	17.1-a	$4$	$6$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	17.1	$17$	$17$	$0.51321$	$(-3a-1)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.1	$1$	$1$	$1$	$44.46323104$	0.436670169	$\frac{3690752}{17} a + \frac{5287232}{17}$	$\bigl[a$ , $a + 1$ , $a + 1$ , $a - 1$ , $-1\bigr]$	${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a-1\right){x}-1$
17.2-a1	17.2-a	$4$	$6$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	17.2	$17$	$17^{6}$	$0.51321$	$(3a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.2	$1$	$2$	$1$	$2.470179502$	0.436670169	$-\frac{1522678220544}{24137569} a - \frac{2147745195712}{24137569}$	$\bigl[a$ , $a + 1$ , $a + 1$ , $-14 a - 19$ , $-48 a - 69\bigr]$	${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-14a-19\right){x}-48a-69$
17.2-a2	17.2-a	$4$	$6$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	17.2	$17$	$17^{2}$	$0.51321$	$(3a-1)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.1	$1$	$2$	$1$	$22.23161552$	0.436670169	$\frac{94464}{289} a + \frac{58688}{289}$	$\bigl[a$ , $a + 1$ , $a + 1$ , $a + 1$ , $0\bigr]$	${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+1\right){x}$
17.2-a3	17.2-a	$4$	$6$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	17.2	$17$	$17$	$0.51321$	$(3a-1)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.1	$1$	$1$	$1$	$44.46323104$	0.436670169	$-\frac{3690752}{17} a + \frac{5287232}{17}$	$\bigl[a$ , $-a + 1$ , $a + 1$ , $-2 a - 1$ , $-a - 1\bigr]$	${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2a-1\right){x}-a-1$
17.2-a4	17.2-a	$4$	$6$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	17.2	$17$	$17^{3}$	$0.51321$	$(3a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.2	$1$	$1$	$1$	$4.940359004$	0.436670169	$\frac{55615383938816}{4913} a + \frac{78652069441856}{4913}$	$\bigl[a$ , $-1$ , $a + 1$ , $33 a - 53$ , $112 a - 163\bigr]$	${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(33a-53\right){x}+112a-163$
28.1-a1	28.1-a	$8$	$12$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	28.1	$2^{2} \cdot 7$	$- 2^{8} \cdot 7^{12}$	$0.58140$	$(a), (-2a+1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.2	$1$	$2^{2} \cdot 3$	$1$	$2.155441053$	0.571547619	$-\frac{29518306565684}{13841287201} a + \frac{41622722395132}{13841287201}$	$\bigl[a$ , $-a + 1$ , $0$ , $-9 a - 18$ , $-320 a - 467\bigr]$	${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-9a-18\right){x}-320a-467$
28.1-a2	28.1-a	$8$	$12$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	28.1	$2^{2} \cdot 7$	$- 2^{8} \cdot 7^{4}$	$0.58140$	$(a), (-2a+1)$	0	$\Z/12\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.1	$1$	$2^{2} \cdot 3$	$1$	$19.39896948$	0.571547619	$-\frac{861093316}{2401} a + \frac{1217791012}{2401}$	$\bigl[a$ , $-a + 1$ , $0$ , $a + 2$ , $12 a + 17\bigr]$	${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(a+2\right){x}+12a+17$
28.1-a3	28.1-a	$8$	$12$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	28.1	$2^{2} \cdot 7$	$2^{8} \cdot 7^{3}$	$0.58140$	$(a), (-2a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.2	$1$	$3$	$1$	$2.155441053$	0.571547619	$-\frac{91481168031853524}{343} a + \frac{129373908533024396}{343}$	$\bigl[a$ , $-a + 1$ , $0$ , $-19 a - 148$ , $318 a - 201\bigr]$	${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-19a-148\right){x}+318a-201$
28.1-a4	28.1-a	$8$	$12$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	28.1	$2^{2} \cdot 7$	$- 2^{8} \cdot 7$	$0.58140$	$(a), (-2a+1)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.1	$1$	$3$	$1$	$19.39896948$	0.571547619	$\frac{4096}{7} a + \frac{16384}{7}$	$\bigl[0$ , $-1$ , $0$ , $-2 a + 3$ , $0\bigr]$	${y}^2={x}^{3}-{x}^{2}+\left(-2a+3\right){x}$
