| 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 |
$6$ |
$8$ |
4.4.13025.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( - 5^{4} \) |
$12.47093$ |
$(1/4a^3+1/2a^2-1/2a-7/4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$2.232009039$ |
$5.797857315$ |
1.814239378 |
\( \frac{16082288916904829}{100} a^{3} + \frac{20149313773541919}{50} a^{2} - \frac{25854498830773729}{50} a - \frac{133033069069667459}{100} \) |
\( \bigl[-\frac{1}{4} a^{3} + \frac{3}{2} a^{2} + \frac{1}{2} a - \frac{25}{4}\) , \( -\frac{1}{4} a^{3} + \frac{3}{2} a^{2} + \frac{1}{2} a - \frac{29}{4}\) , \( a^{2} - a - 5\) , \( -\frac{2735}{4} a^{3} + \frac{3951}{2} a^{2} + \frac{8953}{2} a - \frac{42055}{4}\) , \( -\frac{89057}{4} a^{3} + \frac{128577}{2} a^{2} + \frac{291655}{2} a - \frac{1368261}{4}\bigr] \) |
${y}^2+\left(-\frac{1}{4}a^{3}+\frac{3}{2}a^{2}+\frac{1}{2}a-\frac{25}{4}\right){x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}+\frac{3}{2}a^{2}+\frac{1}{2}a-\frac{29}{4}\right){x}^{2}+\left(-\frac{2735}{4}a^{3}+\frac{3951}{2}a^{2}+\frac{8953}{2}a-\frac{42055}{4}\right){x}-\frac{89057}{4}a^{3}+\frac{128577}{2}a^{2}+\frac{291655}{2}a-\frac{1368261}{4}$ |
| 5.1-a2 |
5.1-a |
$6$ |
$8$ |
4.4.13025.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{4} \) |
$12.47093$ |
$(1/4a^3+1/2a^2-1/2a-7/4)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.558002259$ |
$1484.251472$ |
1.814239378 |
\( \frac{8521}{100} a^{3} + \frac{21131}{50} a^{2} - \frac{321}{50} a + \frac{159009}{100} \) |
\( \bigl[-\frac{1}{4} a^{3} + \frac{3}{2} a^{2} + \frac{1}{2} a - \frac{25}{4}\) , \( -\frac{1}{4} a^{3} + \frac{3}{2} a^{2} + \frac{1}{2} a - \frac{29}{4}\) , \( a^{2} - a - 5\) , \( -\frac{15}{2} a^{3} + 23 a^{2} + 44 a - \frac{225}{2}\) , \( \frac{43}{4} a^{3} - \frac{59}{2} a^{2} - \frac{153}{2} a + \frac{671}{4}\bigr] \) |
${y}^2+\left(-\frac{1}{4}a^{3}+\frac{3}{2}a^{2}+\frac{1}{2}a-\frac{25}{4}\right){x}{y}+\left(a^{2}-a-5\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}+\frac{3}{2}a^{2}+\frac{1}{2}a-\frac{29}{4}\right){x}^{2}+\left(-\frac{15}{2}a^{3}+23a^{2}+44a-\frac{225}{2}\right){x}+\frac{43}{4}a^{3}-\frac{59}{2}a^{2}-\frac{153}{2}a+\frac{671}{4}$ |
| 5.1-a3 |
5.1-a |
$6$ |
$8$ |
4.4.13025.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{2} \) |
$12.47093$ |
$(1/4a^3+1/2a^2-1/2a-7/4)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.116004519$ |
$1484.251472$ |
1.814239378 |
\( \frac{5130041}{20} a^{3} - \frac{7395123}{10} a^{2} - \frac{16865103}{10} a + \frac{78990821}{20} \) |
\( \bigl[1\) , \( \frac{1}{4} a^{3} - \frac{3}{2} a^{2} - \frac{3}{2} a + \frac{25}{4}\) , \( \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{5}{4}\) , \( \frac{61}{2} a^{3} - 103 a^{2} - 113 a + \frac{721}{2}\) , \( -\frac{1013}{4} a^{3} + \frac{1731}{2} a^{2} + \frac{1871}{2} a - \frac{12089}{4}\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{5}{4}\right){y}={x}^{3}+\left(\frac{1}{4}a^{3}-\frac{3}{2}a^{2}-\frac{3}{2}a+\frac{25}{4}\right){x}^{2}+\left(\frac{61}{2}a^{3}-103a^{2}-113a+\frac{721}{2}\right){x}-\frac{1013}{4}a^{3}+\frac{1731}{2}a^{2}+\frac{1871}{2}a-\frac{12089}{4}$ |
