| 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 |
| 51.2-a1 |
51.2-a |
$6$ |
$8$ |
\(\Q(\zeta_{9})^+\) |
$3$ |
$[3, 0]$ |
51.2 |
\( 3 \cdot 17 \) |
\( 3^{32} \cdot 17^{2} \) |
$1.54873$ |
$(-a^2+1), (2a^2-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$1.476452004$ |
0.656200891 |
\( -\frac{78367545688483633}{51195483} a^{2} + \frac{9072229692338704}{17065161} a + \frac{225650240167014325}{51195483} \) |
\( \bigl[a^{2} - 1\) , \( -a^{2} - a + 1\) , \( a^{2} + a - 2\) , \( 1510 a^{2} - 544 a - 4379\) , \( 37713 a^{2} - 13154 a - 108667\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(1510a^{2}-544a-4379\right){x}+37713a^{2}-13154a-108667$ |
| 51.2-a2 |
51.2-a |
$6$ |
$8$ |
\(\Q(\zeta_{9})^+\) |
$3$ |
$[3, 0]$ |
51.2 |
\( 3 \cdot 17 \) |
\( 3^{16} \cdot 17^{4} \) |
$1.54873$ |
$(-a^2+1), (2a^2-a-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$11.81161603$ |
0.656200891 |
\( \frac{257154008212105}{60886809} a^{2} + \frac{479060149170145}{60886809} a + \frac{139274003304361}{60886809} \) |
\( \bigl[a^{2} - 1\) , \( -a^{2} - a + 1\) , \( a^{2} + a - 2\) , \( 85 a^{2} - 49 a - 279\) , \( 599 a^{2} - 256 a - 1795\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(85a^{2}-49a-279\right){x}+599a^{2}-256a-1795$ |
| 51.2-a3 |
51.2-a |
$6$ |
$8$ |
\(\Q(\zeta_{9})^+\) |
$3$ |
$[3, 0]$ |
51.2 |
\( 3 \cdot 17 \) |
\( - 3^{8} \cdot 17^{8} \) |
$1.54873$ |
$(-a^2+1), (2a^2-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.476452004$ |
0.656200891 |
\( \frac{25375998117751504521013489}{188345450907} a^{2} + \frac{15897092117456459660800496}{62781816969} a + \frac{13502286575551170220498043}{188345450907} \) |
\( \bigl[a^{2} - 1\) , \( -a^{2} - a + 1\) , \( a^{2} + a - 2\) , \( -140 a^{2} - 274 a - 99\) , \( -139 a^{2} - 3802 a - 5503\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(-140a^{2}-274a-99\right){x}-139a^{2}-3802a-5503$ |
| 51.2-a4 |
51.2-a |
$6$ |
$8$ |
\(\Q(\zeta_{9})^+\) |
$3$ |
$[3, 0]$ |
51.2 |
\( 3 \cdot 17 \) |
\( 3^{8} \cdot 17^{2} \) |
$1.54873$ |
$(-a^2+1), (2a^2-a-3)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$94.49292831$ |
0.656200891 |
\( \frac{41087226515522}{7803} a^{2} - \frac{20983089393023}{2601} a - \frac{26817760341455}{7803} \) |
\( \bigl[a^{2} - 1\) , \( -a^{2} - a + 1\) , \( a^{2} + a - 2\) , \( 10 a^{2} - 4 a - 34\) , \( -5 a^{2} + 15\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(10a^{2}-4a-34\right){x}-5a^{2}+15$ |
| 51.2-a5 |
51.2-a |
$6$ |
$8$ |
\(\Q(\zeta_{9})^+\) |
$3$ |
$[3, 0]$ |
51.2 |
\( 3 \cdot 17 \) |
\( 3^{4} \cdot 17 \) |
$1.54873$ |
$(-a^2+1), (2a^2-a-3)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$188.9858566$ |
0.656200891 |
\( -\frac{97672967}{153} a^{2} + \frac{40445950}{153} a + \frac{269027197}{153} \) |
\( \bigl[a^{2} - 1\) , \( -a^{2} - a + 1\) , \( a^{2} + a - 2\) , \( 10 a^{2} - 4 a - 29\) , \( -15 a^{2} + 4 a + 41\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(10a^{2}-4a-29\right){x}-15a^{2}+4a+41$ |
| 51.2-a6 |
51.2-a |
$6$ |
$8$ |
\(\Q(\zeta_{9})^+\) |
$3$ |
$[3, 0]$ |
51.2 |
\( 3 \cdot 17 \) |
\( - 3^{4} \cdot 17 \) |
$1.54873$ |
$(-a^2+1), (2a^2-a-3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$47.24646415$ |
0.656200891 |
\( \frac{17146134462759887246327}{153} a^{2} - \frac{26269402052336030317921}{153} a - \frac{11191344455779074334681}{153} \) |
\( \bigl[a^{2} - 1\) , \( -a^{2} - a + 1\) , \( a^{2} + a - 2\) , \( -65 a^{2} + 41 a + 131\) , \( -29 a^{2} - 60 a + 261\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(-65a^{2}+41a+131\right){x}-29a^{2}-60a+261$ |
*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.
