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 |
41.2-a1 |
41.2-a |
$2$ |
$7$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
41.2 |
\( 41 \) |
\( 41^{7} \) |
$0.50561$ |
$(a-7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.3 |
$1$ |
\( 1 \) |
$1$ |
$0.944099560$ |
0.422214159 |
\( -\frac{7215644871110656}{194754273881} a - \frac{4677283260284928}{194754273881} \) |
\( \bigl[0\) , \( \phi - 1\) , \( \phi + 1\) , \( -10 \phi - 30\) , \( -32 \phi - 82\bigr] \) |
${y}^2+\left(\phi+1\right){y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(-10\phi-30\right){x}-32\phi-82$ |
1025.2-a1 |
1025.2-a |
$2$ |
$7$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
1025.2 |
\( 5^{2} \cdot 41 \) |
\( 5^{6} \cdot 41^{7} \) |
$1.13059$ |
$(-2a+1), (a-7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B.6.3 |
$1$ |
\( 1 \) |
$1$ |
$3.626679053$ |
1.621900179 |
\( -\frac{7215644871110656}{194754273881} a - \frac{4677283260284928}{194754273881} \) |
\( \bigl[0\) , \( 1\) , \( \phi + 1\) , \( 105 \phi - 258\) , \( -1417 \phi + 2585\bigr] \) |
${y}^2+\left(\phi+1\right){y}={x}^{3}+{x}^{2}+\left(105\phi-258\right){x}-1417\phi+2585$ |
1681.2-b1 |
1681.2-b |
$2$ |
$7$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
1681.2 |
\( 41^{2} \) |
\( 41^{13} \) |
$1.27943$ |
$(a-7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B.6.3 |
$1$ |
\( 2 \) |
$1$ |
$1.266491262$ |
1.132784221 |
\( -\frac{7215644871110656}{194754273881} a - \frac{4677283260284928}{194754273881} \) |
\( \bigl[0\) , \( -\phi - 1\) , \( \phi + 1\) , \( 42 \phi - 1407\) , \( -5914 \phi + 29839\bigr] \) |
${y}^2+\left(\phi+1\right){y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(42\phi-1407\right){x}-5914\phi+29839$ |
3321.2-f1 |
3321.2-f |
$2$ |
$7$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3321.2 |
\( 3^{4} \cdot 41 \) |
\( 3^{12} \cdot 41^{7} \) |
$1.51684$ |
$(a-7), (3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B.6.3 |
$1$ |
\( 2 \cdot 7 \) |
$0.061854725$ |
$2.703166965$ |
2.093720884 |
\( -\frac{7215644871110656}{194754273881} a - \frac{4677283260284928}{194754273881} \) |
\( \bigl[0\) , \( 0\) , \( \phi + 1\) , \( -87 \phi - 276\) , \( 569 \phi + 2386\bigr] \) |
${y}^2+\left(\phi+1\right){y}={x}^{3}+\left(-87\phi-276\right){x}+569\phi+2386$ |
4961.3-e1 |
4961.3-e |
$2$ |
$7$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
4961.3 |
\( 11^{2} \cdot 41 \) |
\( 11^{6} \cdot 41^{7} \) |
$1.67693$ |
$(-3a+2), (a-7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B.6.3 |
$1$ |
\( 7 \) |
$1$ |
$0.676968964$ |
2.119248073 |
\( -\frac{7215644871110656}{194754273881} a - \frac{4677283260284928}{194754273881} \) |
\( \bigl[0\) , \( \phi - 1\) , \( \phi + 1\) , \( -5 \phi - 369\) , \( -3327 \phi + 2455\bigr] \) |
${y}^2+\left(\phi+1\right){y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(-5\phi-369\right){x}-3327\phi+2455$ |
4961.5-f1 |
4961.5-f |
$2$ |
$7$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
4961.5 |
\( 11^{2} \cdot 41 \) |
\( 11^{6} \cdot 41^{7} \) |
$1.67693$ |
$(-3a+1), (a-7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B.6.3 |
$1$ |
\( 2 \) |
$1.648191839$ |
$1.028135789$ |
3.031330060 |
\( -\frac{7215644871110656}{194754273881} a - \frac{4677283260284928}{194754273881} \) |
\( \bigl[0\) , \( \phi - 1\) , \( \phi + 1\) , \( -218 \phi - 335\) , \( 3808 \phi + 710\bigr] \) |
${y}^2+\left(\phi+1\right){y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(-218\phi-335\right){x}+3808\phi+710$ |
*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.