| 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 |
| 51.3-a1 |
51.3-a |
$6$ |
$8$ |
\(\Q(\zeta_{9})^+\) |
$3$ |
$[3, 0]$ |
51.3 |
\( 3 \cdot 17 \) |
\( 3^{32} \cdot 17^{2} \) |
$1.54873$ |
$(-a^2+1), (a^2+a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$1.476452004$ |
0.656200891 |
\( \frac{9072229692338704}{17065161} a^{2} + \frac{51150856611467521}{51195483} a + \frac{14481770636014835}{51195483} \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( -543 a^{2} - 968 a - 272\) , \( -13153 a^{2} - 24560 a - 6934\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+\left(-543a^{2}-968a-272\right){x}-13153a^{2}-24560a-6934$ |
| 51.3-a2 |
51.3-a |
$6$ |
$8$ |
\(\Q(\zeta_{9})^+\) |
$3$ |
$[3, 0]$ |
51.3 |
\( 3 \cdot 17 \) |
\( - 3^{4} \cdot 17 \) |
$1.54873$ |
$(-a^2+1), (a^2+a-3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$47.24646415$ |
0.656200891 |
\( -\frac{26269402052336030317921}{153} a^{2} + \frac{9123267589576143071594}{153} a + \frac{75639728574412760793815}{153} \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( 42 a^{2} + 22 a - 82\) , \( -59 a^{2} + 88 a + 322\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+\left(42a^{2}+22a-82\right){x}-59a^{2}+88a+322$ |
| 51.3-a3 |
51.3-a |
$6$ |
$8$ |
\(\Q(\zeta_{9})^+\) |
$3$ |
$[3, 0]$ |
51.3 |
\( 3 \cdot 17 \) |
\( 3^{16} \cdot 17^{4} \) |
$1.54873$ |
$(-a^2+1), (a^2+a-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$11.81161603$ |
0.656200891 |
\( \frac{479060149170145}{60886809} a^{2} - \frac{736214157382250}{60886809} a - \frac{304538278611719}{60886809} \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( -48 a^{2} - 38 a - 12\) , \( -255 a^{2} - 344 a - 86\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+\left(-48a^{2}-38a-12\right){x}-255a^{2}-344a-86$ |
| 51.3-a4 |
51.3-a |
$6$ |
$8$ |
\(\Q(\zeta_{9})^+\) |
$3$ |
$[3, 0]$ |
51.3 |
\( 3 \cdot 17 \) |
\( 3^{8} \cdot 17^{2} \) |
$1.54873$ |
$(-a^2+1), (a^2+a-3)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$94.49292831$ |
0.656200891 |
\( -\frac{20983089393023}{2601} a^{2} + \frac{21862041663547}{7803} a + \frac{181255229047727}{7803} \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( -3 a^{2} - 8 a - 7\) , \( a^{2} + 4 a + 4\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+\left(-3a^{2}-8a-7\right){x}+a^{2}+4a+4$ |
| 51.3-a5 |
51.3-a |
$6$ |
$8$ |
\(\Q(\zeta_{9})^+\) |
$3$ |
$[3, 0]$ |
51.3 |
\( 3 \cdot 17 \) |
\( 3^{4} \cdot 17 \) |
$1.54873$ |
$(-a^2+1), (a^2+a-3)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$188.9858566$ |
0.656200891 |
\( \frac{40445950}{153} a^{2} + \frac{57227017}{153} a - \frac{7210637}{153} \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( -3 a^{2} - 8 a - 2\) , \( 5 a^{2} + 10 a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+\left(-3a^{2}-8a-2\right){x}+5a^{2}+10a+2$ |
| 51.3-a6 |
51.3-a |
$6$ |
$8$ |
\(\Q(\zeta_{9})^+\) |
$3$ |
$[3, 0]$ |
51.3 |
\( 3 \cdot 17 \) |
\( - 3^{8} \cdot 17^{8} \) |
$1.54873$ |
$(-a^2+1), (a^2+a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.476452004$ |
0.656200891 |
\( \frac{15897092117456459660800496}{62781816969} a^{2} - \frac{73067274470120883503414977}{188345450907} a - \frac{31128269893684578702277955}{188345450907} \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( -273 a^{2} + 412 a + 168\) , \( -3801 a^{2} + 3940 a + 1822\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+\left(-273a^{2}+412a+168\right){x}-3801a^{2}+3940a+1822$ |
*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.