28.1-a5	28.1-a	$8$	$12$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	28.1	$2^{2} \cdot 7$	$2^{4} \cdot 7^{2}$	$0.58140$	$(a), (-2a+1)$	0	$\Z/2\Z\oplus\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2Cs, 3B.1.1	$1$	$2 \cdot 3$	$1$	$38.79793896$	0.571547619	$\frac{435744}{49} a + \frac{712688}{49}$	$\bigl[a$ , $-1$ , $a$ , $2 a - 4$ , $-a + 1\bigr]$	${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(2a-4\right){x}-a+1$
28.1-a6	28.1-a	$8$	$12$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	28.1	$2^{2} \cdot 7$	$2^{4} \cdot 7^{6}$	$0.58140$	$(a), (-2a+1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2Cs, 3B.1.2	$1$	$2 \cdot 3$	$1$	$4.310882107$	0.571547619	$-\frac{1137747277344}{117649} a + \frac{1622386617968}{117649}$	$\bigl[a$ , $-a + 1$ , $0$ , $-34 a - 53$ , $-133 a - 203\bigr]$	${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-34a-53\right){x}-133a-203$
28.1-a7	28.1-a	$8$	$12$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	28.1	$2^{2} \cdot 7$	$2^{8} \cdot 7$	$0.58140$	$(a), (-2a+1)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.1	$1$	$3$	$1$	$19.39896948$	0.571547619	$\frac{1720664028}{7} a + \frac{2434028852}{7}$	$\bigl[a$ , $-1$ , $a$ , $17 a - 29$ , $49 a - 66\bigr]$	${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(17a-29\right){x}+49a-66$
28.1-a8	28.1-a	$8$	$12$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	28.1	$2^{2} \cdot 7$	$- 2^{8} \cdot 7^{3}$	$0.58140$	$(a), (-2a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.2	$1$	$3$	$1$	$2.155441053$	0.571547619	$\frac{1545435312128}{343} a + \frac{2185574023168}{343}$	$\bigl[0$ , $-1$ , $0$ , $18 a - 37$ , $68 a - 108\bigr]$	${y}^2={x}^{3}-{x}^{2}+\left(18a-37\right){x}+68a-108$
28.2-a1	28.2-a	$8$	$12$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	28.2	$2^{2} \cdot 7$	$- 2^{8} \cdot 7^{3}$	$0.58140$	$(a), (2a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.2	$1$	$3$	$1$	$2.155441053$	0.571547619	$-\frac{1545435312128}{343} a + \frac{2185574023168}{343}$	$\bigl[0$ , $-1$ , $0$ , $-18 a - 37$ , $-68 a - 108\bigr]$	${y}^2={x}^{3}-{x}^{2}+\left(-18a-37\right){x}-68a-108$
28.2-a2	28.2-a	$8$	$12$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	28.2	$2^{2} \cdot 7$	$- 2^{8} \cdot 7$	$0.58140$	$(a), (2a+1)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.1	$1$	$3$	$1$	$19.39896948$	0.571547619	$-\frac{4096}{7} a + \frac{16384}{7}$	$\bigl[0$ , $-1$ , $0$ , $2 a + 3$ , $0\bigr]$	${y}^2={x}^{3}-{x}^{2}+\left(2a+3\right){x}$
28.2-a3	28.2-a	$8$	$12$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	28.2	$2^{2} \cdot 7$	$2^{4} \cdot 7^{2}$	$0.58140$	$(a), (2a+1)$	0	$\Z/2\Z\oplus\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2Cs, 3B.1.1	$1$	$2 \cdot 3$	$1$	$38.79793896$	0.571547619	$-\frac{435744}{49} a + \frac{712688}{49}$	$\bigl[a$ , $-1$ , $a$ , $-2 a - 4$ , $a + 1\bigr]$	${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-2a-4\right){x}+a+1$
28.2-a4	28.2-a	$8$	$12$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	28.2	$2^{2} \cdot 7$	$- 2^{8} \cdot 7^{12}$	$0.58140$	$(a), (2a+1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.2	$1$	$2^{2} \cdot 3$	$1$	$2.155441053$	0.571547619	$\frac{29518306565684}{13841287201} a + \frac{41622722395132}{13841287201}$	$\bigl[a$ , $a + 1$ , $0$ , $9 a - 18$ , $320 a - 467\bigr]$	${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(9a-18\right){x}+320a-467$
