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Base field Conductor norm Rank* Torsion order CM discriminant
Base field degree/signature Bad \(p\) $\Q$-curves Torsion structure CM
Analytic order of Ш* Cyclic isogeny degree Isogeny class size Reduction Galois image
Base change of Regulator* Curves per isogeny class Isogeny class degree Nonmax$\ \ell$
j-invariant
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Results (10 matches)

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Label Class Base field Conductor norm Rank Torsion CM Sato-Tate Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
25.1-a1 25.1-a 4.4.8000.1 \( 5^{2} \) 0 $\Z/2\Z$ $-40$ $N(\mathrm{U}(1))$ $1$ $77.62503566$ 1.735748564 \( -95178240 a^{2} + 688737600 \) \( \bigl[\frac{1}{2} a^{3} - 2 a\) , \( \frac{1}{2} a^{2} - 2\) , \( \frac{1}{2} a^{3} - 2 a + 1\) , \( -\frac{33}{2} a^{3} + \frac{47}{2} a^{2} + 116 a - 182\) , \( -93 a^{3} + 138 a^{2} + 661 a - 1053\bigr] \) ${y}^2+\left(\frac{1}{2}a^{3}-2a\right){x}{y}+\left(\frac{1}{2}a^{3}-2a+1\right){y}={x}^{3}+\left(\frac{1}{2}a^{2}-2\right){x}^{2}+\left(-\frac{33}{2}a^{3}+\frac{47}{2}a^{2}+116a-182\right){x}-93a^{3}+138a^{2}+661a-1053$
25.1-a2 25.1-a 4.4.8000.1 \( 5^{2} \) 0 $\Z/10\Z$ $-40$ $N(\mathrm{U}(1))$ $1$ $1940.625891$ 1.735748564 \( -95178240 a^{2} + 688737600 \) \( \bigl[\frac{1}{2} a^{3} - 3 a\) , \( -\frac{1}{2} a^{2} + 2\) , \( \frac{1}{2} a^{3} - 2 a + 1\) , \( 16 a^{3} - 44 a^{2} - 41 a + 113\) , \( -\frac{259}{2} a^{3} + 347 a^{2} + 356 a - 955\bigr] \) ${y}^2+\left(\frac{1}{2}a^{3}-3a\right){x}{y}+\left(\frac{1}{2}a^{3}-2a+1\right){y}={x}^{3}+\left(-\frac{1}{2}a^{2}+2\right){x}^{2}+\left(16a^{3}-44a^{2}-41a+113\right){x}-\frac{259}{2}a^{3}+347a^{2}+356a-955$
121.10-a1 121.10-a 4.4.8000.1 \( 11^{2} \) $1$ $\Z/2\Z$ $-40$ $N(\mathrm{U}(1))$ $0.433025007$ $413.7428302$ 4.006161577 \( -95178240 a^{2} + 688737600 \) \( \bigl[\frac{1}{2} a^{3} - 3 a\) , \( -\frac{1}{2} a^{2} + 2\) , \( 1\) , \( \frac{19}{2} a^{3} + 6 a^{2} - 111 a - 157\) , \( 195 a^{3} + 381 a^{2} - 1164 a - 2092\bigr] \) ${y}^2+\left(\frac{1}{2}a^{3}-3a\right){x}{y}+{y}={x}^{3}+\left(-\frac{1}{2}a^{2}+2\right){x}^{2}+\left(\frac{19}{2}a^{3}+6a^{2}-111a-157\right){x}+195a^{3}+381a^{2}-1164a-2092$
121.10-a3 121.10-a 4.4.8000.1 \( 11^{2} \) $1$ $\Z/2\Z$ $-40$ $N(\mathrm{U}(1))$ $1.082562519$ $165.4971321$ 4.006161577 \( -95178240 a^{2} + 688737600 \) \( \bigl[a\) , \( \frac{1}{2} a^{2} - 3\) , \( 1\) , \( -2 a^{3} + 4 a^{2} + 17 a - 37\) , \( -6 a^{3} + 6 a^{2} + 51 a - 66\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}+\left(\frac{1}{2}a^{2}-3\right){x}^{2}+\left(-2a^{3}+4a^{2}+17a-37\right){x}-6a^{3}+6a^{2}+51a-66$
