| 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 |
| 25.1-b1 |
25.1-b |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{2} \) |
$4.48185$ |
$(-a-1)$ |
0 |
$\Z/5\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5Cs.1.1[2] |
$1$ |
\( 1 \) |
$1$ |
$603.3662684$ |
0.719556262 |
\( 0 \) |
\( \bigl[0\) , \( -a^{3} + a^{2} + 3 a - 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -a + 2\) , \( -a^{2} - a + 3\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-2\right){x}^{2}+\left(-a+2\right){x}-a^{2}-a+3$ |
| 25.1-b2 |
25.1-b |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{2} \) |
$4.48185$ |
$(-a-1)$ |
0 |
$\Z/5\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5Cs.1.1[2] |
$1$ |
\( 1 \) |
$1$ |
$603.3662684$ |
0.719556262 |
\( 0 \) |
\( \bigl[0\) , \( a^{3} - 2 a + 1\) , \( a^{2} - 1\) , \( a^{3} + a^{2} - 3 a\) , \( -11017 a^{3} - 9112 a^{2} + 27421 a + 6030\bigr] \) |
${y}^2+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-2a+1\right){x}^{2}+\left(a^{3}+a^{2}-3a\right){x}-11017a^{3}-9112a^{2}+27421a+6030$ |
| 841.10-b1 |
841.10-b |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
841.10 |
\( 29^{2} \) |
\( 29^{10} \) |
$6.95528$ |
$(-a^3+a^2+4a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$47.68821090$ |
1.421787750 |
\( 0 \) |
\( \bigl[0\) , \( a^{2} + a - 1\) , \( a^{3} - 3 a\) , \( a^{3} + a^{2} - 2 a\) , \( 216805 a^{3} + 179876 a^{2} - 538381 a - 118433\bigr] \) |
${y}^2+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(a^{3}+a^{2}-2a\right){x}+216805a^{3}+179876a^{2}-538381a-118433$ |
| 841.10-b2 |
841.10-b |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
841.10 |
\( 29^{2} \) |
\( 29^{10} \) |
$6.95528$ |
$(-a^3+a^2+4a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$47.68821090$ |
1.421787750 |
\( 0 \) |
\( \bigl[0\) , \( -a^{3} - a^{2} + 4 a + 3\) , \( a^{2} + a - 1\) , \( -2 a^{3} - a^{2} + 7 a + 3\) , \( 32 a^{3} + a^{2} - 116 a + 9\bigr] \) |
${y}^2+\left(a^{2}+a-1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+4a+3\right){x}^{2}+\left(-2a^{3}-a^{2}+7a+3\right){x}+32a^{3}+a^{2}-116a+9$ |
| 841.7-b1 |
841.7-b |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
841.7 |
\( 29^{2} \) |
\( 29^{10} \) |
$6.95528$ |
$(-a^3-a^2+2a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$47.68821090$ |
1.421787750 |
\( 0 \) |
\( \bigl[0\) , \( a^{3} + a^{2} - 4 a - 3\) , \( a^{3} - 2 a + 1\) , \( -2 a^{3} - a^{2} + 7 a + 3\) , \( 18371836 a^{3} + 15195215 a^{2} - 45724292 a - 10055225\bigr] \) |
${y}^2+\left(a^{3}-2a+1\right){y}={x}^{3}+\left(a^{3}+a^{2}-4a-3\right){x}^{2}+\left(-2a^{3}-a^{2}+7a+3\right){x}+18371836a^{3}+15195215a^{2}-45724292a-10055225$ |
| 841.7-b2 |
841.7-b |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
841.7 |
\( 29^{2} \) |
\( 29^{10} \) |
$6.95528$ |
$(-a^3-a^2+2a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$47.68821090$ |
1.421787750 |
\( 0 \) |
\( \bigl[0\) , \( a^{3} - 2 a + 1\) , \( a\) , \( a^{3} + a^{2} - 3 a\) , \( a^{3} - 3 a^{2} - 6 a + 6\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a^{3}-2a+1\right){x}^{2}+\left(a^{3}+a^{2}-3a\right){x}+a^{3}-3a^{2}-6a+6$ |
