| 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 |
| 9.3-a1 |
9.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{3}, \sqrt{7})\) |
$4$ |
$[4, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{14} \) |
$9.87866$ |
$(-a-1), (-a^3+5a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$0.232242074$ |
$131.1093345$ |
1.449957322 |
\( \frac{968192}{729} a^{3} - \frac{1936384}{243} a + \frac{2296256}{729} \) |
\( \bigl[a^{3} + a^{2} - 5 a - 2\) , \( a^{3} - 4 a + 1\) , \( 1\) , \( -2 a^{2} + 4 a + 4\) , \( a^{3} - 3 a^{2} + a + 1\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-5a-2\right){x}{y}+{y}={x}^{3}+\left(a^{3}-4a+1\right){x}^{2}+\left(-2a^{2}+4a+4\right){x}+a^{3}-3a^{2}+a+1$ |
| 9.3-a2 |
9.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{3}, \sqrt{7})\) |
$4$ |
$[4, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{10} \) |
$9.87866$ |
$(-a-1), (-a^3+5a-1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.077414024$ |
$1179.984011$ |
1.449957322 |
\( \frac{4301312}{27} a^{3} - \frac{8602624}{9} a + \frac{11388736}{27} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( 1\) , \( -22 a^{3} + 11 a^{2} + 106 a - 47\) , \( 98 a^{3} - 43 a^{2} - 467 a + 213\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(-22a^{3}+11a^{2}+106a-47\right){x}+98a^{3}-43a^{2}-467a+213$ |
| 9.3-a3 |
9.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{3}, \sqrt{7})\) |
$4$ |
$[4, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{14} \) |
$9.87866$ |
$(-a-1), (-a^3+5a-1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.038707012$ |
$1179.984011$ |
1.449957322 |
\( -\frac{968192}{729} a^{3} + \frac{1936384}{243} a + \frac{2296256}{729} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{3} + 4 a + 1\) , \( a^{3} - 4 a\) , \( 3 a^{3} + a^{2} - 15 a - 4\) , \( 0\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+4a+1\right){x}^{2}+\left(3a^{3}+a^{2}-15a-4\right){x}$ |
| 9.3-a4 |
9.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{3}, \sqrt{7})\) |
$4$ |
$[4, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{10} \) |
$9.87866$ |
$(-a-1), (-a^3+5a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$0.116121037$ |
$131.1093345$ |
1.449957322 |
\( -\frac{4301312}{27} a^{3} + \frac{8602624}{9} a + \frac{11388736}{27} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{3} - a^{2} + 4 a + 2\) , \( a^{3} + a^{2} - 5 a - 3\) , \( 22 a^{3} + 10 a^{2} - 107 a - 47\) , \( 121 a^{3} + 54 a^{2} - 578 a - 263\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}+a^{2}-5a-3\right){y}={x}^{3}+\left(-a^{3}-a^{2}+4a+2\right){x}^{2}+\left(22a^{3}+10a^{2}-107a-47\right){x}+121a^{3}+54a^{2}-578a-263$ |
| 9.3-b1 |
9.3-b |
$4$ |
$6$ |
\(\Q(\sqrt{3}, \sqrt{7})\) |
$4$ |
$[4, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{14} \) |
$9.87866$ |
$(-a-1), (-a^3+5a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$0.232242074$ |
$131.1093345$ |
1.449957322 |
\( -\frac{968192}{729} a^{3} + \frac{1936384}{243} a + \frac{2296256}{729} \) |
\( \bigl[a^{3} + a^{2} - 5 a - 2\) , \( a^{3} - 5 a + 1\) , \( 1\) , \( -a^{3} - 2 a^{2} + a + 4\) , \( -a^{3} - 3 a^{2} - a + 1\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-5a-2\right){x}{y}+{y}={x}^{3}+\left(a^{3}-5a+1\right){x}^{2}+\left(-a^{3}-2a^{2}+a+4\right){x}-a^{3}-3a^{2}-a+1$ |
| 9.3-b2 |
9.3-b |
$4$ |
$6$ |
\(\Q(\sqrt{3}, \sqrt{7})\) |
$4$ |
$[4, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{10} \) |
$9.87866$ |
$(-a-1), (-a^3+5a-1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.077414024$ |
$1179.984011$ |
1.449957322 |
\( -\frac{4301312}{27} a^{3} + \frac{8602624}{9} a + \frac{11388736}{27} \) |
\( \bigl[a^{2} + a - 3\) , \( a^{2} - a - 3\) , \( 1\) , \( 22 a^{3} + 11 a^{2} - 107 a - 47\) , \( -98 a^{3} - 43 a^{2} + 467 a + 213\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+{y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(22a^{3}+11a^{2}-107a-47\right){x}-98a^{3}-43a^{2}+467a+213$ |
| 9.3-b3 |
9.3-b |
$4$ |
$6$ |
\(\Q(\sqrt{3}, \sqrt{7})\) |
$4$ |
$[4, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{14} \) |
$9.87866$ |
$(-a-1), (-a^3+5a-1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.038707012$ |
$1179.984011$ |
1.449957322 |
\( \frac{968192}{729} a^{3} - \frac{1936384}{243} a + \frac{2296256}{729} \) |
\( \bigl[a^{3} + a^{2} - 5 a - 2\) , \( a^{3} - 5 a + 1\) , \( a^{3} - 4 a\) , \( a^{3} - a^{2} - 4 a + 1\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-5a-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a^{3}-5a+1\right){x}^{2}+\left(a^{3}-a^{2}-4a+1\right){x}$ |
| 9.3-b4 |
9.3-b |
$4$ |
$6$ |
\(\Q(\sqrt{3}, \sqrt{7})\) |
$4$ |
$[4, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{10} \) |
$9.87866$ |
$(-a-1), (-a^3+5a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$0.116121037$ |
$131.1093345$ |
1.449957322 |
\( \frac{4301312}{27} a^{3} - \frac{8602624}{9} a + \frac{11388736}{27} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{2} - a + 2\) , \( a^{3} + a^{2} - 5 a - 3\) , \( -20 a^{3} + 10 a^{2} + 96 a - 47\) , \( -118 a^{3} + 54 a^{2} + 564 a - 263\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}+a^{2}-5a-3\right){y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(-20a^{3}+10a^{2}+96a-47\right){x}-118a^{3}+54a^{2}+564a-263$ |
*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.
