| 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-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 |
\( \frac{9227278873526453906244312610219036327836}{125} a^{4} + \frac{3459009377990835094505717267240191015816}{125} a^{3} - \frac{41380713886380859304809564621687978514293}{125} a^{2} - \frac{38438451731868504360194666541225269140904}{125} a - \frac{6711393728551335750914883813930912798473}{125} \) |
\( \bigl[a^{2} - a - 1\) , \( -3 a^{4} + 4 a^{3} + 13 a^{2} - 10 a - 9\) , \( 0\) , \( 846 a^{4} - 1524 a^{3} - 3297 a^{2} + 4382 a + 1145\) , \( 20454 a^{4} - 36469 a^{3} - 77364 a^{2} + 102303 a + 27278\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}={x}^{3}+\left(-3a^{4}+4a^{3}+13a^{2}-10a-9\right){x}^{2}+\left(846a^{4}-1524a^{3}-3297a^{2}+4382a+1145\right){x}+20454a^{4}-36469a^{3}-77364a^{2}+102303a+27278$ |
| 25.1-c1 |
25.1-c |
$4$ |
$15$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
25.1 |
\( 5^{2} \) |
\( - 5^{9} \) |
$31.59171$ |
$(-a^2+a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 5$ |
3B, 5B.4.2 |
$1$ |
\( 2^{2} \) |
$0.670481688$ |
$54.38783543$ |
2.84628363 |
\( \frac{9227278873526453906244312610219036327836}{125} a^{4} + \frac{3459009377990835094505717267240191015816}{125} a^{3} - \frac{41380713886380859304809564621687978514293}{125} a^{2} - \frac{38438451731868504360194666541225269140904}{125} a - \frac{6711393728551335750914883813930912798473}{125} \) |
\( \bigl[-a^{4} + 2 a^{3} + 4 a^{2} - 5 a - 3\) , \( -a^{4} + 2 a^{3} + 3 a^{2} - 4 a - 2\) , \( a^{3} - 4 a - 1\) , \( 1015 a^{4} - 1849 a^{3} - 3633 a^{2} + 4940 a + 991\) , \( -23261 a^{4} + 41576 a^{3} + 81774 a^{2} - 113086 a - 22399\bigr] \) |
${y}^2+\left(-a^{4}+2a^{3}+4a^{2}-5a-3\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a^{4}+2a^{3}+3a^{2}-4a-2\right){x}^{2}+\left(1015a^{4}-1849a^{3}-3633a^{2}+4940a+991\right){x}-23261a^{4}+41576a^{3}+81774a^{2}-113086a-22399$ |
| 45.1-b1 |
45.1-b |
$4$ |
$15$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{6} \cdot 5^{3} \) |
$33.50428$ |
$(-a^4+a^3+4a^2-2a-2), (-a^2+a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 5$ |
3B.1.2, 5B.4.2 |
$225$ |
\( 1 \) |
$1$ |
$1.656052230$ |
1.45417284 |
\( \frac{9227278873526453906244312610219036327836}{125} a^{4} + \frac{3459009377990835094505717267240191015816}{125} a^{3} - \frac{41380713886380859304809564621687978514293}{125} a^{2} - \frac{38438451731868504360194666541225269140904}{125} a - \frac{6711393728551335750914883813930912798473}{125} \) |
\( \bigl[-a^{4} + a^{3} + 5 a^{2} - 3 a - 3\) , \( -a^{4} + 5 a^{2} - 2\) , \( a^{3} - 4 a - 1\) , \( -527 a^{4} + 1330 a^{3} + 662 a^{2} - 1670 a - 937\) , \( 18278 a^{4} - 41440 a^{3} - 28524 a^{2} + 47105 a + 27660\bigr] \) |
${y}^2+\left(-a^{4}+a^{3}+5a^{2}-3a-3\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a^{4}+5a^{2}-2\right){x}^{2}+\left(-527a^{4}+1330a^{3}+662a^{2}-1670a-937\right){x}+18278a^{4}-41440a^{3}-28524a^{2}+47105a+27660$ |
| 225.1-a1 |
225.1-a |
$4$ |
$15$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
225.1 |
\( 3^{2} \cdot 5^{2} \) |
\( - 3^{6} \cdot 5^{9} \) |
$39.35476$ |
$(-a^4+a^3+4a^2-2a-2), (-a^2+a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 5$ |
3B, 5B.4.2 |
$25$ |
\( 2 \) |
$1$ |
$9.391251630$ |
1.83253789 |
\( \frac{9227278873526453906244312610219036327836}{125} a^{4} + \frac{3459009377990835094505717267240191015816}{125} a^{3} - \frac{41380713886380859304809564621687978514293}{125} a^{2} - \frac{38438451731868504360194666541225269140904}{125} a - \frac{6711393728551335750914883813930912798473}{125} \) |
\( \bigl[-2 a^{4} + 3 a^{3} + 9 a^{2} - 8 a - 7\) , \( -2 a^{4} + 2 a^{3} + 9 a^{2} - 6 a - 7\) , \( a^{4} - a^{3} - 4 a^{2} + 2 a + 2\) , \( 1196 a^{4} - 2255 a^{3} - 4356 a^{2} + 5795 a + 970\) , \( -39450 a^{4} + 69900 a^{3} + 153230 a^{2} - 182868 a - 68777\bigr] \) |
${y}^2+\left(-2a^{4}+3a^{3}+9a^{2}-8a-7\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+2a+2\right){y}={x}^{3}+\left(-2a^{4}+2a^{3}+9a^{2}-6a-7\right){x}^{2}+\left(1196a^{4}-2255a^{3}-4356a^{2}+5795a+970\right){x}-39450a^{4}+69900a^{3}+153230a^{2}-182868a-68777$ |
*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.
