| 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 |
$1$ |
$1$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{14} \cdot 3^{4} \) |
$2.46941$ |
$(2,a+1), (3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$8.070158351$ |
2.083705926 |
\( -\frac{446631}{128} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -24 a - 93\) , \( -244 a - 945\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-24a-93\right){x}-244a-945$ |
| 162.1-b1 |
162.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{2} \cdot 3^{6} \) |
$2.46941$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$39.86878607$ |
1.143786255 |
\( -\frac{132651}{2} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -3\) , \( 3\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-3{x}+3$ |
| 162.1-b2 |
162.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{30} \cdot 3^{10} \) |
$2.46941$ |
$(2,a+1), (3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$1.476621706$ |
1.143786255 |
\( -\frac{1167051}{512} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -436 a - 1691\) , \( -14309 a - 55421\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-436a-1691\right){x}-14309a-55421$ |
| 162.1-b3 |
162.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{18} \cdot 3^{6} \) |
$2.46941$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$13.28959535$ |
1.143786255 |
\( \frac{9261}{8} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 44 a + 169\) , \( 331 a + 1279\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(44a+169\right){x}+331a+1279$ |
| 162.1-c1 |
162.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{2} \cdot 3^{6} \) |
$2.46941$ |
$(2,a+1), (3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$3.185925848$ |
2.467807550 |
\( -\frac{132651}{2} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -6\) , \( -8\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-6{x}-8$ |
| 162.1-c2 |
162.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{30} \cdot 3^{10} \) |
$2.46941$ |
$(2,a+1), (3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$9.557777544$ |
2.467807550 |
\( -\frac{1167051}{512} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -435 a - 1683\) , \( 11742 a + 45477\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-435a-1683\right){x}+11742a+45477$ |
| 162.1-c3 |
162.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{18} \cdot 3^{6} \) |
$2.46941$ |
$(2,a+1), (3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$1$ |
$9.557777544$ |
2.467807550 |
\( \frac{9261}{8} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 45 a + 177\) , \( -78 a - 303\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(45a+177\right){x}-78a-303$ |
| 162.1-d1 |
162.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{26} \cdot 3^{4} \) |
$2.46941$ |
$(2,a+1), (3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 2 \cdot 7 \) |
$0.151355205$ |
$8.070158351$ |
4.415316328 |
\( -\frac{446631}{128} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -2 a - 17\) , \( 7 a - 15\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2a-17\right){x}+7a-15$ |
| 162.1-e1 |
162.1-e |
$3$ |
$9$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{14} \cdot 3^{6} \) |
$2.46941$ |
$(2,a+1), (3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$0.077107140$ |
$39.86878607$ |
4.762480698 |
\( -\frac{132651}{2} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -99 a - 381\) , \( 822 a + 3183\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-99a-381\right){x}+822a+3183$ |
| 162.1-e2 |
162.1-e |
$3$ |
$9$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{18} \cdot 3^{10} \) |
$2.46941$ |
$(2,a+1), (3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.693964260$ |
$1.476621706$ |
4.762480698 |
\( -\frac{1167051}{512} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -6789 a - 26294\) , \( -792843 a - 3070668\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-6789a-26294\right){x}-792843a-3070668$ |
| 162.1-e3 |
162.1-e |
$3$ |
$9$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{6} \cdot 3^{6} \) |
$2.46941$ |
$(2,a+1), (3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3Cs |
$1$ |
\( 2 \cdot 3 \) |
$0.231321420$ |
$13.28959535$ |
4.762480698 |
\( \frac{9261}{8} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 651 a + 2521\) , \( 12327 a + 47742\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(651a+2521\right){x}+12327a+47742$ |
| 162.1-f1 |
162.1-f |
$3$ |
$9$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{14} \cdot 3^{6} \) |
$2.46941$ |
$(2,a+1), (3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$2.936556370$ |
$3.185925848$ |
4.831237322 |
\( -\frac{132651}{2} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -100 a - 389\) , \( -1415 a - 5483\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-100a-389\right){x}-1415a-5483$ |
| 162.1-f2 |
162.1-f |
$3$ |
$9$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{18} \cdot 3^{10} \) |
$2.46941$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/9\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{3} \) |
$2.936556370$ |
$9.557777544$ |
4.831237322 |
\( -\frac{1167051}{512} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -6789 a - 26289\) , \( 786054 a + 3044376\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-6789a-26289\right){x}+786054a+3044376$ |
| 162.1-f3 |
162.1-f |
$3$ |
$9$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{6} \cdot 3^{6} \) |
$2.46941$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.978852123$ |
$9.557777544$ |
4.831237322 |
\( \frac{9261}{8} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 651 a + 2526\) , \( -11676 a - 45219\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(651a+2526\right){x}-11676a-45219$ |
| 162.1-g1 |
162.1-g |
$1$ |
$1$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{14} \cdot 3^{4} \) |
$2.46941$ |
$(2,a+1), (3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$8.070158351$ |
2.083705926 |
\( -\frac{446631}{128} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( 24 a - 93\) , \( 244 a - 945\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(24a-93\right){x}+244a-945$ |
| 162.1-h1 |
162.1-h |
$1$ |
$1$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{26} \cdot 3^{4} \) |
$2.46941$ |
$(2,a+1), (3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 2 \cdot 7 \) |
$0.151355205$ |
$8.070158351$ |
4.415316328 |
\( -\frac{446631}{128} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -17\) , \( -8 a - 15\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}-17{x}-8a-15$ |
*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.
