| 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 |
| 29.2-a1 |
29.2-a |
$2$ |
$3$ |
4.4.5725.1 |
$4$ |
$[4, 0]$ |
29.2 |
\( 29 \) |
\( 29 \) |
$10.29973$ |
$(2/3a^3+a^2-10/3a-16/3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$126.0142097$ |
1.665450674 |
\( -\frac{33442546748576150}{29} a^{3} + \frac{116382822171561015}{29} a^{2} - \frac{21098628237379125}{29} a - \frac{148328941525828343}{29} \) |
\( \bigl[a^{2} + a - 4\) , \( \frac{1}{3} a^{3} - a^{2} - \frac{5}{3} a + \frac{10}{3}\) , \( \frac{1}{3} a^{3} - \frac{2}{3} a + \frac{4}{3}\) , \( -\frac{44}{3} a^{3} + 28 a^{2} + \frac{268}{3} a - \frac{518}{3}\) , \( -\frac{362}{3} a^{3} + 232 a^{2} + \frac{2242}{3} a - \frac{4247}{3}\bigr] \) |
${y}^2+\left(a^{2}+a-4\right){x}{y}+\left(\frac{1}{3}a^{3}-\frac{2}{3}a+\frac{4}{3}\right){y}={x}^{3}+\left(\frac{1}{3}a^{3}-a^{2}-\frac{5}{3}a+\frac{10}{3}\right){x}^{2}+\left(-\frac{44}{3}a^{3}+28a^{2}+\frac{268}{3}a-\frac{518}{3}\right){x}-\frac{362}{3}a^{3}+232a^{2}+\frac{2242}{3}a-\frac{4247}{3}$ |
| 29.2-a2 |
29.2-a |
$2$ |
$3$ |
4.4.5725.1 |
$4$ |
$[4, 0]$ |
29.2 |
\( 29 \) |
\( 29^{3} \) |
$10.29973$ |
$(2/3a^3+a^2-10/3a-16/3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$126.0142097$ |
1.665450674 |
\( \frac{128286191540280}{24389} a^{3} - \frac{248047721770401}{24389} a^{2} - \frac{794734781419895}{24389} a + \frac{1511638253800750}{24389} \) |
\( \bigl[a^{2} - 3\) , \( -\frac{1}{3} a^{3} + a^{2} + \frac{8}{3} a - \frac{10}{3}\) , \( \frac{1}{3} a^{3} + a^{2} - \frac{5}{3} a - \frac{8}{3}\) , \( -46 a^{3} - 32 a^{2} + 294 a + 225\) , \( \frac{361}{3} a^{3} + 115 a^{2} - \frac{2312}{3} a - \frac{2348}{3}\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(\frac{1}{3}a^{3}+a^{2}-\frac{5}{3}a-\frac{8}{3}\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}+a^{2}+\frac{8}{3}a-\frac{10}{3}\right){x}^{2}+\left(-46a^{3}-32a^{2}+294a+225\right){x}+\frac{361}{3}a^{3}+115a^{2}-\frac{2312}{3}a-\frac{2348}{3}$ |
| 29.2-b1 |
29.2-b |
$1$ |
$1$ |
4.4.5725.1 |
$4$ |
$[4, 0]$ |
29.2 |
\( 29 \) |
\( 29 \) |
$10.29973$ |
$(2/3a^3+a^2-10/3a-16/3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.066923655$ |
$671.2139007$ |
2.374722369 |
\( -\frac{1165144}{87} a^{3} - \frac{605414}{29} a^{2} + \frac{4662815}{87} a + \frac{5138546}{87} \) |
\( \bigl[\frac{1}{3} a^{3} + a^{2} - \frac{5}{3} a - \frac{8}{3}\) , \( -a^{2} + a + 5\) , \( \frac{1}{3} a^{3} - \frac{2}{3} a + \frac{1}{3}\) , \( -\frac{1}{3} a^{3} + a^{2} + \frac{8}{3} a + \frac{2}{3}\) , \( -\frac{7}{3} a^{3} - a^{2} + \frac{53}{3} a + \frac{35}{3}\bigr] \) |
