| 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 |
| 2.2-a1 |
2.2-a |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
2.2 |
\( 2 \) |
\( 2^{4} \) |
$1.78301$ |
$(-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$171.4772930$ |
0.602896961 |
\( \frac{200481}{16} a^{2} - \frac{325027}{8} a + \frac{233973}{8} \) |
\( \bigl[a^{2} + a - 3\) , \( -a - 1\) , \( a^{2} - 3\) , \( 971347319857519 a^{2} - 514150241464297 a - 4127391273410075\) , \( 22425156860537519167317 a^{2} - 11870007338272500196506 a - 95287643090023949417928\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(971347319857519a^{2}-514150241464297a-4127391273410075\right){x}+22425156860537519167317a^{2}-11870007338272500196506a-95287643090023949417928$ |
| 16.5-b1 |
16.5-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
16.5 |
\( 2^{4} \) |
\( 2^{16} \) |
$2.52156$ |
$(-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$200.5904149$ |
0.705255777 |
\( \frac{200481}{16} a^{2} - \frac{325027}{8} a + \frac{233973}{8} \) |
\( \bigl[a^{2} - 3\) , \( a^{2} - 3\) , \( a^{2} - 3\) , \( 100434688208 a^{2} - 53161745688 a - 426761105098\) , \( -23577610985047683 a^{2} + 12480020414213596 a + 100184582628814686\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(100434688208a^{2}-53161745688a-426761105098\right){x}-23577610985047683a^{2}+12480020414213596a+100184582628814686$ |
| 32.1-c1 |
32.1-c |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{16} \) |
$2.83035$ |
$(a), (-a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.066041164$ |
$242.5005074$ |
1.351372542 |
\( \frac{200481}{16} a^{2} - \frac{325027}{8} a + \frac{233973}{8} \) |
\( \bigl[a\) , \( a^{2} - 4\) , \( a^{2} + a - 2\) , \( 124431276076006 a^{2} - 65863537513620 a - 528725979385749\) , \( -1028177435994703261869 a^{2} + 544231364186354985038 a + 4368870423675193690762\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(124431276076006a^{2}-65863537513620a-528725979385749\right){x}-1028177435994703261869a^{2}+544231364186354985038a+4368870423675193690762$ |
| 64.7-a4 |
64.7-a |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{22} \) |
$3.17696$ |
$(-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$114.5661335$ |
1.611212134 |
\( \frac{200481}{16} a^{2} - \frac{325027}{8} a + \frac{233973}{8} \) |
\( \bigl[a + 1\) , \( a^{2} - a - 2\) , \( a + 1\) , \( 2665494177 a^{2} - 1410890263 a - 11326059360\) , \( 101937460779857 a^{2} - 53957188127012 a - 433146596953681\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(2665494177a^{2}-1410890263a-11326059360\right){x}+101937460779857a^{2}-53957188127012a-433146596953681$ |
| 64.7-d5 |
64.7-d |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{22} \) |
$3.17696$ |
$(-a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.810264127$ |
$75.05861525$ |
2.565930457 |
\( \frac{200481}{16} a^{2} - \frac{325027}{8} a + \frac{233973}{8} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} + 3\) , \( 0\) , \( 2665494175 a^{2} - 1410890261 a - 11326059353\) , \( -101934795285681 a^{2} + 53955777236750 a + 433135270894324\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(2665494175a^{2}-1410890261a-11326059353\right){x}-101934795285681a^{2}+53955777236750a+433135270894324$ |
| 242.3-b8 |
242.3-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( 2^{4} \cdot 11^{6} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$42.76690299$ |
1.202913127 |
\( \frac{200481}{16} a^{2} - \frac{325027}{8} a + \frac{233973}{8} \) |
\( \bigl[1\) , \( -a^{2} + a + 3\) , \( 0\) , \( 41761256065998246 a^{2} - 22104925242738941 a - 177449446073241693\) , \( 6321701762221270525137686 a^{2} - 3346181557373420332949796 a - 26861799227819409800534166\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(41761256065998246a^{2}-22104925242738941a-177449446073241693\right){x}+6321701762221270525137686a^{2}-3346181557373420332949796a-26861799227819409800534166$ |
*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.
