| 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 |
| 507.2-a1 |
507.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
507.2 |
\( 3 \cdot 13^{2} \) |
\( 3^{2} \cdot 13^{2} \) |
$0.73443$ |
$(-2a+1), (-4a+1), (4a-3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.282583906$ |
$7.561180171$ |
0.616802873 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+{x}$ |
| 13689.2-a1 |
13689.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
13689.2 |
\( 3^{4} \cdot 13^{2} \) |
\( 3^{14} \cdot 13^{2} \) |
$1.93313$ |
$(-3a-2), (2a+3), (3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.130335626$ |
$2.520393390$ |
1.424445220 |
\( \frac{12167}{39} \) |
\( \bigl[i\) , \( 1\) , \( i\) , \( 5\) , \( -6\bigr] \) |
${y}^2+i{x}{y}+i{y}={x}^{3}+{x}^{2}+5{x}-6$ |
| 13689.1-a1 |
13689.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
13689.1 |
\( 3^{4} \cdot 13^{2} \) |
\( 3^{14} \cdot 13^{2} \) |
$2.55729$ |
$(3), (13)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.067363554$ |
$2.520393390$ |
2.033581945 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 4\) , \( 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+4{x}+6$ |
| 13689.3-a1 |
13689.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
13689.3 |
\( 3^{4} \cdot 13^{2} \) |
\( 3^{14} \cdot 13^{2} \) |
$2.73386$ |
$(-a-1), (a-1), (13)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.537762007$ |
$2.520393390$ |
3.833570386 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 4\) , \( 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+4{x}+6$ |
| 13689.3-a1 |
13689.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
13689.3 |
\( 3^{4} \cdot 13^{2} \) |
\( 3^{14} \cdot 13^{2} \) |
$3.20574$ |
$(-a), (a-1), (13)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.676365415$ |
$2.520393390$ |
4.111907808 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 4\) , \( 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+4{x}+6$ |
| 507.1-a1 |
507.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 2^{12} \cdot 3^{2} \cdot 13^{2} \) |
$1.64224$ |
$(3,a+1), (13)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1.130335626$ |
$7.561180171$ |
1.103370523 |
\( \frac{12167}{39} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -3 a - 3\) , \( -3 a + 6\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2+\left(-3a-3\right){x}-3a+6$ |
| 13689.1-a1 |
13689.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
13689.1 |
\( 3^{4} \cdot 13^{2} \) |
\( 3^{14} \cdot 13^{2} \) |
$4.21316$ |
$(3), (13)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.615322880$ |
$2.520393390$ |
1.868017205 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 4\) , \( 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+4{x}+6$ |
| 507.1-a1 |
507.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-6}) \) |
$2$ |
$[0, 1]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 2^{12} \cdot 3^{2} \cdot 13^{2} \) |
$2.07729$ |
$(3,a), (13)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1.130335626$ |
$7.561180171$ |
0.872290989 |
\( \frac{12167}{39} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 5\) , \( -1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+5{x}-1$ |
| 39.1-a3 |
39.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{14} \cdot 13^{2} \) |
$1.39456$ |
$(3,a+1), (13,a+6)$ |
$2$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.432523755$ |
$7.561180171$ |
0.867219670 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 4\) , \( 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+4{x}+6$ |
| 13689.2-a1 |
13689.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{-43}) \) |
$2$ |
$[0, 1]$ |
13689.2 |
\( 3^{4} \cdot 13^{2} \) |
\( 3^{14} \cdot 13^{2} \) |
$6.33820$ |
$(a+1), (a-2), (3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1.130335626$ |
$2.520393390$ |
1.737806877 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 4\) , \( 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+4{x}+6$ |
| 507.2-b1 |
507.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
507.2 |
\( 3 \cdot 13^{2} \) |
\( 3^{14} \cdot 13^{2} \) |
$3.02814$ |
$(3,a+1), (a), (a-1)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$4.375145310$ |
$7.561180171$ |
4.632303228 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 4\) , \( 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+4{x}+6$ |
| 507.1-a1 |
507.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-21}) \) |
$2$ |
$[0, 1]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{14} \cdot 13^{2} \) |
$3.88625$ |
$(3,a), (13)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$2.656445675$ |
$7.561180171$ |
