| 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 |
| 81.1-a1 |
81.1-a |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{12} \) |
$5.19129$ |
$(-a^3+a^2+3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3, 5$ |
3B.1.2, 5B.4.1[2] |
$1$ |
\( 2 \) |
$1$ |
$62.01676032$ |
0.924491278 |
\( -16554983445 a^{3} + 49664950335 a + 10231546590 \) |
\( \bigl[a^{2} + a - 1\) , \( 0\) , \( a\) , \( -5 a^{3} - 23 a^{2} - 28 a - 6\) , \( 264 a^{3} + 140 a^{2} - 825 a - 176\bigr] \) |
${y}^2+\left(a^{2}+a-1\right){x}{y}+a{y}={x}^{3}+\left(-5a^{3}-23a^{2}-28a-6\right){x}+264a^{3}+140a^{2}-825a-176$ |
| 81.1-a2 |
81.1-a |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{12} \) |
$5.19129$ |
$(-a^3+a^2+3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3, 5$ |
3B.1.2, 5B.4.1[2] |
$1$ |
\( 2 \) |
$1$ |
$62.01676032$ |
0.924491278 |
\( 85995 a^{3} - 257985 a - 52515 \) |
\( \bigl[a^{2} + a - 1\) , \( 0\) , \( a\) , \( 10 a^{3} + 7 a^{2} - 28 a - 6\) , \( 30 a^{3} + 23 a^{2} - 78 a - 17\bigr] \) |
${y}^2+\left(a^{2}+a-1\right){x}{y}+a{y}={x}^{3}+\left(10a^{3}+7a^{2}-28a-6\right){x}+30a^{3}+23a^{2}-78a-17$ |
| 81.1-a3 |
81.1-a |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{12} \) |
$5.19129$ |
$(-a^3+a^2+3a-2)$ |
0 |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3, 5$ |
3B.1.1, 5B.4.1[2] |
$1$ |
\( 2 \) |
$1$ |
$558.1508429$ |
0.924491278 |
\( -16554983445 a^{3} + 49664950335 a + 10231546590 \) |
\( \bigl[a^{3} - 3 a + 1\) , \( -a^{3} + 3 a + 1\) , \( 1\) , \( -14 a^{3} + 42 a - 26\) , \( -32 a^{3} + 96 a - 51\bigr] \) |
${y}^2+\left(a^{3}-3a+1\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+3a+1\right){x}^{2}+\left(-14a^{3}+42a-26\right){x}-32a^{3}+96a-51$ |
| 81.1-a4 |
81.1-a |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{12} \) |
$5.19129$ |
$(-a^3+a^2+3a-2)$ |
0 |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3, 5$ |
3B.1.1, 5B.4.1[2] |
$1$ |
\( 2 \) |
$1$ |
$558.1508429$ |
0.924491278 |
\( 85995 a^{3} - 257985 a - 52515 \) |
\( \bigl[a^{3} - 3 a + 1\) , \( -a^{3} + 3 a + 1\) , \( 1\) , \( a^{3} - 3 a + 4\) , \( -2 a^{3} + 6 a - 3\bigr] \) |
${y}^2+\left(a^{3}-3a+1\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+3a+1\right){x}^{2}+\left(a^{3}-3a+4\right){x}-2a^{3}+6a-3$ |
| 81.1-a5 |
81.1-a |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{12} \) |
$5.19129$ |
$(-a^3+a^2+3a-2)$ |
0 |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3, 5$ |
3B.1.1, 5B.4.1[2] |
$1$ |
\( 2 \) |
$1$ |
$558.1508429$ |
0.924491278 |
\( 16554983445 a^{3} - 49664950335 a + 26786530035 \) |
\( \bigl[1\) , \( -1\) , \( a^{3} - 3 a + 1\) , \( -17 a^{3} + 51 a - 32\) , \( 52 a^{3} - 156 a + 82\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}-{x}^{2}+\left(-17a^{3}+51a-32\right){x}+52a^{3}-156a+82$ |
| 81.1-a6 |
81.1-a |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{12} \) |
$5.19129$ |
$(-a^3+a^2+3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3, 5$ |
3B.1.2, 5B.4.1[2] |
$1$ |
\( 2 \) |
$1$ |
$62.01676032$ |
0.924491278 |
\( -85995 a^{3} + 257985 a - 138510 \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 2\) , \( a^{2} - 1\) , \( -161 a^{3} - 131 a^{2} + 401 a + 86\) , \( -1640 a^{3} - 1355 a^{2} + 4084 a + 893\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(-161a^{3}-131a^{2}+401a+86\right){x}-1640a^{3}-1355a^{2}+4084a+893$ |
| 81.1-a7 |
81.1-a |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{12} \) |
$5.19129$ |
$(-a^3+a^2+3a-2)$ |
0 |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3, 5$ |
3B.1.1, 5B.4.1[2] |
$1$ |
\( 2 \) |
$1$ |
$558.1508429$ |
0.924491278 |
\( -85995 a^{3} + 257985 a - 138510 \) |
\( \bigl[1\) , \( -1\) , \( a^{3} - 3 a + 1\) , \( -2 a^{3} + 6 a - 2\) , \( a^{3} - 3 a + 1\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}-{x}^{2}+\left(-2a^{3}+6a-2\right){x}+a^{3}-3a+1$ |
| 81.1-a8 |
81.1-a |
$8$ |
$30$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{12} \) |
$5.19129$ |
$(-a^3+a^2+3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3, 5$ |
3B.1.2, 5B.4.1[2] |
$1$ |
\( 2 \) |
$1$ |
$62.01676032$ |
0.924491278 |
\( 16554983445 a^{3} - 49664950335 a + 26786530035 \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 2\) , \( a^{2} - 1\) , \( -2561 a^{3} - 2096 a^{2} + 6386 a + 1346\) , \( -98327 a^{3} - 81248 a^{2} + 244768 a + 53612\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(-2561a^{3}-2096a^{2}+6386a+1346\right){x}-98327a^{3}-81248a^{2}+244768a+53612$ |
*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.
