| 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 |
| 9.1-a1 |
9.1-a |
$4$ |
$6$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{10} \) |
$21.26627$ |
$(a^2-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$386.5609799$ |
1.01171790 |
\( \frac{1246362404}{81} a^{4} - \frac{1041117076}{27} a^{3} - \frac{719458786}{27} a^{2} + \frac{7323102649}{81} a - \frac{2459871980}{81} \) |
\( \bigl[a^{4} - 2 a^{3} - 3 a^{2} + 6 a + 1\) , \( a^{4} - a^{3} - 4 a^{2} + a + 2\) , \( a^{4} - a^{3} - 4 a^{2} + 2 a + 3\) , \( 106 a^{4} - 70 a^{3} - 413 a^{2} - 23 a + 81\) , \( 2837 a^{4} - 1889 a^{3} - 11031 a^{2} - 533 a + 2126\bigr] \) |
${y}^2+\left(a^{4}-2a^{3}-3a^{2}+6a+1\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+2a+3\right){y}={x}^{3}+\left(a^{4}-a^{3}-4a^{2}+a+2\right){x}^{2}+\left(106a^{4}-70a^{3}-413a^{2}-23a+81\right){x}+2837a^{4}-1889a^{3}-11031a^{2}-533a+2126$ |
| 9.1-a2 |
9.1-a |
$4$ |
$6$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{30} \) |
$21.26627$ |
$(a^2-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$193.2804899$ |
1.01171790 |
\( \frac{42958432423020244475506}{282429536481} a^{4} + \frac{4551898895594886045259}{94143178827} a^{3} - \frac{32400118789856343855629}{94143178827} a^{2} - \frac{10536177229996279192588}{282429536481} a + \frac{18530426122247352513662}{282429536481} \) |
\( \bigl[a\) , \( -a^{3} + 5 a\) , \( a^{4} - a^{3} - 4 a^{2} + 2 a + 3\) , \( 76 a^{4} - 83 a^{3} - 258 a^{2} + 73 a - 19\) , \( -302 a^{4} + 116 a^{3} + 1335 a^{2} + 348 a - 480\bigr] \) |
${y}^2+a{x}{y}+\left(a^{4}-a^{3}-4a^{2}+2a+3\right){y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(76a^{4}-83a^{3}-258a^{2}+73a-19\right){x}-302a^{4}+116a^{3}+1335a^{2}+348a-480$ |
| 9.1-a3 |
9.1-a |
$4$ |
$6$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{18} \) |
$21.26627$ |
$(a^2-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$386.5609799$ |
1.01171790 |
\( \frac{679068463130598740}{531441} a^{4} - \frac{359821343885971159}{177147} a^{3} - \frac{826730295123875959}{177147} a^{2} + \frac{2377533227165763589}{531441} a + \frac{1654749596218451752}{531441} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a^{2} - a - 2\) , \( -a^{4} - 2 a^{3} + 6 a^{2} + 8 a - 11\) , \( 36 a^{4} - 88 a^{3} - 60 a^{2} + 206 a - 75\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(-a^{4}-2a^{3}+6a^{2}+8a-11\right){x}+36a^{4}-88a^{3}-60a^{2}+206a-75$ |
| 9.1-a4 |
9.1-a |
$4$ |
$6$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{14} \) |
$21.26627$ |
$(a^2-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$193.2804899$ |
1.01171790 |
\( -\frac{7766624836025326355}{6561} a^{4} + \frac{6488464292861171239}{2187} a^{3} + \frac{4481598027790652077}{2187} a^{2} - \frac{45640052133196144267}{6561} a + \frac{15340352105491554221}{6561} \) |
\( \bigl[2 a^{4} - 3 a^{3} - 7 a^{2} + 8 a + 3\) , \( a^{2} - a - 3\) , \( a^{4} - a^{3} - 4 a^{2} + 3 a + 3\) , \( 71 a^{4} - 36 a^{3} - 299 a^{2} - 24 a + 56\) , \( 497 a^{4} - 316 a^{3} - 1964 a^{2} - 104 a + 377\bigr] \) |
${y}^2+\left(2a^{4}-3a^{3}-7a^{2}+8a+3\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+3a+3\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(71a^{4}-36a^{3}-299a^{2}-24a+56\right){x}+497a^{4}-316a^{3}-1964a^{2}-104a+377$ |