${y}^2+\left(\frac{1}{3}a^{3}+a^{2}-\frac{5}{3}a-\frac{8}{3}\right){x}{y}+\left(\frac{1}{3}a^{3}-\frac{2}{3}a+\frac{1}{3}\right){y}={x}^{3}+\left(-a^{2}+a+5\right){x}^{2}+\left(-\frac{1}{3}a^{3}+a^{2}+\frac{8}{3}a+\frac{2}{3}\right){x}-\frac{7}{3}a^{3}-a^{2}+\frac{53}{3}a+\frac{35}{3}$ |
| 29.2-c1 |
29.2-c |
$2$ |
$3$ |
4.4.5725.1 |
$4$ |
$[4, 0]$ |
29.2 |
\( 29 \) |
\( 29 \) |
$10.29973$ |
$(2/3a^3+a^2-10/3a-16/3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.081055783$ |
$538.6246620$ |
2.308035528 |
\( \frac{5834057}{87} a^{3} - \frac{1357306}{29} a^{2} - \frac{64073665}{87} a + \frac{92919212}{87} \) |
\( \bigl[a^{2} - 4\) , \( \frac{1}{3} a^{3} - \frac{2}{3} a - \frac{2}{3}\) , \( a^{2} - 4\) , \( \frac{2}{3} a^{3} + 2 a^{2} - \frac{13}{3} a - \frac{37}{3}\) , \( a^{3} + 2 a^{2} - 6 a - 14\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(\frac{1}{3}a^{3}-\frac{2}{3}a-\frac{2}{3}\right){x}^{2}+\left(\frac{2}{3}a^{3}+2a^{2}-\frac{13}{3}a-\frac{37}{3}\right){x}+a^{3}+2a^{2}-6a-14$ |
| 29.2-c2 |
29.2-c |
$2$ |
$3$ |
4.4.5725.1 |
$4$ |
$[4, 0]$ |
29.2 |
\( 29 \) |
\( 29^{3} \) |
$10.29973$ |
$(2/3a^3+a^2-10/3a-16/3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.027018594$ |
$538.6246620$ |
2.308035528 |
\( \frac{5696335}{73167} a^{3} - \frac{3104056}{24389} a^{2} - \frac{90025829}{73167} a + \frac{272497447}{73167} \) |
\( \bigl[a\) , \( -\frac{1}{3} a^{3} + \frac{8}{3} a - \frac{4}{3}\) , \( a^{2} + a - 3\) , \( -a^{3} + 5 a - 3\) , \( \frac{1}{3} a^{3} - 3 a^{2} - \frac{11}{3} a + \frac{43}{3}\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}+\frac{8}{3}a-\frac{4}{3}\right){x}^{2}+\left(-a^{3}+5a-3\right){x}+\frac{1}{3}a^{3}-3a^{2}-\frac{11}{3}a+\frac{43}{3}$ |
| 29.2-d1 |
29.2-d |
$1$ |
$1$ |
4.4.5725.1 |
$4$ |
$[4, 0]$ |
29.2 |
\( 29 \) |
\( 29^{5} \) |
$10.29973$ |
$(2/3a^3+a^2-10/3a-16/3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5Nn.2.2[2] |
$1$ |
\( 1 \) |
$1$ |
$88.95160728$ |
1.175617532 |
\( -\frac{209272945351749}{20511149} a^{3} + \frac{730103522587178}{20511149} a^{2} - \frac{135071723250478}{20511149} a - \frac{932259985706165}{20511149} \) |
\( \bigl[\frac{1}{3} a^{3} - \frac{5}{3} a + \frac{4}{3}\) , \( -a^{2} - a + 5\) , \( 1\) , \( a^{3} - 4 a^{2} + 7\) , \( -4 a^{3} + 9 a^{2} + 12 a - 28\bigr] \) |
${y}^2+\left(\frac{1}{3}a^{3}-\frac{5}{3}a+\frac{4}{3}\right){x}{y}+{y}={x}^{3}+\left(-a^{2}-a+5\right){x}^{2}+\left(a^{3}-4a^{2}+7\right){x}-4a^{3}+9a^{2}+12a-28$ |
*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.
