| 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 |
| 4.1-a1 |
4.1-a |
$2$ |
$3$ |
4.4.17600.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{6} \) |
$14.09783$ |
$(-1/2a^3+4a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$9$ |
\( 1 \) |
$1$ |
$24.10346410$ |
1.635180271 |
\( -\frac{121623412147097347}{8} a^{3} - 33192402438219336 a^{2} + \frac{280847373358114115}{2} a + \frac{613083368541824541}{2} \) |
\( \bigl[\frac{1}{2} a^{2} + a - 3\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - 4 a - 5\) , \( 1\) , \( \frac{101}{2} a^{3} - 151 a^{2} - 250 a + 749\) , \( \frac{1969}{2} a^{3} - \frac{5391}{2} a^{2} - 6233 a + 16203\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{2}+a-3\right){x}{y}+{y}={x}^{3}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-4a-5\right){x}^{2}+\left(\frac{101}{2}a^{3}-151a^{2}-250a+749\right){x}+\frac{1969}{2}a^{3}-\frac{5391}{2}a^{2}-6233a+16203$ |
| 4.1-a2 |
4.1-a |
$2$ |
$3$ |
4.4.17600.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{18} \) |
$14.09783$ |
$(-1/2a^3+4a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$216.9311769$ |
1.635180271 |
\( -\frac{2103593}{64} a^{3} - \frac{1259703}{16} a^{2} + \frac{2476531}{8} a + \frac{11343627}{16} \) |
\( \bigl[\frac{1}{2} a^{3} - 4 a + 1\) , \( -\frac{1}{2} a^{3} + 3 a + 1\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - 3 a - 4\) , \( -2 a^{3} + 3 a^{2} + 16 a - 27\) , \( -\frac{461}{2} a^{3} + \frac{1003}{2} a^{2} + 2128 a - 4637\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-4a+1\right){x}{y}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-3a-4\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+3a+1\right){x}^{2}+\left(-2a^{3}+3a^{2}+16a-27\right){x}-\frac{461}{2}a^{3}+\frac{1003}{2}a^{2}+2128a-4637$ |
| 4.1-b1 |
4.1-b |
$2$ |
$3$ |
4.4.17600.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{6} \) |
$14.09783$ |
$(-1/2a^3+4a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 3 \) |
$1.590794367$ |
$2.231941677$ |
2.890442326 |
\( \frac{121623412147097347}{8} a^{3} - 33192402438219336 a^{2} - \frac{280847373358114115}{2} a + \frac{613083368541824541}{2} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - 4 a - 4\) , \( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + 3 a + 5\) , \( \frac{1}{2} a^{3} - 3 a\) , \( -\frac{55}{2} a^{3} + \frac{95}{2} a^{2} + 251 a - 454\) , \( -\frac{449}{2} a^{3} + 467 a^{2} + 2117 a - 4469\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-4a-4\right){x}{y}+\left(\frac{1}{2}a^{3}-3a\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+3a+5\right){x}^{2}+\left(-\frac{55}{2}a^{3}+\frac{95}{2}a^{2}+251a-454\right){x}-\frac{449}{2}a^{3}+467a^{2}+2117a-4469$ |
| 4.1-b2 |
4.1-b |
$2$ |
$3$ |
4.4.17600.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{18} \) |
$14.09783$ |
$(-1/2a^3+4a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$0.530264789$ |
$180.7872758$ |
2.890442326 |
\( \frac{2103593}{64} a^{3} - \frac{1259703}{16} a^{2} - \frac{2476531}{8} a + \frac{11343627}{16} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - 4 a - 4\) , \( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + 3 a + 5\) , \( \frac{1}{2} a^{3} - 3 a\) , \( -\frac{5}{2} a^{3} - \frac{5}{2} a^{2} + 21 a + 26\) , \( -\frac{7}{2} a^{3} - 3 a^{2} + 29 a + 31\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-4a-4\right){x}{y}+\left(\frac{1}{2}a^{3}-3a\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+3a+5\right){x}^{2}+\left(-\frac{5}{2}a^{3}-\frac{5}{2}a^{2}+21a+26\right){x}-\frac{7}{2}a^{3}-3a^{2}+29a+31$ |
