| 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 |
| 13.1-a1 |
13.1-a |
$4$ |
$4$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
13.1 |
\( 13 \) |
\( 13^{2} \) |
$2.07355$ |
$(2a^2-a-6)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$153.1181165$ |
1.264791243 |
\( -\frac{2911512}{169} a^{2} + \frac{1025865}{169} a + \frac{12442564}{169} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - 4\) , \( a^{2} - 3\) , \( -2 a^{2} + 3 a + 1\) , \( 2 a^{2} - 3 a - 4\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-2a^{2}+3a+1\right){x}+2a^{2}-3a-4$ |
| 13.1-a2 |
13.1-a |
$4$ |
$4$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
13.1 |
\( 13 \) |
\( - 13^{4} \) |
$2.07355$ |
$(2a^2-a-6)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$19.13976456$ |
1.264791243 |
\( -\frac{37002219875353}{28561} a^{2} + \frac{9402056060606}{28561} a + \frac{145620309456504}{28561} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - 4\) , \( a^{2} - 3\) , \( -12 a^{2} + 23 a + 6\) , \( -40 a^{2} + 75 a + 19\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-12a^{2}+23a+6\right){x}-40a^{2}+75a+19$ |
| 13.1-a3 |
13.1-a |
$4$ |
$4$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
13.1 |
\( 13 \) |
\( -13 \) |
$2.07355$ |
$(2a^2-a-6)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$76.55905826$ |
1.264791243 |
\( \frac{109709855}{13} a^{2} + \frac{205310236}{13} a + \frac{45085962}{13} \) |
\( \bigl[a\) , \( -a^{2} + a + 4\) , \( a\) , \( -7 a^{2} - 11 a + 2\) , \( -21 a^{2} - 44 a - 10\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(-7a^{2}-11a+2\right){x}-21a^{2}-44a-10$ |
| 13.1-a4 |
13.1-a |
$4$ |
$4$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
13.1 |
\( 13 \) |
\( -13 \) |
$2.07355$ |
$(2a^2-a-6)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$306.2362330$ |
1.264791243 |
\( \frac{4451}{13} a^{2} + \frac{17718}{13} a + \frac{31630}{13} \) |
\( \bigl[1\) , \( -a^{2} + 4\) , \( a^{2} - 3\) , \( -2 a^{2} + 7\) , \( 0\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-2a^{2}+7\right){x}$ |
| 13.1-b1 |
13.1-b |
$4$ |
$4$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
13.1 |
\( 13 \) |
\( -13 \) |
$2.07355$ |
$(2a^2-a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.236027789$ |
$56.91991814$ |
0.665841604 |
\( \frac{109709855}{13} a^{2} + \frac{205310236}{13} a + \frac{45085962}{13} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - a - 4\) , \( 0\) , \( -a^{2} - 3 a - 1\) , \( -31 a^{2} + 4 a + 115\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-a^{2}-3a-1\right){x}-31a^{2}+4a+115$ |
| 13.1-b2 |
13.1-b |
$4$ |
$4$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
13.1 |
\( 13 \) |
\( -13 \) |
$2.07355$ |
$(2a^2-a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.059006947$ |
$227.6796725$ |
0.665841604 |
\( \frac{4451}{13} a^{2} + \frac{17718}{13} a + \frac{31630}{13} \) |
\( \bigl[a\) , \( a^{2} - a - 2\) , \( a^{2} + a - 2\) , \( 2 a^{2} - 3 a - 7\) , \( -2 a - 1\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(2a^{2}-3a-7\right){x}-2a-1$ |
| 13.1-b3 |
13.1-b |
$4$ |
$4$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
13.1 |
\( 13 \) |
\( - 13^{4} \) |
$2.07355$ |
$(2a^2-a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.059006947$ |
$113.8398362$ |
0.665841604 |
\( -\frac{37002219875353}{28561} a^{2} + \frac{9402056060606}{28561} a + \frac{145620309456504}{28561} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( 6 a - 11\) , \( 4 a^{2} - 14 a + 12\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(6a-11\right){x}+4a^{2}-14a+12$ |
| 13.1-b4 |
13.1-b |
$4$ |
$4$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
13.1 |
\( 13 \) |
\( 13^{2} \) |
$2.07355$ |
$(2a^2-a-6)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.118013894$ |
$227.6796725$ |
0.665841604 |
\( -\frac{2911512}{169} a^{2} + \frac{1025865}{169} a + \frac{12442564}{169} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( a - 1\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(a-1\right){x}$ |
*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.
