| 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 |
$2$ |
$2$ |
3.3.1489.1 |
$3$ |
$[3, 0]$ |
13.1 |
\( 13 \) |
\( 13^{2} \) |
$5.28741$ |
$(a^2-3a-5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.798508442$ |
$113.7406028$ |
3.530522212 |
\( \frac{1940676136}{169} a^{2} - \frac{3539834321}{169} a - \frac{16489182614}{169} \) |
\( \bigl[a + 1\) , \( -a^{2} + 3 a + 5\) , \( a + 1\) , \( -a^{2} + 4 a + 5\) , \( a + 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+3a+5\right){x}^{2}+\left(-a^{2}+4a+5\right){x}+a+1$ |
| 13.1-a2 |
13.1-a |
$2$ |
$2$ |
3.3.1489.1 |
$3$ |
$[3, 0]$ |
13.1 |
\( 13 \) |
\( 13 \) |
$5.28741$ |
$(a^2-3a-5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1.597016884$ |
$113.7406028$ |
3.530522212 |
\( -\frac{10824428774162257}{13} a^{2} + \frac{19741105456676281}{13} a + \frac{91982454588646569}{13} \) |
\( \bigl[a + 1\) , \( -a^{2} + 3 a + 5\) , \( a + 1\) , \( -6 a^{2} + 14 a + 15\) , \( 21 a^{2} - 92 a - 90\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+3a+5\right){x}^{2}+\left(-6a^{2}+14a+15\right){x}+21a^{2}-92a-90$ |
| 13.1-b1 |
13.1-b |
$2$ |
$7$ |
3.3.1489.1 |
$3$ |
$[3, 0]$ |
13.1 |
\( 13 \) |
\( 13 \) |
$5.28741$ |
$(a^2-3a-5)$ |
$1$ |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.1 |
$1$ |
\( 1 \) |
$6.086570486$ |
$299.9195632$ |
2.896378243 |
\( \frac{5287458}{13} a^{2} - \frac{9665008}{13} a - \frac{44823817}{13} \) |
\( \bigl[a^{2} - 2 a - 5\) , \( a^{2} - 3 a - 5\) , \( a^{2} - a - 5\) , \( 3 a^{2} - 10 a - 15\) , \( 2 a^{2} - 8 a - 11\bigr] \) |
${y}^2+\left(a^{2}-2a-5\right){x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}+\left(a^{2}-3a-5\right){x}^{2}+\left(3a^{2}-10a-15\right){x}+2a^{2}-8a-11$ |
| 13.1-b2 |
13.1-b |
$2$ |
$7$ |
3.3.1489.1 |
$3$ |
$[3, 0]$ |
13.1 |
\( 13 \) |
\( 13^{7} \) |
$5.28741$ |
$(a^2-3a-5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.3 |
$1$ |
\( 1 \) |
$42.60599340$ |
$0.874401058$ |
2.896378243 |
\( \frac{59458245109305170602}{62748517} a^{2} - \frac{187197696596990380780}{62748517} a - \frac{194551739766325064917}{62748517} \) |
\( \bigl[a^{2} - 2 a - 5\) , \( a^{2} - 3 a - 5\) , \( a^{2} - a - 5\) , \( -527 a^{2} + 1465 a + 1560\) , \( -14200 a^{2} + 42364 a + 44530\bigr] \) |
${y}^2+\left(a^{2}-2a-5\right){x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}+\left(a^{2}-3a-5\right){x}^{2}+\left(-527a^{2}+1465a+1560\right){x}-14200a^{2}+42364a+44530$ |
| 13.1-c1 |
13.1-c |
$2$ |
$7$ |
3.3.1489.1 |
$3$ |
$[3, 0]$ |
13.1 |
\( 13 \) |
\( 13^{7} \) |
$5.28741$ |
$(a^2-3a-5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.6.3 |
$1$ |
\( 1 \) |
$3.684300274$ |
$9.959558664$ |
2.852784842 |
\( \frac{59458245109305170602}{62748517} a^{2} - \frac{187197696596990380780}{62748517} a - \frac{194551739766325064917}{62748517} \) |
\( \bigl[1\) , \( -a - 1\) , \( a^{2} - 2 a - 6\) , \( -1118 a^{2} - 3252 a - 1918\) , \( 78452 a^{2} + 232952 a + 138657\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-2a-6\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1118a^{2}-3252a-1918\right){x}+78452a^{2}+232952a+138657$ |
| 13.1-c2 |
13.1-c |
$2$ |
$7$ |
3.3.1489.1 |
$3$ |
$[3, 0]$ |
13.1 |
\( 13 \) |
\( 13 \) |
$5.28741$ |
$(a^2-3a-5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.6.1 |
$1$ |
\( 1 \) |
$0.526328610$ |
$69.71691065$ |
2.852784842 |
\( \frac{5287458}{13} a^{2} - \frac{9665008}{13} a - \frac{44823817}{13} \) |
\( \bigl[1\) , \( -a - 1\) , \( a^{2} - 2 a - 6\) , \( 2 a^{2} + 8 a + 7\) , \( -a^{2} - 4 a - 6\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-2a-6\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(2a^{2}+8a+7\right){x}-a^{2}-4a-6$ |
| 13.1-d1 |
13.1-d |
$2$ |
$2$ |
3.3.1489.1 |
$3$ |
$[3, 0]$ |
13.1 |
\( 13 \) |
\( 13 \) |
$5.28741$ |
$(a^2-3a-5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$4.559800508$ |
$36.07630523$ |
3.197288384 |
\( -\frac{10824428774162257}{13} a^{2} + \frac{19741105456676281}{13} a + \frac{91982454588646569}{13} \) |
\( \bigl[a^{2} - 2 a - 6\) , \( -a^{2} + a + 6\) , \( 1\) , \( -39 a^{2} - 99 a - 35\) , \( 500 a^{2} + 1491 a + 899\bigr] \) |
${y}^2+\left(a^{2}-2a-6\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+a+6\right){x}^{2}+\left(-39a^{2}-99a-35\right){x}+500a^{2}+1491a+899$ |
| 13.1-d2 |
13.1-d |
$2$ |
$2$ |
3.3.1489.1 |
$3$ |
$[3, 0]$ |
13.1 |
\( 13 \) |
\( 13^{2} \) |
$5.28741$ |
$(a^2-3a-5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.279900254$ |
$36.07630523$ |
3.197288384 |
\( \frac{1940676136}{169} a^{2} - \frac{3539834321}{169} a - \frac{16489182614}{169} \) |
\( \bigl[a^{2} - 2 a - 6\) , \( -a^{2} + a + 6\) , \( 1\) , \( -4 a^{2} + 6 a + 30\) , \( 15 a^{2} + 58 a + 55\bigr] \) |
${y}^2+\left(a^{2}-2a-6\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+a+6\right){x}^{2}+\left(-4a^{2}+6a+30\right){x}+15a^{2}+58a+55$ |
*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.