28.2-a5	28.2-a	$8$	$12$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	28.2	$2^{2} \cdot 7$	$2^{8} \cdot 7$	$0.58140$	$(a), (2a+1)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.1	$1$	$3$	$1$	$19.39896948$	0.571547619	$-\frac{1720664028}{7} a + \frac{2434028852}{7}$	$\bigl[a$ , $-1$ , $a$ , $-17 a - 29$ , $-49 a - 66\bigr]$	${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-17a-29\right){x}-49a-66$
28.2-a6	28.2-a	$8$	$12$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	28.2	$2^{2} \cdot 7$	$- 2^{8} \cdot 7^{4}$	$0.58140$	$(a), (2a+1)$	0	$\Z/12\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.1	$1$	$2^{2} \cdot 3$	$1$	$19.39896948$	0.571547619	$\frac{861093316}{2401} a + \frac{1217791012}{2401}$	$\bigl[a$ , $a + 1$ , $0$ , $-a + 2$ , $-12 a + 17\bigr]$	${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-a+2\right){x}-12a+17$
28.2-a7	28.2-a	$8$	$12$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	28.2	$2^{2} \cdot 7$	$2^{4} \cdot 7^{6}$	$0.58140$	$(a), (2a+1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2Cs, 3B.1.2	$1$	$2 \cdot 3$	$1$	$4.310882107$	0.571547619	$\frac{1137747277344}{117649} a + \frac{1622386617968}{117649}$	$\bigl[a$ , $a + 1$ , $0$ , $34 a - 53$ , $133 a - 203\bigr]$	${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(34a-53\right){x}+133a-203$
28.2-a8	28.2-a	$8$	$12$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	28.2	$2^{2} \cdot 7$	$2^{8} \cdot 7^{3}$	$0.58140$	$(a), (2a+1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$					$2, 3$	2B, 3B.1.2	$1$	$3$	$1$	$2.155441053$	0.571547619	$\frac{91481168031853524}{343} a + \frac{129373908533024396}{343}$	$\bigl[a$ , $a + 1$ , $0$ , $19 a - 148$ , $-318 a - 201\bigr]$	${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(19a-148\right){x}-318a-201$
31.1-a1	31.1-a	$4$	$4$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	31.1	$31$	$- 31^{4}$	$0.59638$	$(-4a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	$2$	$1$	$2.544454463$	0.449800251	$-\frac{78588372777605}{923521} a + \frac{111138046783591}{923521}$	$\bigl[a + 1$ , $a - 1$ , $a + 1$ , $-7 a - 21$ , $-31 a - 52\bigr]$	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-7a-21\right){x}-31a-52$
31.1-a2	31.1-a	$4$	$4$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	31.1	$31$	$-31$	$0.59638$	$(-4a-1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	$1$	$1$	$20.35563570$	0.449800251	$\frac{47780}{31} a + \frac{69151}{31}$	$\bigl[a + 1$ , $a - 1$ , $a + 1$ , $-2 a - 1$ , $-a - 2\bigr]$	${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2a-1\right){x}-a-2$
31.1-a3	31.1-a	$4$	$4$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	31.1	$31$	$31^{2}$	$0.59638$	$(-4a-1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2Cs	$1$	$2$	$1$	$10.17781785$	0.449800251	$\frac{4423034250}{961} a + \frac{6270751283}{961}$	$\bigl[1$ , $-a - 1$ , $1$ , $10 a - 14$ , $21 a - 30\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(10a-14\right){x}+21a-30$
31.1-a4	31.1-a	$4$	$4$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	31.1	$31$	$31$	$0.59638$	$(-4a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	$1$	$1$	$5.088908926$	0.449800251	$\frac{1861812264958875}{31} a + \frac{2633000155833143}{31}$	$\bigl[1$ , $-a - 1$ , $1$ , $25 a - 49$ , $-79 a + 100\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(25a-49\right){x}-79a+100$
