| 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 |
$8$ |
$20$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{10} \) |
$5.64495$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$285.8329267$ |
1.488713160 |
\( -\frac{320418987231990328}{27} a^{3} + \frac{1602094936159951640}{27} a + 29069000837710152 \) |
\( \bigl[a^{3} + a^{2} - 4 a - 2\) , \( a\) , \( a\) , \( -2463 a^{3} + 1265 a^{2} + 9170 a - 4780\) , \( 84852 a^{3} - 43876 a^{2} - 316598 a + 163923\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-2\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-2463a^{3}+1265a^{2}+9170a-4780\right){x}+84852a^{3}-43876a^{2}-316598a+163923$ |
| 9.1-a2 |
9.1-a |
$8$ |
$20$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \) |
$5.64495$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$285.8329267$ |
1.488713160 |
\( \frac{467951528}{3} a^{3} - \frac{2339757640}{3} a + 382083912 \) |
\( \bigl[a^{3} + a^{2} - 4 a - 2\) , \( a^{3} + a^{2} - 3 a - 3\) , \( a^{3} + a^{2} - 4 a - 1\) , \( -2 a^{3} + a^{2} + 4 a - 14\) , \( -6 a^{3} + 2 a^{2} + 24 a - 7\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-2\right){x}{y}+\left(a^{3}+a^{2}-4a-1\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-3\right){x}^{2}+\left(-2a^{3}+a^{2}+4a-14\right){x}-6a^{3}+2a^{2}+24a-7$ |
| 9.1-a3 |
9.1-a |
$8$ |
$20$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{20} \) |
$5.64495$ |
$(a^2-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2Cs, 5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$571.6658535$ |
1.488713160 |
\( \frac{58591911104}{243} \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 3 a - 1\) , \( a^{3} + a^{2} - 3 a - 2\) , \( -161 a^{3} + 482 a - 243\) , \( 1495 a^{3} - 4486 a + 2128\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}+a^{2}-3a-2\right){y}={x}^{3}+\left(-a^{3}+3a-1\right){x}^{2}+\left(-161a^{3}+482a-243\right){x}+1495a^{3}-4486a+2128$ |
| 9.1-a4 |
9.1-a |
$8$ |
$20$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$5.64495$ |
$(a^2-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2Cs, 5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$571.6658535$ |
1.488713160 |
\( \frac{85184}{3} \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 3 a - 1\) , \( a^{3} + a^{2} - 3 a - 2\) , \( -a^{3} + 2 a - 3\) , \( -a - 2\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}+a^{2}-3a-2\right){y}={x}^{3}+\left(-a^{3}+3a-1\right){x}^{2}+\left(-a^{3}+2a-3\right){x}-a-2$ |
| 9.1-a5 |
9.1-a |
$8$ |
$20$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$5.64495$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$142.9164633$ |
1.488713160 |
\( \frac{64}{9} \) |
\( \bigl[a^{3} - 3 a\) , \( a^{3} - 3 a\) , \( a^{3} + a^{2} - 3 a - 2\) , \( a^{3} - 4 a\) , \( -a - 2\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}+a^{2}-3a-2\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(a^{3}-4a\right){x}-a-2$ |
| 9.1-a6 |
9.1-a |
$8$ |
$20$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{40} \) |
$5.64495$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$142.9164633$ |
1.488713160 |
\( -\frac{873722816}{59049} \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 3 a\) , \( a^{3} + a^{2} - 3 a - 2\) , \( 40 a^{3} - 121 a - 60\) , \( -152 a^{3} + 455 a + 218\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}+a^{2}-3a-2\right){y}={x}^{3}+\left(-a^{3}+3a\right){x}^{2}+\left(40a^{3}-121a-60\right){x}-152a^{3}+455a+218$ |
| 9.1-a7 |
9.1-a |
$8$ |
$20$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \) |
$5.64495$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$285.8329267$ |
1.488713160 |
\( -\frac{467951528}{3} a^{3} + \frac{2339757640}{3} a + 382083912 \) |
\( \bigl[a + 1\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} - 4 a\) , \( -67 a^{3} + 35 a^{2} + 251 a - 129\) , \( 381 a^{3} - 197 a^{2} - 1422 a + 735\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(-67a^{3}+35a^{2}+251a-129\right){x}+381a^{3}-197a^{2}-1422a+735$ |
| 9.1-a8 |
9.1-a |
$8$ |
$20$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{10} \) |
$5.64495$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$285.8329267$ |
1.488713160 |
\( \frac{320418987231990328}{27} a^{3} - \frac{1602094936159951640}{27} a + 29069000837710152 \) |
\( \bigl[a + 1\) , \( -a^{2} + 3\) , \( a^{3} - 4 a\) , \( -372 a^{3} + 150 a^{2} + 1225 a - 878\) , \( 5455 a^{3} - 2171 a^{2} - 18929 a + 10866\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-372a^{3}+150a^{2}+1225a-878\right){x}+5455a^{3}-2171a^{2}-18929a+10866$ |