2.191547473 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 4\) , \( 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+4{x}+6$ |
| 507.2-a1 |
507.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{-87}) \) |
$2$ |
$[0, 1]$ |
507.2 |
\( 3 \cdot 13^{2} \) |
\( 3^{14} \cdot 13^{2} \) |
$3.95503$ |
$(3,a+1), (13,a+5), (13,a+7)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$3.312841673$ |
$7.561180171$ |
2.685533913 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 4\) , \( 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+4{x}+6$ |
| 507.1-c1 |
507.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{-111}) \) |
$2$ |
$[0, 1]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{14} \cdot 13^{2} \) |
$4.46737$ |
$(3,a+1), (13)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1.983811828$ |
$15.12236034$ |
1.423733069 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 4\) , \( 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+4{x}+6$ |
| 507.2-a1 |
507.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{-30}) \) |
$2$ |
$[0, 1]$ |
507.2 |
\( 3 \cdot 13^{2} \) |
\( 3^{14} \cdot 13^{2} \) |
$4.64495$ |
$(3,a), (13,a+3), (13,a+10)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2 \) |
$1.130335626$ |
$7.561180171$ |
1.560401558 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 4\) , \( 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+4{x}+6$ |
| 39.1-d1 |
39.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{-195}) \) |
$2$ |
$[0, 1]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{14} \cdot 13^{2} \) |
$3.11833$ |
$(3,a+1), (13,a+6)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.130335626$ |
$7.561180171$ |
0.612039845 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 4\) , \( 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+4{x}+6$ |
| 39.1-d1 |
39.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{-78}) \) |
$2$ |
$[0, 1]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{14} \cdot 13^{2} \) |
$3.94441$ |
$(3,a), (13,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$9$ |
\( 2^{2} \) |
$1.130335626$ |
$7.561180171$ |
4.354739846 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 4\) , \( 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+4{x}+6$ |
| 39.1-b1 |
39.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-663}) \) |
$2$ |
$[0, 1]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{14} \cdot 13^{2} \) |
$5.74992$ |
$(3,a+1), (13,a+6)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1.130335626$ |
$15.12236034$ |
1.327700839 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 4\) , \( 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+4{x}+6$ |
| 507.1-e2 |
507.1-e |
$6$ |
$8$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{2} \cdot 13^{2} \) |
$1.46886$ |
$(a), (a+4), (a-4)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.130335626$ |
$20.91395031$ |
1.706054394 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( -27 a + 47\) , \( -230 a + 398\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-27a+47\right){x}-230a+398$ |
| 1053.1-h2 |
1053.1-h |
$6$ |
$8$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1053.1 |
\( 3^{4} \cdot 13 \) |
\( 3^{14} \cdot 13^{2} \) |
$1.83534$ |
$(-a), (-a+1), (-2a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.282583906$ |
$6.971316772$ |
2.185498723 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 4\) , \( 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+4{x}+6$ |
| 507.1-b1 |
507.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{2} \cdot 13^{2} \) |
$1.94312$ |
$(-a+2), (13)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.250644754$ |
$20.91395031$ |
2.853845087 |
\( \frac{12167}{39} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 11\) , \( -5 a + 21\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+11{x}-5a+21$ |
| 507.1-c1 |
507.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{2} \cdot 13^{2} \) |
$2.07729$ |
$(a+3), (13)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.130335626$ |
$20.91395031$ |
1.206362631 |
\( \frac{12167}{39} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 10 a + 26\) , \( 67 a + 164\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(10a+26\right){x}+67a+164$ |
| 507.1-b1 |
507.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
507.1 |
\( 3 \cdot 13^{2} \) |
\( 3^{2} \cdot 13^{2} \) |
$2.43583$ |
$(-2a+7), (13)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.130335626$ |
$20.91395031$ |
1.028789508 |
\( \frac{12167}{39} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -177 a + 597\) , \( -4419 a + 14898\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-177a+597\right){x}-4419a+14898$ |
*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.
