| 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 |
$8$ |
$30$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{6} \) |
$0.59944$ |
$(3)$ |
0 |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$23.62521625$ |
0.586973216 |
\( -85995 a - 52515 \) |
\( \bigl[1\) , \( -1\) , \( \phi\) , \( -2 \phi\) , \( \phi\bigr] \) |
${y}^2+{x}{y}+\phi{y}={x}^{3}-{x}^{2}-2\phi{x}+\phi$ |
| 81.1-a2 |
81.1-a |
$8$ |
$30$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{18} \) |
$0.59944$ |
$(3)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.625024027$ |
0.586973216 |
\( -85995 a - 52515 \) |
\( \bigl[1\) , \( -1\) , \( \phi + 1\) , \( -14 \phi - 2\) , \( -21 \phi - 6\bigr] \) |
${y}^2+{x}{y}+\left(\phi+1\right){y}={x}^{3}-{x}^{2}+\left(-14\phi-2\right){x}-21\phi-6$ |
| 81.1-a3 |
81.1-a |
$8$ |
$30$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{6} \) |
$0.59944$ |
$(3)$ |
0 |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$23.62521625$ |
0.586973216 |
\( 85995 a - 138510 \) |
\( \bigl[1\) , \( -1\) , \( \phi + 1\) , \( \phi - 2\) , \( -2 \phi + 1\bigr] \) |
${y}^2+{x}{y}+\left(\phi+1\right){y}={x}^{3}-{x}^{2}+\left(\phi-2\right){x}-2\phi+1$ |
| 81.1-a4 |
81.1-a |
$8$ |
$30$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{18} \) |
$0.59944$ |
$(3)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-15$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.625024027$ |
0.586973216 |
\( 85995 a - 138510 \) |
\( \bigl[1\) , \( -1\) , \( \phi\) , \( 13 \phi - 15\) , \( 20 \phi - 26\bigr] \) |
${y}^2+{x}{y}+\phi{y}={x}^{3}-{x}^{2}+\left(13\phi-15\right){x}+20\phi-26$ |
| 81.1-a5 |
81.1-a |
$8$ |
$30$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{6} \) |
$0.59944$ |
$(3)$ |
0 |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$23.62521625$ |
0.586973216 |
\( -16554983445 a + 26786530035 \) |
\( \bigl[\phi\) , \( -\phi - 1\) , \( \phi + 1\) , \( -13 \phi - 14\) , \( -20 \phi - 6\bigr] \) |
${y}^2+\phi{x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(-13\phi-14\right){x}-20\phi-6$ |
| 81.1-a6 |
81.1-a |
$8$ |
$30$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{18} \) |
$0.59944$ |
$(3)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.625024027$ |
0.586973216 |
\( -16554983445 a + 26786530035 \) |
\( \bigl[\phi\) , \( -\phi - 1\) , \( \phi\) , \( -113 \phi - 125\) , \( 867 \phi + 384\bigr] \) |
${y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(-113\phi-125\right){x}+867\phi+384$ |
| 81.1-a7 |
81.1-a |
$8$ |
$30$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{6} \) |
$0.59944$ |
$(3)$ |
0 |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$23.62521625$ |
0.586973216 |
\( 16554983445 a + 10231546590 \) |
\( \bigl[\phi + 1\) , \( 1\) , \( 1\) , \( 13 \phi - 26\) , \( 32 \phi - 51\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(13\phi-26\right){x}+32\phi-51$ |
| 81.1-a8 |
81.1-a |
$8$ |
$30$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
81.1 |
\( 3^{4} \) |
\( 3^{18} \) |
$0.59944$ |
$(3)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-60$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.625024027$ |
0.586973216 |
\( 16554983445 a + 10231546590 \) |
\( \bigl[\phi + 1\) , \( 1\) , \( 0\) , \( 114 \phi - 237\) , \( -754 \phi + 1014\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}={x}^{3}+{x}^{2}+\left(114\phi-237\right){x}-754\phi+1014$ |
*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.
