| 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 |
| 41.1-a1 |
41.1-a |
$1$ |
$1$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
41.1 |
\( 41 \) |
\( 41^{4} \) |
$10.17615$ |
$(a^3-a^2-4a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.009312090$ |
$1120.903857$ |
2.332859573 |
\( -\frac{71073792}{1681} a^{3} + \frac{148373504}{1681} a^{2} + \frac{213286912}{1681} a - \frac{439844864}{1681} \) |
\( \bigl[0\) , \( -a^{2} + 2 a + 3\) , \( a^{2} - 4\) , \( -6 a^{3} - 2 a^{2} + 25 a + 22\) , \( 8 a^{3} + 7 a^{2} - 30 a - 33\bigr] \) |
${y}^2+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}+2a+3\right){x}^{2}+\left(-6a^{3}-2a^{2}+25a+22\right){x}+8a^{3}+7a^{2}-30a-33$ |
| 41.1-b1 |
41.1-b |
$1$ |
$1$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
41.1 |
\( 41 \) |
\( 41^{4} \) |
$10.17615$ |
$(a^3-a^2-4a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.485050994$ |
$21.35387477$ |
2.314926137 |
\( \frac{71073792}{1681} a^{3} - \frac{64847872}{1681} a^{2} - \frac{296812544}{1681} a - \frac{149258240}{1681} \) |
\( \bigl[0\) , \( -a^{3} + a^{2} + 4 a\) , \( a^{3} - a^{2} - 3 a\) , \( -9 a^{3} - 7 a^{2} + 37 a + 39\) , \( -31 a^{3} - 23 a^{2} + 126 a + 126\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a\right){x}^{2}+\left(-9a^{3}-7a^{2}+37a+39\right){x}-31a^{3}-23a^{2}+126a+126$ |
| 41.1-c1 |
41.1-c |
$2$ |
$7$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
41.1 |
\( 41 \) |
\( 41^{2} \) |
$10.17615$ |
$(a^3-a^2-4a-3)$ |
0 |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$7$ |
7B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$2140.068876$ |
1.220154235 |
\( \frac{176128}{41} a^{2} - \frac{176128}{41} a - \frac{815104}{41} \) |
\( \bigl[0\) , \( a^{2} - a - 4\) , \( a^{2} - a - 4\) , \( 0\) , \( 0\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}$ |
| 41.1-c2 |
41.1-c |
$2$ |
$7$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
41.1 |
\( 41 \) |
\( 41^{14} \) |
$10.17615$ |
$(a^3-a^2-4a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$7$ |
7B.1.3 |
$49$ |
\( 2 \) |
$1$ |
$0.891323980$ |
1.220154235 |
\( -\frac{7215644871110656}{194754273881} a^{2} + \frac{7215644871110656}{194754273881} a + \frac{16969651353047040}{194754273881} \) |
\( \bigl[0\) , \( a^{2} - a - 4\) , \( a^{2} - a - 4\) , \( -10 a^{2} + 10 a\) , \( -31 a^{2} + 31 a + 11\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-10a^{2}+10a\right){x}-31a^{2}+31a+11$ |
| 41.1-d1 |
41.1-d |
$2$ |
$7$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
41.1 |
\( 41 \) |
\( 41^{14} \) |
$10.17615$ |
$(a^3-a^2-4a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$7$ |
7B.6.3 |
$1$ |
\( 2 \) |
$1$ |
$65.76400479$ |
1.837262932 |
\( -\frac{7215644871110656}{194754273881} a^{2} + \frac{7215644871110656}{194754273881} a + \frac{16969651353047040}{194754273881} \) |
\( \bigl[0\) , \( -a^{3} + 2 a^{2} + 4 a - 4\) , \( a^{3} - 4 a - 3\) , \( 144 a^{3} - 153 a^{2} - 559 a - 225\) , \( -1890 a^{3} + 1126 a^{2} + 8698 a + 5670\bigr] \) |
${y}^2+\left(a^{3}-4a-3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-4\right){x}^{2}+\left(144a^{3}-153a^{2}-559a-225\right){x}-1890a^{3}+1126a^{2}+8698a+5670$ |
| 41.1-d2 |
41.1-d |
$2$ |
$7$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
41.1 |
\( 41 \) |
\( 41^{2} \) |
$10.17615$ |
$(a^3-a^2-4a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$7$ |
7B.6.1 |
$1$ |
\( 2 \) |
$1$ |
$65.76400479$ |
1.837262932 |
\( \frac{176128}{41} a^{2} - \frac{176128}{41} a - \frac{815104}{41} \) |
\( \bigl[0\) , \( -a^{3} + 2 a^{2} + 4 a - 4\) , \( a^{3} - 4 a - 3\) , \( 4 a^{3} - 3 a^{2} - 19 a - 5\) , \( 11 a^{3} + a^{2} - 64 a - 61\bigr] \) |
${y}^2+\left(a^{3}-4a-3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-4\right){x}^{2}+\left(4a^{3}-3a^{2}-19a-5\right){x}+11a^{3}+a^{2}-64a-61$ |
| 41.1-e1 |
41.1-e |
$1$ |
$1$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
41.1 |
\( 41 \) |
\( 41^{4} \) |
$10.17615$ |
$(a^3-a^2-4a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.009312090$ |
$1120.903857$ |
2.332859573 |
\( \frac{71073792}{1681} a^{3} - \frac{64847872}{1681} a^{2} - \frac{296812544}{1681} a - \frac{149258240}{1681} \) |
\( \bigl[0\) , \( -a^{2} + 4\) , \( a^{2} - 3\) , \( 6 a^{3} - 20 a^{2} - 3 a + 39\) , \( -9 a^{3} + 32 a^{2} - 5 a - 48\bigr] \) |
${y}^2+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(6a^{3}-20a^{2}-3a+39\right){x}-9a^{3}+32a^{2}-5a-48$ |
| 41.1-f1 |
41.1-f |
$1$ |
$1$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
41.1 |
\( 41 \) |
\( 41^{4} \) |
$10.17615$ |
$(a^3-a^2-4a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.485050994$ |
$21.35387477$ |
2.314926137 |
\( -\frac{71073792}{1681} a^{3} + \frac{148373504}{1681} a^{2} + \frac{213286912}{1681} a - \frac{439844864}{1681} \) |
\( \bigl[0\) , \( a^{3} - 2 a^{2} - 3 a + 4\) , \( a^{3} - 4 a - 3\) , \( 9 a^{3} - 34 a^{2} + 4 a + 60\) , \( 30 a^{3} - 115 a^{2} + 15 a + 198\bigr] \) |
${y}^2+\left(a^{3}-4a-3\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+4\right){x}^{2}+\left(9a^{3}-34a^{2}+4a+60\right){x}+30a^{3}-115a^{2}+15a+198$ |
*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.