| 5.1-a4 |
5.1-a |
$6$ |
$8$ |
4.4.13025.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{2} \) |
$12.47093$ |
$(1/4a^3+1/2a^2-1/2a-7/4)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.279001129$ |
$1484.251472$ |
1.814239378 |
\( -\frac{5243081}{20} a^{3} - \frac{2114437}{10} a^{2} + \frac{27643383}{10} a + \frac{84189419}{20} \) |
\( \bigl[\frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a + \frac{9}{4}\) , \( -a - 1\) , \( a^{2} - 5\) , \( -\frac{7}{2} a^{3} + a^{2} + 12 a - \frac{5}{2}\) , \( \frac{39}{4} a^{3} + \frac{27}{2} a^{2} - \frac{65}{2} a - \frac{177}{4}\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{3}{2}a+\frac{9}{4}\right){x}{y}+\left(a^{2}-5\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-\frac{7}{2}a^{3}+a^{2}+12a-\frac{5}{2}\right){x}+\frac{39}{4}a^{3}+\frac{27}{2}a^{2}-\frac{65}{2}a-\frac{177}{4}$ |
| 5.1-a5 |
5.1-a |
$6$ |
$8$ |
4.4.13025.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{8} \) |
$12.47093$ |
$(1/4a^3+1/2a^2-1/2a-7/4)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.116004519$ |
$92.76571705$ |
1.814239378 |
\( \frac{3544909651}{2500} a^{3} + \frac{1336211879}{250} a^{2} - \frac{5524619889}{1250} a - \frac{44521344553}{2500} \) |
\( \bigl[\frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a + \frac{9}{4}\) , \( -\frac{1}{4} a^{3} + \frac{1}{2} a^{2} + \frac{1}{2} a - \frac{1}{4}\) , \( a^{2} - 5\) , \( -4 a^{2} - 25 a - 29\) , \( -\frac{1}{4} a^{3} - \frac{41}{2} a^{2} - \frac{179}{2} a - \frac{389}{4}\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{3}{2}a+\frac{9}{4}\right){x}{y}+\left(a^{2}-5\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}+\frac{1}{2}a^{2}+\frac{1}{2}a-\frac{1}{4}\right){x}^{2}+\left(-4a^{2}-25a-29\right){x}-\frac{1}{4}a^{3}-\frac{41}{2}a^{2}-\frac{179}{2}a-\frac{389}{4}$ |
| 5.1-a6 |
5.1-a |
$6$ |
$8$ |
4.4.13025.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( - 5^{16} \) |
$12.47093$ |
$(1/4a^3+1/2a^2-1/2a-7/4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$2.232009039$ |
$5.797857315$ |
1.814239378 |
\( -\frac{393118916044291297}{312500} a^{3} + \frac{3366097771500863977}{781250} a^{2} + \frac{3630721974525889001}{781250} a - \frac{23505911313760565541}{1562500} \) |
\( \bigl[\frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a + \frac{9}{4}\) , \( -\frac{1}{4} a^{3} + \frac{1}{2} a^{2} + \frac{1}{2} a - \frac{1}{4}\) , \( a^{2} - 5\) , \( -\frac{15}{4} a^{3} - \frac{23}{2} a^{2} + \frac{45}{2} a + \frac{249}{4}\) , \( 32 a^{3} - 11 a^{2} - 411 a - 521\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{3}{2}a+\frac{9}{4}\right){x}{y}+\left(a^{2}-5\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}+\frac{1}{2}a^{2}+\frac{1}{2}a-\frac{1}{4}\right){x}^{2}+\left(-\frac{15}{4}a^{3}-\frac{23}{2}a^{2}+\frac{45}{2}a+\frac{249}{4}\right){x}+32a^{3}-11a^{2}-411a-521$ |
*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.
