| 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 |
| 8.2-a1 |
8.2-a |
$4$ |
$6$ |
4.4.13968.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{10} \) |
$13.69594$ |
$(1/2a^2+1/2a-1), (-1/2a^3+1/2a^2+4a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$207.3911203$ |
2.632172233 |
\( \frac{2520855}{4} a^{3} - \frac{3033369}{2} a^{2} - \frac{15361245}{4} a + 6566157 \) |
\( \bigl[\frac{1}{2} a^{2} + \frac{1}{2} a - 2\) , \( \frac{1}{2} a^{2} - \frac{3}{2} a - 3\) , \( 0\) , \( -\frac{3}{2} a^{3} + \frac{7}{2} a^{2} + 9 a - 10\) , \( -a^{3} + 3 a^{2} + 5 a - 12\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-2\right){x}{y}={x}^{3}+\left(\frac{1}{2}a^{2}-\frac{3}{2}a-3\right){x}^{2}+\left(-\frac{3}{2}a^{3}+\frac{7}{2}a^{2}+9a-10\right){x}-a^{3}+3a^{2}+5a-12$ |
| 8.2-a2 |
8.2-a |
$4$ |
$6$ |
4.4.13968.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{5} \) |
$13.69594$ |
$(1/2a^2+1/2a-1), (-1/2a^3+1/2a^2+4a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$1$ |
$414.7822406$ |
2.632172233 |
\( \frac{1650794283}{4} a^{3} + 527191659 a^{2} - \frac{4644226125}{4} a - \frac{1007365491}{2} \) |
\( \bigl[\frac{1}{2} a^{2} + \frac{1}{2} a - 2\) , \( -\frac{1}{2} a^{3} + \frac{1}{2} a^{2} + 4 a - 1\) , \( \frac{1}{2} a^{3} - \frac{5}{2} a\) , \( -\frac{7}{2} a^{3} + 2 a^{2} + \frac{51}{2} a + 7\) , \( -14 a^{3} + 8 a^{2} + 108 a + 39\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-2\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{5}{2}a\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}+4a-1\right){x}^{2}+\left(-\frac{7}{2}a^{3}+2a^{2}+\frac{51}{2}a+7\right){x}-14a^{3}+8a^{2}+108a+39$ |
| 8.2-a3 |
8.2-a |
$4$ |
$6$ |
4.4.13968.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{14} \) |
$13.69594$ |
$(1/2a^2+1/2a-1), (-1/2a^3+1/2a^2+4a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$207.3911203$ |
2.632172233 |
\( -\frac{1492047}{16} a^{3} + \frac{452385}{8} a^{2} + \frac{11736333}{16} a + \frac{1089153}{4} \) |
\( \bigl[\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - 2 a + 1\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 3\) , \( 0\) , \( a^{3} + 2 a^{2} - 2 a - 3\) , \( 2 a^{3} + a^{2} - 6 a + 1\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-2a+1\right){x}{y}={x}^{3}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-3\right){x}^{2}+\left(a^{3}+2a^{2}-2a-3\right){x}+2a^{3}+a^{2}-6a+1$ |
| 8.2-a4 |
8.2-a |
$4$ |
$6$ |
4.4.13968.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{7} \) |
$13.69594$ |
$(1/2a^2+1/2a-1), (-1/2a^3+1/2a^2+4a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$1$ |
$414.7822406$ |
2.632172233 |
\( -\frac{1552959}{8} a^{3} + 1299159 a^{2} - \frac{21733839}{8} a + \frac{7144713}{4} \) |
\( \bigl[\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - 2 a + 1\) , \( -1\) , \( a + 1\) , \( 2 a^{3} + \frac{3}{2} a^{2} - \frac{29}{2} a - 19\) , \( \frac{29}{2} a^{3} - 10 a^{2} - \frac{213}{2} a - 20\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-2a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(2a^{3}+\frac{3}{2}a^{2}-\frac{29}{2}a-19\right){x}+\frac{29}{2}a^{3}-10a^{2}-\frac{213}{2}a-20$ |
| 8.2-b1 |
8.2-b |
$4$ |
$6$ |
4.4.13968.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{7} \) |
$13.69594$ |
$(1/2a^2+1/2a-1), (-1/2a^3+1/2a^2+4a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$0.064705518$ |
$1280.249639$ |