| 9.1-b1 |
9.1-b |
$8$ |
$20$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{10} \) |
$5.64495$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$25$ |
\( 1 \) |
$1$ |
$3.210189511$ |
0.417993425 |
\( \frac{320418987231990328}{27} a^{3} - \frac{1602094936159951640}{27} a + 29069000837710152 \) |
\( \bigl[a^{3} + a^{2} - 4 a - 2\) , \( a^{3} + a^{2} - 4 a - 2\) , \( a^{3} + a^{2} - 3 a - 2\) , \( -370 a^{3} + 152 a^{2} + 1216 a - 885\) , \( -5826 a^{3} + 2322 a^{2} + 20149 a - 11749\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-2\right){x}{y}+\left(a^{3}+a^{2}-3a-2\right){y}={x}^{3}+\left(a^{3}+a^{2}-4a-2\right){x}^{2}+\left(-370a^{3}+152a^{2}+1216a-885\right){x}-5826a^{3}+2322a^{2}+20149a-11749$ |
| 9.1-b2 |
9.1-b |
$8$ |
$20$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \) |
$5.64495$ |
$(a^2-2)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$2006.368444$ |
0.417993425 |
\( -\frac{467951528}{3} a^{3} + \frac{2339757640}{3} a + 382083912 \) |
\( \bigl[a^{3} + a^{2} - 4 a - 2\) , \( -a^{3} - a^{2} + 4 a + 1\) , \( a^{2} - 2\) , \( -71 a^{3} + 34 a^{2} + 265 a - 128\) , \( -251 a^{3} + 128 a^{2} + 937 a - 478\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}-a^{2}+4a+1\right){x}^{2}+\left(-71a^{3}+34a^{2}+265a-128\right){x}-251a^{3}+128a^{2}+937a-478$ |
| 9.1-b3 |
9.1-b |
$8$ |
$20$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{40} \) |
$5.64495$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$25$ |
\( 2 \) |
$1$ |
$1.605094755$ |
0.417993425 |
\( -\frac{873722816}{59049} \) |
\( \bigl[a^{3} - 3 a\) , \( a^{3} - 3 a - 1\) , \( a^{3} - 3 a + 1\) , \( 39 a^{3} - 117 a - 60\) , \( 152 a^{3} - 456 a - 220\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(a^{3}-3a-1\right){x}^{2}+\left(39a^{3}-117a-60\right){x}+152a^{3}-456a-220$ |
| 9.1-b4 |
9.1-b |
$8$ |
$20$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$5.64495$ |
$(a^2-2)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$1003.184222$ |
0.417993425 |
\( \frac{64}{9} \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 3 a - 1\) , \( a^{3} - 3 a + 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(-a^{3}+3a-1\right){x}^{2}$ |
| 9.1-b5 |
9.1-b |
$8$ |
$20$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{20} \) |
$5.64495$ |
$(a^2-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2Cs, 5B.1.2 |
$25$ |
\( 2 \) |
$1$ |
$6.420379023$ |
0.417993425 |
\( \frac{58591911104}{243} \) |
\( \bigl[a^{3} - 3 a\) , \( a^{3} - 3 a\) , \( a^{3} - 3 a + 1\) , \( -162 a^{3} + 486 a - 243\) , \( -1495 a^{3} + 4485 a - 2130\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(-162a^{3}+486a-243\right){x}-1495a^{3}+4485a-2130$ |
| 9.1-b6 |
9.1-b |
$8$ |
$20$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$5.64495$ |
$(a^2-2)$ |
0 |
$\Z/2\Z\oplus\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2Cs, 5B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$4012.736889$ |
0.417993425 |
\( \frac{85184}{3} \) |
\( \bigl[a^{3} - 3 a\) , \( a^{3} - 3 a\) , \( a^{3} - 3 a + 1\) , \( -2 a^{3} + 6 a - 3\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(-2a^{3}+6a-3\right){x}$ |
| 9.1-b7 |
9.1-b |
$8$ |
$20$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \) |
$5.64495$ |
$(a^2-2)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$2006.368444$ |
0.417993425 |
\( \frac{467951528}{3} a^{3} - \frac{2339757640}{3} a + 382083912 \) |
\( \bigl[a + 1\) , \( -a^{2} - a + 1\) , \( a^{2} + a - 1\) , \( -4 a^{3} + 9 a - 11\) , \( 6 a^{3} - 3 a^{2} - 25 a + 6\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(-4a^{3}+9a-11\right){x}+6a^{3}-3a^{2}-25a+6$ |
| 9.1-b8 |
9.1-b |
$8$ |
$20$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{10} \) |
$5.64495$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$25$ |
\( 1 \) |
$1$ |
$3.210189511$ |
0.417993425 |
\( -\frac{320418987231990328}{27} a^{3} + \frac{1602094936159951640}{27} a + 29069000837710152 \) |
\( \bigl[a + 1\) , \( a^{3} - 5 a + 1\) , \( a^{3} - 3 a + 1\) , \( -2461 a^{3} + 1265 a^{2} + 9161 a - 4781\) , \( -87314 a^{3} + 45141 a^{2} + 325764 a - 168705\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(a^{3}-5a+1\right){x}^{2}+\left(-2461a^{3}+1265a^{2}+9161a-4781\right){x}-87314a^{3}+45141a^{2}+325764a-168705$ |
*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.
