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 |
51.1-a1 |
51.1-a |
$6$ |
$8$ |
\(\Q(\zeta_{9})^+\) |
$3$ |
$[3, 0]$ |
51.1 |
\( 3 \cdot 17 \) |
\( - 3^{8} \cdot 17^{8} \) |
$1.54873$ |
$(-a^2+1), (a^2-2a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.476452004$ |
0.656200891 |
\( -\frac{73067274470120883503414977}{188345450907} a^{2} + \frac{25375998117751504521013489}{188345450907} a + \frac{70129610583765315423118325}{62781816969} \) |
\( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( 417 a^{2} - 133 a - 1203\) , \( 4217 a^{2} - 237 a - 14449\bigr] \) |
${y}^2+\left(a^{2}+a-1\right){x}{y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(417a^{2}-133a-1203\right){x}+4217a^{2}-237a-14449$ |
51.1-a2 |
51.1-a |
$6$ |
$8$ |
\(\Q(\zeta_{9})^+\) |
$3$ |
$[3, 0]$ |
51.1 |
\( 3 \cdot 17 \) |
\( - 3^{4} \cdot 17 \) |
$1.54873$ |
$(-a^2+1), (a^2-2a-3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$47.24646415$ |
0.656200891 |
\( \frac{9123267589576143071594}{153} a^{2} + \frac{17146134462759887246327}{153} a + \frac{4854389290588414014785}{153} \) |
\( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( 27 a^{2} - 58 a - 43\) , \( 50 a^{2} - 62 a + 9\bigr] \) |
${y}^2+\left(a^{2}+a-1\right){x}{y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(27a^{2}-58a-43\right){x}+50a^{2}-62a+9$ |
51.1-a3 |
51.1-a |
$6$ |
$8$ |
\(\Q(\zeta_{9})^+\) |
$3$ |
$[3, 0]$ |
51.1 |
\( 3 \cdot 17 \) |
\( 3^{16} \cdot 17^{4} \) |
$1.54873$ |
$(-a^2+1), (a^2-2a-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$11.81161603$ |
0.656200891 |
\( -\frac{736214157382250}{60886809} a^{2} + \frac{257154008212105}{60886809} a + \frac{2126010334493071}{60886809} \) |
\( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( -33 a^{2} + 92 a - 33\) , \( -292 a^{2} + 546 a + 23\bigr] \) |
${y}^2+\left(a^{2}+a-1\right){x}{y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(-33a^{2}+92a-33\right){x}-292a^{2}+546a+23$ |
51.1-a4 |
51.1-a |
$6$ |
$8$ |
\(\Q(\zeta_{9})^+\) |
$3$ |
$[3, 0]$ |
51.1 |
\( 3 \cdot 17 \) |
\( 3^{8} \cdot 17^{2} \) |
$1.54873$ |
$(-a^2+1), (a^2-2a-3)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$94.49292831$ |
0.656200891 |
\( \frac{21862041663547}{7803} a^{2} + \frac{41087226515522}{7803} a + \frac{3877536454165}{2601} \) |
\( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( -3 a^{2} + 17 a + 2\) , \( 11 a^{2} - 8 a - 6\bigr] \) |
${y}^2+\left(a^{2}+a-1\right){x}{y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(-3a^{2}+17a+2\right){x}+11a^{2}-8a-6$ |
51.1-a5 |
51.1-a |
$6$ |
$8$ |
\(\Q(\zeta_{9})^+\) |
$3$ |
$[3, 0]$ |
51.1 |
\( 3 \cdot 17 \) |
\( 3^{4} \cdot 17 \) |
$1.54873$ |
$(-a^2+1), (a^2-2a-3)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$188.9858566$ |
0.656200891 |
\( \frac{57227017}{153} a^{2} - \frac{97672967}{153} a - \frac{40772771}{153} \) |
\( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( -3 a^{2} + 17 a + 7\) , \( 17 a^{2} - 13 a - 7\bigr] \) |
${y}^2+\left(a^{2}+a-1\right){x}{y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(-3a^{2}+17a+7\right){x}+17a^{2}-13a-7$ |
51.1-a6 |
51.1-a |
$6$ |
$8$ |
\(\Q(\zeta_{9})^+\) |
$3$ |
$[3, 0]$ |
51.1 |
\( 3 \cdot 17 \) |
\( 3^{32} \cdot 17^{2} \) |
$1.54873$ |
$(-a^2+1), (a^2-2a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$1.476452004$ |
0.656200891 |
\( \frac{51150856611467521}{51195483} a^{2} - \frac{78367545688483633}{51195483} a - \frac{3709618270320887}{5688387} \) |
\( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( -963 a^{2} + 1517 a + 577\) , \( -24013 a^{2} + 36905 a + 15491\bigr] \) |
${y}^2+\left(a^{2}+a-1\right){x}{y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(-963a^{2}+1517a+577\right){x}-24013a^{2}+36905a+15491$ |
*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.