2.102762030 |
\( -\frac{1552959}{8} a^{3} + 1299159 a^{2} - \frac{21733839}{8} a + \frac{7144713}{4} \) |
\( \bigl[\frac{1}{2} a^{2} + \frac{1}{2} a - 2\) , \( -\frac{1}{2} a^{3} + a^{2} + \frac{7}{2} a - 3\) , \( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - 2 a + 1\) , \( 2 a^{3} - 16 a - 14\) , \( -\frac{21}{2} a^{3} + \frac{15}{2} a^{2} + 83 a + 24\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-2\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-2a+1\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+a^{2}+\frac{7}{2}a-3\right){x}^{2}+\left(2a^{3}-16a-14\right){x}-\frac{21}{2}a^{3}+\frac{15}{2}a^{2}+83a+24$ |
| 8.2-b2 |
8.2-b |
$4$ |
$6$ |
4.4.13968.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{14} \) |
$13.69594$ |
$(1/2a^2+1/2a-1), (-1/2a^3+1/2a^2+4a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$0.032352759$ |
$1280.249639$ |
2.102762030 |
\( -\frac{1492047}{16} a^{3} + \frac{452385}{8} a^{2} + \frac{11736333}{16} a + \frac{1089153}{4} \) |
\( \bigl[\frac{1}{2} a^{2} + \frac{1}{2} a - 2\) , \( -\frac{1}{2} a^{3} + \frac{1}{2} a^{2} + 4 a - 1\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 1\) , \( -\frac{1}{2} a^{3} + a^{2} + \frac{9}{2} a - 2\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-2\right){x}{y}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-1\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}+4a-1\right){x}^{2}+\left(-\frac{1}{2}a^{3}+a^{2}+\frac{9}{2}a-2\right){x}+\frac{1}{2}a^{2}+\frac{1}{2}a$ |
| 8.2-b3 |
8.2-b |
$4$ |
$6$ |
4.4.13968.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{5} \) |
$13.69594$ |
$(1/2a^2+1/2a-1), (-1/2a^3+1/2a^2+4a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$0.194116555$ |
$1280.249639$ |
2.102762030 |
\( \frac{1650794283}{4} a^{3} + 527191659 a^{2} - \frac{4644226125}{4} a - \frac{1007365491}{2} \) |
\( \bigl[\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - 2 a + 1\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 3\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 2\) , \( -2 a^{3} + 3 a^{2} + 19 a + 6\) , \( \frac{27}{2} a^{3} - 6 a^{2} - \frac{197}{2} a - 33\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-2a+1\right){x}{y}+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-2\right){y}={x}^{3}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-3\right){x}^{2}+\left(-2a^{3}+3a^{2}+19a+6\right){x}+\frac{27}{2}a^{3}-6a^{2}-\frac{197}{2}a-33$ |
| 8.2-b4 |
8.2-b |
$4$ |
$6$ |
4.4.13968.1 |
$4$ |
$[4, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{10} \) |
$13.69594$ |
$(1/2a^2+1/2a-1), (-1/2a^3+1/2a^2+4a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.097058277$ |
$1280.249639$ |
2.102762030 |
\( \frac{2520855}{4} a^{3} - \frac{3033369}{2} a^{2} - \frac{15361245}{4} a + 6566157 \) |
\( \bigl[\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - 2 a + 1\) , \( -\frac{1}{2} a^{3} + \frac{1}{2} a^{2} + 2 a - 1\) , \( \frac{1}{2} a^{3} - \frac{7}{2} a - 1\) , \( -\frac{3}{2} a^{3} + \frac{7}{2} a^{2} + 5 a - 12\) , \( 2 a^{3} - 4 a^{2} - 15 a + 18\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-2a+1\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{7}{2}a-1\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}+2a-1\right){x}^{2}+\left(-\frac{3}{2}a^{3}+\frac{7}{2}a^{2}+5a-12\right){x}+2a^{3}-4a^{2}-15a+18$ |
*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.