| 9.1-b1 |
9.1-b |
$4$ |
$6$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{10} \) |
$21.26627$ |
$(a^2-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$3087.086684$ |
0.897734128 |
\( \frac{1246362404}{81} a^{4} - \frac{1041117076}{27} a^{3} - \frac{719458786}{27} a^{2} + \frac{7323102649}{81} a - \frac{2459871980}{81} \) |
\( \bigl[a + 1\) , \( a^{4} - a^{3} - 3 a^{2} + a - 1\) , \( 0\) , \( 2 a^{2} - a\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a^{4}-a^{3}-3a^{2}+a-1\right){x}^{2}+\left(2a^{2}-a\right){x}$ |
| 9.1-b2 |
9.1-b |
$4$ |
$6$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{30} \) |
$21.26627$ |
$(a^2-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$19.05609064$ |
0.897734128 |
\( \frac{42958432423020244475506}{282429536481} a^{4} + \frac{4551898895594886045259}{94143178827} a^{3} - \frac{32400118789856343855629}{94143178827} a^{2} - \frac{10536177229996279192588}{282429536481} a + \frac{18530426122247352513662}{282429536481} \) |
\( \bigl[a^{4} - a^{3} - 4 a^{2} + 3 a + 2\) , \( 2 a^{4} - 3 a^{3} - 7 a^{2} + 7 a + 3\) , \( a^{4} - a^{3} - 4 a^{2} + 2 a + 2\) , \( 9 a^{4} + 38 a^{3} - 125 a^{2} - 29 a + 5\) , \( -52 a^{4} + 394 a^{3} - 503 a^{2} - 335 a + 53\bigr] \) |
${y}^2+\left(a^{4}-a^{3}-4a^{2}+3a+2\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+2a+2\right){y}={x}^{3}+\left(2a^{4}-3a^{3}-7a^{2}+7a+3\right){x}^{2}+\left(9a^{4}+38a^{3}-125a^{2}-29a+5\right){x}-52a^{4}+394a^{3}-503a^{2}-335a+53$ |
| 9.1-b3 |
9.1-b |
$4$ |
$6$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{18} \) |
$21.26627$ |
$(a^2-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$38.11218129$ |
0.897734128 |
\( \frac{679068463130598740}{531441} a^{4} - \frac{359821343885971159}{177147} a^{3} - \frac{826730295123875959}{177147} a^{2} + \frac{2377533227165763589}{531441} a + \frac{1654749596218451752}{531441} \) |
\( \bigl[a^{4} - a^{3} - 4 a^{2} + 3 a + 2\) , \( 2 a^{4} - 3 a^{3} - 7 a^{2} + 7 a + 3\) , \( a^{4} - a^{3} - 4 a^{2} + 2 a + 2\) , \( -11 a^{4} + 18 a^{3} + 35 a^{2} - 29 a - 20\) , \( -37 a^{4} + 72 a^{3} + 122 a^{2} - 163 a - 101\bigr] \) |
${y}^2+\left(a^{4}-a^{3}-4a^{2}+3a+2\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+2a+2\right){y}={x}^{3}+\left(2a^{4}-3a^{3}-7a^{2}+7a+3\right){x}^{2}+\left(-11a^{4}+18a^{3}+35a^{2}-29a-20\right){x}-37a^{4}+72a^{3}+122a^{2}-163a-101$ |
| 9.1-b4 |
9.1-b |
$4$ |
$6$ |
5.5.36497.1 |
$5$ |
$[5, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{14} \) |
$21.26627$ |
$(a^2-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$1543.543342$ |
0.897734128 |
\( -\frac{7766624836025326355}{6561} a^{4} + \frac{6488464292861171239}{2187} a^{3} + \frac{4481598027790652077}{2187} a^{2} - \frac{45640052133196144267}{6561} a + \frac{15340352105491554221}{6561} \) |
\( \bigl[a + 1\) , \( a^{4} - a^{3} - 3 a^{2} + a - 1\) , \( 0\) , \( -8 a^{2} + 4 a\) , \( -14 a^{4} + 5 a^{3} + 40 a^{2} - 7 a - 4\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a^{4}-a^{3}-3a^{2}+a-1\right){x}^{2}+\left(-8a^{2}+4a\right){x}-14a^{4}+5a^{3}+40a^{2}-7a-4$ |
*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.