121.7-a1 121.7-a 4.4.8000.1 \( 11^{2} \) $1$ $\Z/2\Z$ $-40$ $N(\mathrm{U}(1))$ $0.433025007$ $413.7428302$ 4.006161577 \( -95178240 a^{2} + 688737600 \) \( \bigl[\frac{1}{2} a^{3} - 3 a\) , \( -\frac{1}{2} a^{2} + 2\) , \( 1\) , \( -10 a^{3} + 6 a^{2} + 114 a - 157\) , \( -195 a^{3} + 381 a^{2} + 1164 a - 2092\bigr] \) ${y}^2+\left(\frac{1}{2}a^{3}-3a\right){x}{y}+{y}={x}^{3}+\left(-\frac{1}{2}a^{2}+2\right){x}^{2}+\left(-10a^{3}+6a^{2}+114a-157\right){x}-195a^{3}+381a^{2}+1164a-2092$
121.7-a3 121.7-a 4.4.8000.1 \( 11^{2} \) $1$ $\Z/2\Z$ $-40$ $N(\mathrm{U}(1))$ $1.082562519$ $165.4971321$ 4.006161577 \( -95178240 a^{2} + 688737600 \) \( \bigl[a\) , \( \frac{1}{2} a^{2} - 3\) , \( 1\) , \( 2 a^{3} + 4 a^{2} - 18 a - 37\) , \( 6 a^{3} + 6 a^{2} - 51 a - 66\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}+\left(\frac{1}{2}a^{2}-3\right){x}^{2}+\left(2a^{3}+4a^{2}-18a-37\right){x}+6a^{3}+6a^{2}-51a-66$
121.8-a1 121.8-a 4.4.8000.1 \( 11^{2} \) $1$ $\Z/2\Z$ $-40$ $N(\mathrm{U}(1))$ $0.216512503$ $827.4856605$ 4.006161577 \( -95178240 a^{2} + 688737600 \) \( \bigl[\frac{1}{2} a^{3} - 3 a\) , \( -\frac{1}{2} a^{2} + 2\) , \( 1\) , \( \frac{9}{2} a^{3} + 7 a^{2} - 31 a - 57\) , \( -\frac{25}{2} a^{3} - 18 a^{2} + 87 a + 139\bigr] \) ${y}^2+\left(\frac{1}{2}a^{3}-3a\right){x}{y}+{y}={x}^{3}+\left(-\frac{1}{2}a^{2}+2\right){x}^{2}+\left(\frac{9}{2}a^{3}+7a^{2}-31a-57\right){x}-\frac{25}{2}a^{3}-18a^{2}+87a+139$
121.8-a3 121.8-a 4.4.8000.1 \( 11^{2} \) $1$ $\Z/2\Z$ $-40$ $N(\mathrm{U}(1))$ $2.165125039$ $82.74856605$ 4.006161577 \( -95178240 a^{2} + 688737600 \) \( \bigl[\frac{1}{2} a^{3} - 2 a\) , \( \frac{1}{2} a^{2} - 2\) , \( 1\) , \( -\frac{59}{2} a^{3} - \frac{75}{2} a^{2} + 121 a + 38\) , \( 87 a^{3} + 400 a^{2} - 72 a - 1386\bigr] \) ${y}^2+\left(\frac{1}{2}a^{3}-2a\right){x}{y}+{y}={x}^{3}+\left(\frac{1}{2}a^{2}-2\right){x}^{2}+\left(-\frac{59}{2}a^{3}-\frac{75}{2}a^{2}+121a+38\right){x}+87a^{3}+400a^{2}-72a-1386$
121.9-a1 121.9-a 4.4.8000.1 \( 11^{2} \) $1$ $\Z/2\Z$ $-40$ $N(\mathrm{U}(1))$ $0.216512503$ $827.4856605$ 4.006161577 \( -95178240 a^{2} + 688737600 \) \( \bigl[\frac{1}{2} a^{3} - 3 a\) , \( -\frac{1}{2} a^{2} + 2\) , \( 1\) , \( -5 a^{3} + 7 a^{2} + 34 a - 57\) , \( \frac{25}{2} a^{3} - 18 a^{2} - 87 a + 139\bigr] \) ${y}^2+\left(\frac{1}{2}a^{3}-3a\right){x}{y}+{y}={x}^{3}+\left(-\frac{1}{2}a^{2}+2\right){x}^{2}+\left(-5a^{3}+7a^{2}+34a-57\right){x}+\frac{25}{2}a^{3}-18a^{2}-87a+139$
121.9-a3 121.9-a 4.4.8000.1 \( 11^{2} \) $1$ $\Z/2\Z$ $-40$ $N(\mathrm{U}(1))$ $2.165125039$ $82.74856605$ 4.006161577 \( -95178240 a^{2} + 688737600 \) \( \bigl[\frac{1}{2} a^{3} - 2 a\) , \( \frac{1}{2} a^{2} - 2\) , \( 1\) , \( 29 a^{3} - \frac{75}{2} a^{2} - 119 a + 38\) , \( -87 a^{3} + 400 a^{2} + 72 a - 1386\bigr] \) ${y}^2+\left(\frac{1}{2}a^{3}-2a\right){x}{y}+{y}={x}^{3}+\left(\frac{1}{2}a^{2}-2\right){x}^{2}+\left(29a^{3}-\frac{75}{2}a^{2}-119a+38\right){x}-87a^{3}+400a^{2}+72a-1386$
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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.