| 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 |
| 162.1-a1 |
162.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{2} \cdot 3^{6} \) |
$1.56179$ |
$(-a+2), (a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$39.86878607$ |
1.808484862 |
\( -\frac{132651}{2} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -3\) , \( 3\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-3{x}+3$ |
| 162.1-a2 |
162.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{18} \cdot 3^{10} \) |
$1.56179$ |
$(-a+2), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$1.476621706$ |
1.808484862 |
\( -\frac{1167051}{512} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -273 a - 666\) , \( -5520 a - 13522\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-273a-666\right){x}-5520a-13522$ |
| 162.1-a3 |
162.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{6} \cdot 3^{6} \) |
$1.56179$ |
$(-a+2), (a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$13.28959535$ |
1.808484862 |
\( \frac{9261}{8} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -26 a + 66\) , \( -54 a + 134\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-26a+66\right){x}-54a+134$ |
| 162.1-b1 |
162.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( - 2^{3} \cdot 3^{6} \) |
$1.56179$ |
$(-a+2), (a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$14.89232845$ |
3.039883816 |
\( \frac{1539}{4} a + \frac{2187}{2} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -3 a + 7\) , \( 15 a - 37\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-3a+7\right){x}+15a-37$ |
| 162.1-b2 |
162.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( - 2^{9} \cdot 3^{10} \) |
$1.56179$ |
$(-a+2), (a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$14.89232845$ |
3.039883816 |
\( \frac{146671347}{32} a + \frac{179632401}{16} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 27 a - 68\) , \( -423 a + 1037\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(27a-68\right){x}-423a+1037$ |
| 162.1-b3 |
162.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( - 2 \cdot 3^{10} \) |
$1.56179$ |
$(-a+2), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$1.654703161$ |
3.039883816 |
\( \frac{533697987}{2} a + 652741521 \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 192 a - 473\) , \( 2229 a - 5461\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(192a-473\right){x}+2229a-5461$ |
| 162.1-c1 |
162.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( - 2 \cdot 3^{10} \) |
$1.56179$ |
$(-a+2), (a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$1.654703161$ |
3.039883816 |
\( -\frac{533697987}{2} a + 652741521 \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -192 a - 473\) , \( -2229 a - 5461\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-192a-473\right){x}-2229a-5461$ |
| 162.1-c2 |
162.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( - 2^{9} \cdot 3^{10} \) |
$1.56179$ |
$(-a+2), (a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$14.89232845$ |
3.039883816 |
\( -\frac{146671347}{32} a + \frac{179632401}{16} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -27 a - 68\) , \( 423 a + 1037\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-27a-68\right){x}+423a+1037$ |
| 162.1-c3 |
162.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( - 2^{3} \cdot 3^{6} \) |
$1.56179$ |
$(-a+2), (a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$14.89232845$ |
3.039883816 |
\( -\frac{1539}{4} a + \frac{2187}{2} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 3 a + 7\) , \( -15 a - 37\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(3a+7\right){x}-15a-37$ |
| 162.1-d1 |
162.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( - 2 \cdot 3^{10} \) |
$1.56179$ |
$(-a+2), (a+3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1.437828763$ |
$24.13040967$ |
1.573815172 |
\( -\frac{533697987}{2} a + 652741521 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 46 a - 120\) , \( -327 a + 800\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(46a-120\right){x}-327a+800$ |
| 162.1-d2 |
162.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( - 2^{9} \cdot 3^{10} \) |
$1.56179$ |
$(-a+2), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1.437828763$ |
$2.681156630$ |
1.573815172 |
\( -\frac{146671347}{32} a + \frac{179632401}{16} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 31 a - 75\) , \( 117 a - 289\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(31a-75\right){x}+117a-289$ |
| 162.1-d3 |
162.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( - 2^{3} \cdot 3^{6} \) |
$1.56179$ |
$(-a+2), (a+3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$0.479276254$ |
$24.13040967$ |
1.573815172 |
\( -\frac{1539}{4} a + \frac{2187}{2} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( a\) , \( 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+a{x}+2$ |
| 162.1-e1 |
162.1-e |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( - 2^{3} \cdot 3^{6} \) |
$1.56179$ |
$(-a+2), (a+3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$0.479276254$ |
$24.13040967$ |
1.573815172 |
\( \frac{1539}{4} a + \frac{2187}{2} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 3\) , \( -1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+3{x}-1$ |
| 162.1-e2 |
162.1-e |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( - 2^{9} \cdot 3^{10} \) |
$1.56179$ |
$(-a+2), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1.437828763$ |
$2.681156630$ |
1.573815172 |
\( \frac{146671347}{32} a + \frac{179632401}{16} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -30 a - 72\) , \( -192 a - 472\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-30a-72\right){x}-192a-472$ |
| 162.1-e3 |
162.1-e |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( - 2 \cdot 3^{10} \) |
$1.56179$ |
$(-a+2), (a+3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1.437828763$ |
$24.13040967$ |
1.573815172 |
\( \frac{533697987}{2} a + 652741521 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -45 a - 117\) , \( 207 a + 527\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-45a-117\right){x}+207a+527$ |
| 162.1-f1 |
162.1-f |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{2} \cdot 3^{6} \) |
$1.56179$ |
$(-a+2), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1.015681695$ |
$3.185925848$ |
2.642090317 |
\( -\frac{132651}{2} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 64 a - 152\) , \( 378 a - 923\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(64a-152\right){x}+378a-923$ |
| 162.1-f2 |
162.1-f |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{18} \cdot 3^{10} \) |
$1.56179$ |
$(-a+2), (a+3)$ |
$1$ |
$\Z/9\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{3} \) |
$1.015681695$ |
$9.557777544$ |
2.642090317 |
\( -\frac{1167051}{512} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -14\) , \( 29\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-14{x}+29$ |
| 162.1-f3 |
162.1-f |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{6} \cdot 3^{6} \) |
$1.56179$ |
$(-a+2), (a+3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.338560565$ |
$9.557777544$ |
2.642090317 |
\( \frac{9261}{8} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 1\) , \( -1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+{x}-1$ |
*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.