31.2-a1	31.2-a	$4$	$4$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	31.2	$31$	$-31$	$0.59638$	$(4a-1)$	0	$\Z/4\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	$1$	$1$	$20.35563570$	0.449800251	$-\frac{47780}{31} a + \frac{69151}{31}$	$\bigl[a + 1$ , $a - 1$ , $1$ , $0$ , $0\bigr]$	${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}$
31.2-a2	31.2-a	$4$	$4$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	31.2	$31$	$31$	$0.59638$	$(4a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	$1$	$1$	$5.088908926$	0.449800251	$-\frac{1861812264958875}{31} a + \frac{2633000155833143}{31}$	$\bigl[1$ , $a - 1$ , $1$ , $-25 a - 49$ , $79 a + 100\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-25a-49\right){x}+79a+100$
31.2-a3	31.2-a	$4$	$4$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	31.2	$31$	$31^{2}$	$0.59638$	$(4a-1)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2Cs	$1$	$2$	$1$	$10.17781785$	0.449800251	$-\frac{4423034250}{961} a + \frac{6270751283}{961}$	$\bigl[1$ , $a - 1$ , $1$ , $-10 a - 14$ , $-21 a - 30\bigr]$	${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-10a-14\right){x}-21a-30$
31.2-a4	31.2-a	$4$	$4$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	31.2	$31$	$- 31^{4}$	$0.59638$	$(4a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	$2$	$1$	$2.544454463$	0.449800251	$\frac{78588372777605}{923521} a + \frac{111138046783591}{923521}$	$\bigl[a + 1$ , $a - 1$ , $1$ , $5 a - 20$ , $10 a - 40\bigr]$	${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(5a-20\right){x}+10a-40$
32.1-a1	32.1-a	$8$	$16$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	32.1	$2^{5}$	$- 2^{9}$	$0.60113$	$(a)$	0	$\Z/2\Z$	$\textsf{potential}$	$-64$	$N(\mathrm{U}(1))$	✓			✓			$1$	$2$	$1$	$3.437592909$	0.607686314	$-29071392966 a + 41113158120$	$\bigl[a$ , $1$ , $0$ , $15 a - 22$ , $46 a - 69\bigr]$	${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(15a-22\right){x}+46a-69$
32.1-a2	32.1-a	$8$	$16$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	32.1	$2^{5}$	$- 2^{9}$	$0.60113$	$(a)$	0	$\Z/4\Z$	$\textsf{potential}$	$-64$	$N(\mathrm{U}(1))$	✓			✓			$1$	$1$	$1$	$27.50074327$	0.607686314	$-29071392966 a + 41113158120$	$\bigl[a$ , $1$ , $a$ , $15 a - 23$ , $-31 a + 46\bigr]$	${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(15a-23\right){x}-31a+46$
32.1-a3	32.1-a	$8$	$16$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	32.1	$2^{5}$	$2^{12}$	$0.60113$	$(a)$	0	$\Z/4\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓	✓		✓			$1$	$2$	$1$	$13.75037163$	0.607686314	$1728$	$\bigl[0$ , $0$ , $0$ , $1$ , $0\bigr]$	${y}^2={x}^{3}+{x}$
32.1-a4	32.1-a	$8$	$16$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	32.1	$2^{5}$	$2^{12}$	$0.60113$	$(a)$	0	$\Z/2\Z\oplus\Z/4\Z$	$\textsf{potential}$	$-4$	$N(\mathrm{U}(1))$	✓	✓		✓	$2$	2Cs	$1$	$2^{2}$	$1$	$27.50074327$	0.607686314	$1728$	$\bigl[0$ , $0$ , $0$ , $-1$ , $0\bigr]$	${y}^2={x}^{3}-{x}$
32.1-a5	32.1-a	$8$	$16$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	32.1	$2^{5}$	$2^{6}$	$0.60113$	$(a)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{potential}$	$-16$	$N(\mathrm{U}(1))$	✓	✓		✓	$2$	2Cs	$1$	$2$	$1$	$13.75037163$	0.607686314	$287496$	$\bigl[a$ , $1$ , $0$ , $-2$ , $-3\bigr]$	${y}^2+a{x}{y}={x}^{3}+{x}^{2}-2{x}-3$
32.1-a6	32.1-a	$8$	$16$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	32.1	$2^{5}$	$2^{6}$	$0.60113$	$(a)$	0	$\Z/2\Z\oplus\Z/4\Z$	$\textsf{potential}$	$-16$	$N(\mathrm{U}(1))$	✓	✓		✓	$2$	2Cs	$1$	$2$	$1$	$55.00148654$	0.607686314	$287496$	$\bigl[a$ , $1$ , $a$ , $-3$ , $0\bigr]$	${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-3{x}$