| 4.1-c1 |
4.1-c |
$2$ |
$3$ |
4.4.17600.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{18} \) |
$14.09783$ |
$(-1/2a^3+4a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$216.9311769$ |
1.635180271 |
\( \frac{2103593}{64} a^{3} - \frac{1259703}{16} a^{2} - \frac{2476531}{8} a + \frac{11343627}{16} \) |
\( \bigl[\frac{1}{2} a^{3} - 4 a + 1\) , \( a + 1\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - 3 a - 4\) , \( 2 a^{3} + 3 a^{2} - 18 a - 27\) , \( \frac{461}{2} a^{3} + \frac{1003}{2} a^{2} - 2129 a - 4637\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-4a+1\right){x}{y}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-3a-4\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2a^{3}+3a^{2}-18a-27\right){x}+\frac{461}{2}a^{3}+\frac{1003}{2}a^{2}-2129a-4637$ |
| 4.1-c2 |
4.1-c |
$2$ |
$3$ |
4.4.17600.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{6} \) |
$14.09783$ |
$(-1/2a^3+4a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$9$ |
\( 1 \) |
$1$ |
$24.10346410$ |
1.635180271 |
\( \frac{121623412147097347}{8} a^{3} - 33192402438219336 a^{2} - \frac{280847373358114115}{2} a + \frac{613083368541824541}{2} \) |
\( \bigl[\frac{1}{2} a^{3} - 4 a + 1\) , \( a + 1\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - 3 a - 4\) , \( -23 a^{3} - 57 a^{2} + 182 a + 433\) , \( -\frac{12443}{2} a^{3} - \frac{27249}{2} a^{2} + 57233 a + 125135\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-4a+1\right){x}{y}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-3a-4\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-23a^{3}-57a^{2}+182a+433\right){x}-\frac{12443}{2}a^{3}-\frac{27249}{2}a^{2}+57233a+125135$ |
| 4.1-d1 |
4.1-d |
$2$ |
$3$ |
4.4.17600.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{6} \) |
$14.09783$ |
$(-1/2a^3+4a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 3 \) |
$1.590794367$ |
$2.231941677$ |
2.890442326 |
\( -\frac{121623412147097347}{8} a^{3} - 33192402438219336 a^{2} + \frac{280847373358114115}{2} a + \frac{613083368541824541}{2} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - 4 a - 4\) , \( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + 4 a + 5\) , \( 0\) , \( \frac{45}{2} a^{3} + 45 a^{2} - 207 a - 432\) , \( 309 a^{3} + 656 a^{2} - 2888 a - 6218\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-4a-4\right){x}{y}={x}^{3}+\left(-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+4a+5\right){x}^{2}+\left(\frac{45}{2}a^{3}+45a^{2}-207a-432\right){x}+309a^{3}+656a^{2}-2888a-6218$ |
| 4.1-d2 |
4.1-d |
$2$ |
$3$ |
4.4.17600.1 |
$4$ |
$[4, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{18} \) |
$14.09783$ |
$(-1/2a^3+4a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$0.530264789$ |
$180.7872758$ |
2.890442326 |
\( -\frac{2103593}{64} a^{3} - \frac{1259703}{16} a^{2} + \frac{2476531}{8} a + \frac{11343627}{16} \) |
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - 4 a - 4\) , \( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + 4 a + 5\) , \( 0\) , \( -\frac{5}{2} a^{3} - 5 a^{2} + 23 a + 48\) , \( -2 a^{3} - 4 a^{2} + 20 a + 42\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-4a-4\right){x}{y}={x}^{3}+\left(-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+4a+5\right){x}^{2}+\left(-\frac{5}{2}a^{3}-5a^{2}+23a+48\right){x}-2a^{3}-4a^{2}+20a+42$ |
*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.
