| 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 |
$1$ |
$1$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
5.1 |
\( 5 \) |
\( -5 \) |
$26.89528$ |
$(-a^2+a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$526.9651791$ |
2.05656008 |
\( \frac{144184611}{5} a^{4} - \frac{255843129}{5} a^{3} - \frac{524006273}{5} a^{2} + \frac{693618211}{5} a + \frac{186593617}{5} \) |
\( \bigl[-a^{4} + 2 a^{3} + 4 a^{2} - 5 a - 3\) , \( -a^{4} + a^{3} + 4 a^{2} - 3 a - 1\) , \( a^{4} - a^{3} - 4 a^{2} + 3 a + 2\) , \( -3 a^{4} + 2 a^{3} + 10 a^{2} - 7 a - 3\) , \( a^{4} + 3 a^{3} - 6 a - 3\bigr] \) |
${y}^2+\left(-a^{4}+2a^{3}+4a^{2}-5a-3\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+3a+2\right){y}={x}^{3}+\left(-a^{4}+a^{3}+4a^{2}-3a-1\right){x}^{2}+\left(-3a^{4}+2a^{3}+10a^{2}-7a-3\right){x}+a^{4}+3a^{3}-6a-3$ |
| 5.1-b1 |
5.1-b |
$4$ |
$15$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
5.1 |
\( 5 \) |
\( - 5^{3} \) |
$26.89528$ |
$(-a^2+a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.2 |
$625$ |
\( 3 \) |
$1$ |
$0.111196110$ |
0.813673832 |
\( \frac{9227278873526453906244312610219036327836}{125} a^{4} + \frac{3459009377990835094505717267240191015816}{125} a^{3} - \frac{41380713886380859304809564621687978514293}{125} a^{2} - \frac{38438451731868504360194666541225269140904}{125} a - \frac{6711393728551335750914883813930912798473}{125} \) |
\( \bigl[a^{2} - a - 1\) , \( -3 a^{4} + 4 a^{3} + 13 a^{2} - 10 a - 9\) , \( 0\) , \( 846 a^{4} - 1524 a^{3} - 3297 a^{2} + 4382 a + 1145\) , \( 20454 a^{4} - 36469 a^{3} - 77364 a^{2} + 102303 a + 27278\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}={x}^{3}+\left(-3a^{4}+4a^{3}+13a^{2}-10a-9\right){x}^{2}+\left(846a^{4}-1524a^{3}-3297a^{2}+4382a+1145\right){x}+20454a^{4}-36469a^{3}-77364a^{2}+102303a+27278$ |
| 5.1-b2 |
5.1-b |
$4$ |
$15$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
5.1 |
\( 5 \) |
\( -5 \) |
$26.89528$ |
$(-a^2+a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.2 |
$625$ |
\( 1 \) |
$1$ |
$0.333588330$ |
0.813673832 |
\( -\frac{660212871515640604}{5} a^{4} + \frac{1794053534107468711}{5} a^{3} + \frac{220022592976562427}{5} a^{2} - \frac{1698345910262749894}{5} a - \frac{384439131601574358}{5} \) |
\( \bigl[-a^{4} + 2 a^{3} + 4 a^{2} - 5 a - 2\) , \( -2 a^{4} + 3 a^{3} + 9 a^{2} - 10 a - 8\) , \( a^{4} - a^{3} - 4 a^{2} + 2 a + 3\) , \( 86 a^{4} + 73 a^{3} - 426 a^{2} - 515 a - 104\) , \( 1231 a^{4} + 500 a^{3} - 5653 a^{2} - 5184 a - 905\bigr] \) |
${y}^2+\left(-a^{4}+2a^{3}+4a^{2}-5a-2\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+2a+3\right){y}={x}^{3}+\left(-2a^{4}+3a^{3}+9a^{2}-10a-8\right){x}^{2}+\left(86a^{4}+73a^{3}-426a^{2}-515a-104\right){x}+1231a^{4}+500a^{3}-5653a^{2}-5184a-905$ |
| 5.1-b3 |
5.1-b |
$4$ |
$15$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
5.1 |
\( 5 \) |
\( - 5^{15} \) |
$26.89528$ |
$(-a^2+a+2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 3 \cdot 5 \) |
$1$ |
$347.4878438$ |
0.813673832 |
\( \frac{175802229236840746}{30517578125} a^{4} + \frac{65933855205252401}{30517578125} a^{3} - \frac{788398234528339998}{30517578125} a^{2} - \frac{732499454042744644}{30517578125} a - \frac{127944679222457853}{30517578125} \) |
\( \bigl[-2 a^{4} + 3 a^{3} + 9 a^{2} - 9 a - 7\) , \( a^{4} - 5 a^{2} - a + 4\) , \( a^{4} - a^{3} - 4 a^{2} + 3 a + 2\) , \( 5 a^{4} - 2 a^{3} - 25 a^{2} - 4 a + 29\) , \( a^{4} + 3 a^{3} - 10 a^{2} - 16 a + 22\bigr] \) |
${y}^2+\left(-2a^{4}+3a^{3}+9a^{2}-9a-7\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+3a+2\right){y}={x}^{3}+\left(a^{4}-5a^{2}-a+4\right){x}^{2}+\left(5a^{4}-2a^{3}-25a^{2}-4a+29\right){x}+a^{4}+3a^{3}-10a^{2}-16a+22$ |
| 5.1-b4 |
5.1-b |
$4$ |
$15$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
5.1 |
\( 5 \) |
\( - 5^{5} \) |
$26.89528$ |
$(-a^2+a+2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3, 5$ |
3B, 5B.1.1 |
$1$ |
\( 5 \) |
$1$ |
$1042.463531$ |
0.813673832 |
\( \frac{81829946}{3125} a^{4} - \frac{143808649}{3125} a^{3} - \frac{300612723}{3125} a^{2} + \frac{393132181}{3125} a + \frac{109010922}{3125} \) |
\( \bigl[-a^{4} + 2 a^{3} + 4 a^{2} - 5 a - 2\) , \( -2 a^{4} + 3 a^{3} + 9 a^{2} - 10 a - 8\) , \( a^{4} - a^{3} - 4 a^{2} + 2 a + 3\) , \( a^{4} - 2 a^{3} - 6 a^{2} + 5 a + 11\) , \( -2 a^{4} + 3 a^{3} + 9 a^{2} - 9 a - 9\bigr] \) |
${y}^2+\left(-a^{4}+2a^{3}+4a^{2}-5a-2\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+2a+3\right){y}={x}^{3}+\left(-2a^{4}+3a^{3}+9a^{2}-10a-8\right){x}^{2}+\left(a^{4}-2a^{3}-6a^{2}+5a+11\right){x}-2a^{4}+3a^{3}+9a^{2}-9a-9$ |
*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.