| 841.8-b1 |
841.8-b |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
841.8 |
\( 29^{2} \) |
\( 29^{10} \) |
$6.95528$ |
$(2a^3+a^2-7a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$47.68821090$ |
1.421787750 |
\( 0 \) |
\( \bigl[0\) , \( -a^{3} + a^{2} + 3 a - 2\) , \( a^{2} - 2\) , \( -a + 2\) , \( 133196725 a^{3} + 110165536 a^{2} - 331504541 a - 72901094\bigr] \) |
${y}^2+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-2\right){x}^{2}+\left(-a+2\right){x}+133196725a^{3}+110165536a^{2}-331504541a-72901094$ |
| 841.8-b2 |
841.8-b |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
841.8 |
\( 29^{2} \) |
\( 29^{10} \) |
$6.95528$ |
$(2a^3+a^2-7a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$47.68821090$ |
1.421787750 |
\( 0 \) |
\( \bigl[0\) , \( a^{2} + a - 1\) , \( a^{3} + a^{2} - 2 a - 2\) , \( a^{3} + a^{2} - 2 a\) , \( -32 a^{3} - 20 a^{2} + 94 a + 20\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(a^{3}+a^{2}-2a\right){x}-32a^{3}-20a^{2}+94a+20$ |
| 841.9-b1 |
841.9-b |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
841.9 |
\( 29^{2} \) |
\( 29^{10} \) |
$6.95528$ |
$(a^2-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$47.68821090$ |
1.421787750 |
\( 0 \) |
\( \bigl[0\) , \( -a^{2} - a + 1\) , \( a^{3} + a^{2} - 2 a - 1\) , \( a^{3} + a^{2} - 2 a\) , \( -16 a^{3} + 18 a^{2} + 49 a - 60\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(a^{3}+a^{2}-2a\right){x}-16a^{3}+18a^{2}+49a-60$ |
| 841.9-b2 |
841.9-b |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
841.9 |
\( 29^{2} \) |
\( 29^{10} \) |
$6.95528$ |
$(a^2-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$47.68821090$ |
1.421787750 |
\( 0 \) |
\( \bigl[0\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 2 a - 2\) , \( -a + 2\) , \( -815183 a^{3} - 674230 a^{2} + 2028854 a + 446165\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(-a+2\right){x}-815183a^{3}-674230a^{2}+2028854a+446165$ |
| 961.10-a1 |
961.10-a |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.10 |
\( 31^{2} \) |
\( 31^{2} \) |
$7.07221$ |
$(-2a^3-2a^2+6a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.084970583$ |
$257.0130160$ |
2.604398559 |
\( 0 \) |
\( \bigl[0\) , \( -a^{2} - a + 1\) , \( a^{2} + a - 1\) , \( a^{3} + a^{2} - 2 a\) , \( -a^{3} - a^{2} + 2 a\bigr] \) |
${y}^2+\left(a^{2}+a-1\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(a^{3}+a^{2}-2a\right){x}-a^{3}-a^{2}+2a$ |
| 961.10-a2 |
961.10-a |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.10 |
\( 31^{2} \) |
\( 31^{2} \) |
$7.07221$ |
$(-2a^3-2a^2+6a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.028323527$ |
$771.0390481$ |
2.604398559 |
\( 0 \) |
\( \bigl[0\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - 3 a\) , \( -a + 2\) , \( -2365813 a^{3} - 1956743 a^{2} + 5888106 a + 1294853\bigr] \) |
${y}^2+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(-a+2\right){x}-2365813a^{3}-1956743a^{2}+5888106a+1294853$ |
| 961.7-a1 |
961.7-a |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.7 |
\( 31^{2} \) |
\( 31^{2} \) |
