| 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 |
| 5.1-a1 |
5.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{5}, \sqrt{6})\) |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{4} \) |
$13.11267$ |
$(3/19a^3+5/19a^2-8/19a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$372.0983287$ |
1.550409703 |
\( -\frac{17408512}{475} a^{3} + \frac{26112768}{475} a^{2} + \frac{322057472}{475} a + \frac{2538048}{5} \) |
\( \bigl[-\frac{2}{19} a^{3} + \frac{3}{19} a^{2} + \frac{37}{19} a - 1\) , \( -\frac{3}{19} a^{3} - \frac{5}{19} a^{2} + \frac{46}{19} a + 1\) , \( -\frac{1}{19} a^{3} + \frac{11}{19} a^{2} + \frac{9}{19} a - 3\) , \( -\frac{454}{19} a^{3} + \frac{1270}{19} a^{2} + \frac{4827}{19} a - 535\) , \( -\frac{2934}{19} a^{3} + \frac{8296}{19} a^{2} + \frac{31232}{19} a - 3525\bigr] \) |
${y}^2+\left(-\frac{2}{19}a^{3}+\frac{3}{19}a^{2}+\frac{37}{19}a-1\right){x}{y}+\left(-\frac{1}{19}a^{3}+\frac{11}{19}a^{2}+\frac{9}{19}a-3\right){y}={x}^{3}+\left(-\frac{3}{19}a^{3}-\frac{5}{19}a^{2}+\frac{46}{19}a+1\right){x}^{2}+\left(-\frac{454}{19}a^{3}+\frac{1270}{19}a^{2}+\frac{4827}{19}a-535\right){x}-\frac{2934}{19}a^{3}+\frac{8296}{19}a^{2}+\frac{31232}{19}a-3525$ |
| 5.1-a2 |
5.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{5}, \sqrt{6})\) |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{8} \) |
$13.11267$ |
$(3/19a^3+5/19a^2-8/19a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$93.02458218$ |
1.550409703 |
\( \frac{426496}{11875} a^{3} - \frac{639744}{11875} a^{2} - \frac{7890176}{11875} a + \frac{59456}{125} \) |
\( \bigl[-\frac{2}{19} a^{3} + \frac{3}{19} a^{2} + \frac{37}{19} a - 1\) , \( \frac{1}{19} a^{3} - \frac{11}{19} a^{2} + \frac{10}{19} a + 5\) , \( \frac{1}{19} a^{3} + \frac{8}{19} a^{2} - \frac{28}{19} a - 2\) , \( -\frac{14}{19} a^{3} - \frac{17}{19} a^{2} + \frac{88}{19} a + 14\) , \( -\frac{32}{19} a^{3} - \frac{104}{19} a^{2} + \frac{174}{19} a + 17\bigr] \) |
${y}^2+\left(-\frac{2}{19}a^{3}+\frac{3}{19}a^{2}+\frac{37}{19}a-1\right){x}{y}+\left(\frac{1}{19}a^{3}+\frac{8}{19}a^{2}-\frac{28}{19}a-2\right){y}={x}^{3}+\left(\frac{1}{19}a^{3}-\frac{11}{19}a^{2}+\frac{10}{19}a+5\right){x}^{2}+\left(-\frac{14}{19}a^{3}-\frac{17}{19}a^{2}+\frac{88}{19}a+14\right){x}-\frac{32}{19}a^{3}-\frac{104}{19}a^{2}+\frac{174}{19}a+17$ |
| 5.1-b1 |
5.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{5}, \sqrt{6})\) |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{4} \) |
$13.11267$ |
$(3/19a^3+5/19a^2-8/19a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.023808089$ |
$2876.367198$ |
2.282693617 |
\( -\frac{17408512}{475} a^{3} + \frac{26112768}{475} a^{2} + \frac{322057472}{475} a + \frac{2538048}{5} \) |
\( \bigl[-\frac{2}{19} a^{3} + \frac{3}{19} a^{2} + \frac{37}{19} a - 1\) , \( \frac{1}{19} a^{3} - \frac{11}{19} a^{2} + \frac{10}{19} a + 5\) , \( -\frac{1}{19} a^{3} + \frac{11}{19} a^{2} + \frac{9}{19} a - 3\) , \( -\frac{40}{19} a^{3} - \frac{92}{19} a^{2} + \frac{227}{19} a + 19\) , \( \frac{136}{19} a^{3} + \frac{290}{19} a^{2} - \frac{616}{19} a - 29\bigr] \) |
${y}^2+\left(-\frac{2}{19}a^{3}+\frac{3}{19}a^{2}+\frac{37}{19}a-1\right){x}{y}+\left(-\frac{1}{19}a^{3}+\frac{11}{19}a^{2}+\frac{9}{19}a-3\right){y}={x}^{3}+\left(\frac{1}{19}a^{3}-\frac{11}{19}a^{2}+\frac{10}{19}a+5\right){x}^{2}+\left(-\frac{40}{19}a^{3}-\frac{92}{19}a^{2}+\frac{227}{19}a+19\right){x}+\frac{136}{19}a^{3}+\frac{290}{19}a^{2}-\frac{616}{19}a-29$ |
| 5.1-b2 |
5.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{5}, \sqrt{6})\) |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{8} \) |
$13.11267$ |
$(3/19a^3+5/19a^2-8/19a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.047616179$ |
$719.0917996$ |
2.282693617 |
\( \frac{426496}{11875} a^{3} - \frac{639744}{11875} a^{2} - \frac{7890176}{11875} a + \frac{59456}{125} \) |
\( \bigl[-\frac{2}{19} a^{3} + \frac{3}{19} a^{2} + \frac{37}{19} a - 1\) , \( \frac{1}{19} a^{3} + \frac{8}{19} a^{2} - \frac{9}{19} a - 2\) , \( \frac{1}{19} a^{3} + \frac{8}{19} a^{2} - \frac{28}{19} a - 2\) , \( -\frac{122}{19} a^{3} + \frac{639}{19} a^{2} - \frac{308}{19} a - 40\) , \( -\frac{1060}{19} a^{3} + \frac{5409}{19} a^{2} - \frac{2753}{19} a - 349\bigr] \) |
${y}^2+\left(-\frac{2}{19}a^{3}+\frac{3}{19}a^{2}+\frac{37}{19}a-1\right){x}{y}+\left(\frac{1}{19}a^{3}+\frac{8}{19}a^{2}-\frac{28}{19}a-2\right){y}={x}^{3}+\left(\frac{1}{19}a^{3}+\frac{8}{19}a^{2}-\frac{9}{19}a-2\right){x}^{2}+\left(-\frac{122}{19}a^{3}+\frac{639}{19}a^{2}-\frac{308}{19}a-40\right){x}-\frac{1060}{19}a^{3}+\frac{5409}{19}a^{2}-\frac{2753}{19}a-349$ |
*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.