32.1-a7	32.1-a	$8$	$16$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	32.1	$2^{5}$	$- 2^{9}$	$0.60113$	$(a)$	0	$\Z/2\Z$	$\textsf{potential}$	$-64$	$N(\mathrm{U}(1))$	✓			✓			$1$	$2$	$1$	$3.437592909$	0.607686314	$29071392966 a + 41113158120$	$\bigl[a$ , $1$ , $0$ , $-15 a - 22$ , $-46 a - 69\bigr]$	${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-15a-22\right){x}-46a-69$
32.1-a8	32.1-a	$8$	$16$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	32.1	$2^{5}$	$- 2^{9}$	$0.60113$	$(a)$	0	$\Z/4\Z$	$\textsf{potential}$	$-64$	$N(\mathrm{U}(1))$	✓			✓			$1$	$1$	$1$	$27.50074327$	0.607686314	$29071392966 a + 41113158120$	$\bigl[a$ , $1$ , $a$ , $-15 a - 23$ , $31 a + 46\bigr]$	${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-15a-23\right){x}+31a+46$
34.1-a1	34.1-a	$4$	$6$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	34.1	$2 \cdot 17$	$2^{18} \cdot 17^{3}$	$0.61031$	$(a), (-3a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.2	$1$	$2$	$1$	$2.790055734$	0.493216832	$-\frac{9756993259}{1257728} a - \frac{25455932221}{2515456}$	$\bigl[a + 1$ , $a - 1$ , $a$ , $-16 a + 19$ , $-25 a + 31\bigr]$	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-16a+19\right){x}-25a+31$
34.1-a2	34.1-a	$4$	$6$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	34.1	$2 \cdot 17$	$2^{6} \cdot 17$	$0.61031$	$(a), (-3a-1)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.1	$1$	$2$	$1$	$25.11050160$	0.493216832	$\frac{2939047}{68} a - \frac{8089117}{136}$	$\bigl[1$ , $a - 1$ , $0$ , $2 a + 4$ , $0\bigr]$	${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(2a+4\right){x}$
34.1-a3	34.1-a	$4$	$6$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	34.1	$2 \cdot 17$	$2^{3} \cdot 17^{2}$	$0.61031$	$(a), (-3a-1)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.1	$1$	$2$	$1$	$25.11050160$	0.493216832	$-\frac{6018090689657}{1156} a + \frac{4255433785057}{578}$	$\bigl[1$ , $a - 1$ , $0$ , $-8 a - 16$ , $-10 a - 4\bigr]$	${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-8a-16\right){x}-10a-4$
34.1-a4	34.1-a	$4$	$6$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	34.1	$2 \cdot 17$	$2^{9} \cdot 17^{6}$	$0.61031$	$(a), (-3a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.2	$1$	$2$	$1$	$2.790055734$	0.493216832	$\frac{130087595511310753}{772402208} a + \frac{91987087285468997}{386201104}$	$\bigl[a + 1$ , $a - 1$ , $a$ , $64 a - 141$ , $-457 a + 479\bigr]$	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(64a-141\right){x}-457a+479$
34.2-a1	34.2-a	$4$	$6$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	34.2	$2 \cdot 17$	$2^{6} \cdot 17$	$0.61031$	$(a), (3a-1)$	0	$\Z/6\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.1	$1$	$2$	$1$	$25.11050160$	0.493216832	$-\frac{2939047}{68} a - \frac{8089117}{136}$	$\bigl[1$ , $-a - 1$ , $0$ , $-2 a + 4$ , $0\bigr]$	${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2a+4\right){x}$
34.2-a2	34.2-a	$4$	$6$	$\Q(\sqrt{2})$	$2$	$[2, 0]$	34.2	$2 \cdot 17$	$2^{18} \cdot 17^{3}$	$0.61031$	$(a), (3a-1)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2, 3$	2B, 3B.1.2	$1$	$2$	$1$	$2.790055734$	0.493216832	$\frac{9756993259}{1257728} a - \frac{25455932221}{2515456}$	$\bigl[a + 1$ , $a - 1$ , $0$ , $15 a + 20$ , $44 a + 62\bigr]$	${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(15a+20\right){x}+44a+62$

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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.