$7.07221$ |
$(a^3+2a^2-3a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.084970583$ |
$257.0130160$ |
2.604398559 |
\( 0 \) |
\( \bigl[0\) , \( a^{3} + a^{2} - 4 a - 3\) , \( a^{3} + a^{2} - 3 a - 2\) , \( -2 a^{3} - a^{2} + 7 a + 3\) , \( -7541548 a^{3} - 6237551 a^{2} + 18769617 a + 4127623\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){y}={x}^{3}+\left(a^{3}+a^{2}-4a-3\right){x}^{2}+\left(-2a^{3}-a^{2}+7a+3\right){x}-7541548a^{3}-6237551a^{2}+18769617a+4127623$ |
| 961.7-a2 |
961.7-a |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.7 |
\( 31^{2} \) |
\( 31^{2} \) |
$7.07221$ |
$(a^3+2a^2-3a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.028323527$ |
$771.0390481$ |
2.604398559 |
\( 0 \) |
\( \bigl[0\) , \( a^{3} - 2 a + 1\) , \( a^{3} - 3 a + 1\) , \( a^{3} + a^{2} - 3 a\) , \( a^{3} + a^{2} - 2 a - 1\bigr] \) |
${y}^2+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(a^{3}-2a+1\right){x}^{2}+\left(a^{3}+a^{2}-3a\right){x}+a^{3}+a^{2}-2a-1$ |
| 961.8-a1 |
961.8-a |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.8 |
\( 31^{2} \) |
\( 31^{2} \) |
$7.07221$ |
$(-a^3+5a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.028323527$ |
$771.0390481$ |
2.604398559 |
\( 0 \) |
\( \bigl[0\) , \( -a^{3} + a^{2} + 3 a - 2\) , \( a^{3} - 3 a\) , \( -a + 2\) , \( a^{2} - a - 1\bigr] \) |
${y}^2+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-2\right){x}^{2}+\left(-a+2\right){x}+a^{2}-a-1$ |
| 961.8-a2 |
961.8-a |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.8 |
\( 31^{2} \) |
\( 31^{2} \) |
$7.07221$ |
$(-a^3+5a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.084970583$ |
$257.0130160$ |
2.604398559 |
\( 0 \) |
\( \bigl[0\) , \( a^{3} - 2 a + 1\) , \( a^{2} + a - 2\) , \( a^{3} + a^{2} - 3 a\) , \( -1139511 a^{3} - 942479 a^{2} + 2836049 a + 623674\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{3}-2a+1\right){x}^{2}+\left(a^{3}+a^{2}-3a\right){x}-1139511a^{3}-942479a^{2}+2836049a+623674$ |
| 961.9-a1 |
961.9-a |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.9 |
\( 31^{2} \) |
\( 31^{2} \) |
$7.07221$ |
$(2a^3-8a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.028323527$ |
$771.0390481$ |
2.604398559 |
\( 0 \) |
\( \bigl[0\) , \( -a^{3} + 2 a - 1\) , \( a^{3} - 3 a + 1\) , \( a^{3} + a^{2} - 3 a\) , \( -a^{2} - a + 3\bigr] \) |
${y}^2+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(-a^{3}+2a-1\right){x}^{2}+\left(a^{3}+a^{2}-3a\right){x}-a^{2}-a+3$ |
| 961.9-a2 |
961.9-a |
$2$ |
$3$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
961.9 |
\( 31^{2} \) |
\( 31^{2} \) |
$7.07221$ |
$(2a^3-8a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.084970583$ |
$257.0130160$ |
2.604398559 |
\( 0 \) |
\( \bigl[0\) , \( -a^{3} - a^{2} + 4 a + 3\) , \( a^{3} + a^{2} - 3 a - 2\) , \( -2 a^{3} - a^{2} + 7 a + 3\) , \( -6908 a^{3} - 5713 a^{2} + 17194 a + 3781\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){y}={x}^{3}+\left(-a^{3}-a^{2}+4a+3\right){x}^{2}+\left(-2a^{3}-a^{2}+7a+3\right){x}-6908a^{3}-5713a^{2}+17194a+3781$ |
*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.
