| 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 |
| 23.2-a1 |
23.2-a |
$1$ |
$1$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
23.2 |
\( 23 \) |
\( -23 \) |
$4.15705$ |
$(a^2-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$41.27345117$ |
1.496161436 |
\( -\frac{1812432}{23} a^{2} + \frac{2863257}{23} a + \frac{9076295}{23} \) |
\( \bigl[a^{2} - a - 3\) , \( -a - 1\) , \( a^{2} - 4\) , \( -a - 6\) , \( -a^{2} + 2 a - 4\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-a-6\right){x}-a^{2}+2a-4$ |
| 23.2-b1 |
23.2-b |
$1$ |
$1$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
23.2 |
\( 23 \) |
\( -23 \) |
$4.15705$ |
$(a^2-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.066849611$ |
$314.3535004$ |
2.285315242 |
\( -\frac{1812432}{23} a^{2} + \frac{2863257}{23} a + \frac{9076295}{23} \) |
\( \bigl[1\) , \( a^{2} - a - 4\) , \( a^{2} - a - 4\) , \( 3 a^{2} - 3 a - 18\) , \( -12 a^{2} + 14 a + 69\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(3a^{2}-3a-18\right){x}-12a^{2}+14a+69$ |
| 23.2-c1 |
23.2-c |
$2$ |
$5$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
23.2 |
\( 23 \) |
\( - 23^{5} \) |
$4.15705$ |
$(a^2-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$0.346350743$ |
$79.51304173$ |
2.994907528 |
\( -\frac{961139734845}{6436343} a^{2} + \frac{1173762893721}{6436343} a + \frac{5459934581087}{6436343} \) |
\( \bigl[a^{2} - a - 4\) , \( -a - 1\) , \( 0\) , \( -2 a^{2} + 7 a - 1\) , \( 5 a^{2} - 12 a - 10\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2a^{2}+7a-1\right){x}+5a^{2}-12a-10$ |
| 23.2-c2 |
23.2-c |
$2$ |
$5$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
23.2 |
\( 23 \) |
\( -23 \) |
$4.15705$ |
$(a^2-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1.731753715$ |
$15.90260834$ |
2.994907528 |
\( -\frac{90444121625531417484414}{23} a^{2} + \frac{261560279167054112306301}{23} a + \frac{47804598082098941998736}{23} \) |
\( \bigl[a^{2} - a - 4\) , \( -a - 1\) , \( 0\) , \( -1617 a^{2} + 4672 a + 719\) , \( 73180 a^{2} - 211965 a - 38243\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1617a^{2}+4672a+719\right){x}+73180a^{2}-211965a-38243$ |
| 23.2-d1 |
23.2-d |
$2$ |
$5$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
23.2 |
\( 23 \) |
\( -23 \) |
$4.15705$ |
$(a^2-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 1 \) |
$1$ |
$0.417196637$ |
0.378084157 |
\( -\frac{90444121625531417484414}{23} a^{2} + \frac{261560279167054112306301}{23} a + \frac{47804598082098941998736}{23} \) |
\( \bigl[a^{2} - 4\) , \( a\) , \( a + 1\) , \( 374 a^{2} + 347 a - 4567\) , \( 12103 a^{2} + 4128 a - 126350\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(374a^{2}+347a-4567\right){x}+12103a^{2}+4128a-126350$ |
| 23.2-d2 |
23.2-d |
$2$ |
$5$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
23.2 |
\( 23 \) |
\( - 23^{5} \) |
$4.15705$ |
$(a^2-2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5 \) |
$1$ |
$52.14957970$ |
0.378084157 |
\( -\frac{961139734845}{6436343} a^{2} + \frac{1173762893721}{6436343} a + \frac{5459934581087}{6436343} \) |
\( \bigl[a^{2} - 4\) , \( a\) , \( a + 1\) , \( 29 a^{2} - 28 a - 162\) , \( -105 a^{2} + 131 a + 621\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(29a^{2}-28a-162\right){x}-105a^{2}+131a+621$ |
*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.
