| 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 |
| 180.2-a1 |
180.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
180.2 |
\( 2^{2} \cdot 3^{2} \cdot 5 \) |
\( 2^{12} \cdot 3^{18} \cdot 5^{2} \) |
$1.26766$ |
$(2,a), (2,a+1), (3,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{5} \) |
$0.198498654$ |
$2.819962089$ |
1.156232559 |
\( -\frac{1860867}{320} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -69\) , \( -235\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-69{x}-235$ |
| 180.2-b1 |
180.2-b |
$8$ |
$12$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
180.2 |
\( 2^{2} \cdot 3^{2} \cdot 5 \) |
\( 2^{12} \cdot 3^{6} \cdot 5^{2} \) |
$1.26766$ |
$(2,a), (2,a+1), (3,a+1), (5,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2^{5} \cdot 3^{2} \) |
$1$ |
$2.819962089$ |
2.912444322 |
\( -\frac{1860867}{320} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -8\) , \( 11\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2-8{x}+11$ |
| 640.2-a1 |
640.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
640.2 |
\( 2^{7} \cdot 5 \) |
\( 2^{30} \cdot 5^{2} \) |
$1.74072$ |
$(2,a), (2,a+1), (5,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.726867053$ |
0.891750311 |
\( -\frac{1860867}{320} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -8 a - 9\) , \( -16 a + 5\bigr] \) |
${y}^2+a{x}{y}={x}^3-{x}^2+\left(-8a-9\right){x}-16a+5$ |
| 640.2-b1 |
640.2-b |
$8$ |
$12$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
640.2 |
\( 2^{7} \cdot 5 \) |
\( 2^{30} \cdot 3^{12} \cdot 5^{2} \) |
$1.74072$ |
$(2,a), (2,a+1), (5,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$1.726867053$ |
2.675250935 |
\( -\frac{1860867}{320} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -69 a - 93\) , \( 573 a + 43\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(-a+1\right){x}^2+\left(-69a-93\right){x}+573a+43$ |
| 640.7-a1 |
640.7-a |
$8$ |
$12$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
640.7 |
\( 2^{7} \cdot 5 \) |
\( 2^{30} \cdot 5^{2} \) |
$1.74072$ |
$(2,a), (2,a+1), (5,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.726867053$ |
0.891750311 |
\( -\frac{1860867}{320} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 8 a - 17\) , \( 16 a - 11\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-a-1\right){x}^2+\left(8a-17\right){x}+16a-11$ |
| 640.7-b1 |
640.7-b |
$8$ |
$12$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
640.7 |
\( 2^{7} \cdot 5 \) |
\( 2^{30} \cdot 3^{12} \cdot 5^{2} \) |
$1.74072$ |
$(2,a), (2,a+1), (5,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$1.726867053$ |
2.675250935 |
\( -\frac{1860867}{320} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 69 a - 162\) , \( -573 a + 616\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(69a-162\right){x}-573a+616$ |
| 800.2-b1 |
800.2-b |
$8$ |
$12$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
800.2 |
\( 2^{5} \cdot 5^{2} \) |
\( 2^{24} \cdot 3^{12} \cdot 5^{8} \) |
$1.84059$ |
$(2,a), (2,a+1), (5,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{5} \) |
$1$ |
$1.092166620$ |
1.127984835 |
\( -\frac{1860867}{320} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 115 a - 460\) , \( 1628 a - 3808\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2+\left(115a-460\right){x}+1628a-3808$ |
| 800.2-e1 |
800.2-e |
$8$ |
$12$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
800.2 |
\( 2^{5} \cdot 5^{2} \) |
\( 2^{24} \cdot 5^{8} \) |
$1.84059$ |
$(2,a), (2,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{5} \cdot 3 \) |
$0.598387813$ |
$1.092166620$ |
4.049834279 |
\( -\frac{1860867}{320} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 12 a - 48\) , \( -69 a + 176\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(12a-48\right){x}-69a+176$ |
| 800.5-b1 |
800.5-b |
$8$ |
$12$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
800.5 |
\( 2^{5} \cdot 5^{2} \) |
\( 2^{24} \cdot 3^{12} \cdot 5^{8} \) |
$1.84059$ |
$(2,a), (2,a+1), (5,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{5} \) |
$1$ |
$1.092166620$ |
1.127984835 |
\( -\frac{1860867}{320} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -117 a - 345\) , \( -1629 a - 2180\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-117a-345\right){x}-1629a-2180$ |
| 800.5-e1 |
800.5-e |
$8$ |
$12$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
800.5 |
\( 2^{5} \cdot 5^{2} \) |
\( 2^{24} \cdot 5^{8} \) |
$1.84059$ |
$(2,a), (2,a+1), (5,a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{5} \cdot 3 \) |
$0.598387813$ |
$1.092166620$ |
4.049834279 |
\( -\frac{1860867}{320} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -14 a - 36\) , \( 68 a + 107\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(-14a-36\right){x}+68a+107$ |
*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.
