| 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 |
| 55.1-a1 |
55.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-165}) \) |
$2$ |
$[0, 1]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{14} \cdot 11^{2} \) |
$6.25174$ |
$(5,a), (11,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$2.384490492$ |
$14.16497688$ |
2.629477707 |
\( \frac{59319}{55} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 20\) , \( 22\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+20{x}+22$ |
| 55.1-b1 |
55.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-165}) \) |
$2$ |
$[0, 1]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 3^{12} \cdot 5^{2} \cdot 11^{2} \) |
$6.25174$ |
$(5,a), (11,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$3.452752300$ |
$14.16497688$ |
7.614989644 |
\( \frac{59319}{55} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 7\) , \( -8\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+7{x}-8$ |
| 55.1-c1 |
55.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{-165}) \) |
$2$ |
$[0, 1]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{2} \cdot 11^{14} \) |
$6.25174$ |
$(5,a), (11,a)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$15.68344211$ |
$14.16497688$ |
8.647394808 |
\( \frac{59319}{55} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 98\) , \( -316\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+98{x}-316$ |
| 55.1-d1 |
55.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{-165}) \) |
$2$ |
$[0, 1]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{2} \cdot 11^{2} \) |
$6.25174$ |
$(5,a), (11,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$5.189627184$ |
$14.16497688$ |
5.722819627 |
\( \frac{59319}{55} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 568\) , \( -2611\bigr] \) |
${y}^2+a{x}{y}={x}^3+568{x}-2611$ |
| 55.1-e1 |
55.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{-165}) \) |
$2$ |
$[0, 1]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 3^{12} \cdot 5^{2} \cdot 11^{2} \) |
$6.25174$ |
$(5,a), (11,a)$ |
$1 \le r \le 3$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1.664591700$ |
$14.16497688$ |
7.342460423 |
\( \frac{59319}{55} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 657\) , \( -2653\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+657{x}-2653$ |
| 55.1-f1 |
55.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{-165}) \) |
$2$ |
$[0, 1]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{14} \cdot 11^{2} \) |
$6.25174$ |
$(5,a), (11,a)$ |
$1 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
|
\( 2^{2} \) |
|
$14.16497688$ |
15.28985518 |
\( \frac{59319}{55} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 670\) , \( -2865\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+670{x}-2865$ |
| 55.1-g1 |
55.1-g |
$4$ |
$4$ |
\(\Q(\sqrt{-165}) \) |
$2$ |
$[0, 1]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{2} \cdot 11^{2} \) |
$6.25174$ |
$(5,a), (11,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$14.16497688$ |
2.205483910 |
\( \frac{59319}{55} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2+{x}$ |
| 55.1-h1 |
55.1-h |
$4$ |
$4$ |
\(\Q(\sqrt{-165}) \) |
$2$ |
$[0, 1]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{2} \cdot 11^{14} \) |
$6.25174$ |
$(5,a), (11,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$4.314990037$ |
$14.16497688$ |
4.758320550 |
\( \frac{59319}{55} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 748\) , \( -3619\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+748{x}-3619$ |
*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.
