| 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 |
| 16.4-a1 |
16.4-a |
$8$ |
$16$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{4} \) |
$12.15283$ |
$(a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 1 \) |
$0.539799292$ |
$2312.344697$ |
3.244897502 |
\( 24225 a^{2} - 4222 \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{2} + a - 4\) , \( a\) , \( 3 a^{3} + 3 a^{2} - 5 a + 2\) , \( 2 a^{3} + 6 a^{2} + a - 4\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+a{y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(3a^{3}+3a^{2}-5a+2\right){x}+2a^{3}+6a^{2}+a-4$ |
| 16.4-a2 |
16.4-a |
$8$ |
$16$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{8} \) |
$12.15283$ |
$(a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1.079598584$ |
$578.0861742$ |
3.244897502 |
\( 2701312025 a^{2} - 1184382322 \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{2} + a - 4\) , \( a\) , \( 3 a^{3} - 7 a^{2} - 5 a + 7\) , \( -8 a^{3} + 16 a^{2} + 6 a - 9\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+a{y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(3a^{3}-7a^{2}-5a+7\right){x}-8a^{3}+16a^{2}+6a-9$ |
| 16.4-a3 |
16.4-a |
$8$ |
$16$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{10} \) |
$12.15283$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.134949823$ |
$578.0861742$ |
3.244897502 |
\( 343 a^{2} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{2} + a - 3\) , \( a^{3} + a^{2} - 4 a - 2\) , \( 4 a^{3} + 6 a^{2} - 9 a - 4\) , \( 5 a^{3} + 10 a^{2} - 4 a - 7\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{3}+a^{2}-4a-2\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(4a^{3}+6a^{2}-9a-4\right){x}+5a^{3}+10a^{2}-4a-7$ |
| 16.4-a4 |
16.4-a |
$8$ |
$16$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{11} \) |
$12.15283$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.079598584$ |
$72.26077178$ |
3.244897502 |
\( -1409277718922882 a^{3} - 933097728375808 a^{2} + 6428663656193677 a + 4256735329251048 \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{2} + a + 2\) , \( a\) , \( 424 a^{3} + 319 a^{2} - 1933 a - 1447\) , \( -7799 a^{3} - 5343 a^{2} + 35576 a + 24369\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(424a^{3}+319a^{2}-1933a-1447\right){x}-7799a^{3}-5343a^{2}+35576a+24369$ |
| 16.4-a5 |
16.4-a |
$8$ |
$16$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{11} \) |
$12.15283$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.079598584$ |
$72.26077178$ |
3.244897502 |
\( 1409277718922882 a^{3} - 933097728375808 a^{2} - 6428663656193677 a + 4256735329251048 \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{2} + a + 2\) , \( a\) , \( -426 a^{3} + 319 a^{2} + 1947 a - 1447\) , \( 8435 a^{3} - 5713 a^{2} - 38472 a + 26069\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(-426a^{3}+319a^{2}+1947a-1447\right){x}+8435a^{3}-5713a^{2}-38472a+26069$ |
| 16.4-a6 |
16.4-a |
$8$ |
$16$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{10} \) |
$12.15283$ |
$(a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.539799292$ |
$578.0861742$ |
3.244897502 |
\( -21069823 a^{2} + 96961280 \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{2} + a + 2\) , \( a\) , \( -a^{3} + 39 a^{2} + 7 a - 167\) , \( 38 a^{3} - 142 a^{2} - 168 a + 655\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(-a^{3}+39a^{2}+7a-167\right){x}+38a^{3}-142a^{2}-168a+655$ |
| 16.4-a7 |
16.4-a |
$8$ |
$16$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{8} \) |
$12.15283$ |
$(a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.269899646$ |
$2312.344697$ |
3.244897502 |
\( -1995 a^{2} + 11006 \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{2} + a + 2\) , \( a\) , \( -a^{3} + 4 a^{2} + 7 a - 7\) , \( 3 a^{3} + a^{2} - 8 a + 3\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(-a^{3}+4a^{2}+7a-7\right){x}+3a^{3}+a^{2}-8a+3$ |
| 16.4-a8 |
16.4-a |
$8$ |
$16$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{8} \) |
$12.15283$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1.079598584$ |
$289.0430871$ |
3.244897502 |
\( -7659605 a^{2} + 34939686 \) |
\( \bigl[a^{3} - 4 a\) , \( a^{2} - 2\) , \( 0\) , \( 5 a^{2} - 20\) , \( 7 a^{2} - 31\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(5a^{2}-20\right){x}+7a^{2}-31$ |
| 16.4-b1 |
16.4-b |
$8$ |
$16$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{8} \) |
$12.15283$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1.963122304$ |
$85.28948525$ |
1.741080855 |
\( -7659605 a^{2} + 34939686 \) |
\( \bigl[a^{2} - 2\) , \( -1\) , \( a^{2} - 2\) , \( -a^{2} - 1\) , \( -6 a^{2} + 2\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}-{x}^{2}+\left(-a^{2}-1\right){x}-6a^{2}+2$ |
| 16.4-b2 |
16.4-b |
$8$ |
$16$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{4} \) |
$12.15283$ |
$(a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 1 \) |
$0.981561152$ |
$682.3158820$ |
1.741080855 |
\( 24225 a^{2} - 4222 \) |
\( \bigl[a\) , \( a^{2} - 2\) , \( a\) , \( 4 a^{2} - 16\) , \( 3 a^{2} - 13\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(4a^{2}-16\right){x}+3a^{2}-13$ |
| 16.4-b3 |
16.4-b |
$8$ |
$16$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{8} \) |
$12.15283$ |
$(a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1.963122304$ |
$42.64474262$ |
1.741080855 |
\( 2701312025 a^{2} - 1184382322 \) |
\( \bigl[a\) , \( a^{2} - 2\) , \( a\) , \( -6 a^{2} + 29\) , \( 28 a^{2} - 128\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(-6a^{2}+29\right){x}+28a^{2}-128$ |
| 16.4-b4 |
16.4-b |
$8$ |
$16$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{11} \) |
$12.15283$ |
$(a)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1.963122304$ |
$682.3158820$ |
1.741080855 |
\( -1409277718922882 a^{3} - 933097728375808 a^{2} + 6428663656193677 a + 4256735329251048 \) |
\( \bigl[a\) , \( -a^{2} + 2\) , \( a\) , \( -15 a^{3} - 46 a^{2} - 20 a + 1\) , \( 166 a^{3} + 392 a^{2} - 22 a - 138\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-15a^{3}-46a^{2}-20a+1\right){x}+166a^{3}+392a^{2}-22a-138$ |
| 16.4-b5 |
16.4-b |
$8$ |
$16$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{10} \) |
$12.15283$ |
$(a)$ |
$1$ |
$\Z/2\Z\oplus\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.981561152$ |
$2729.263528$ |
1.741080855 |
\( -21069823 a^{2} + 96961280 \) |
\( \bigl[a\) , \( -a^{2} + 2\) , \( a\) , \( -6 a^{2} + 1\) , \( 14 a^{2} - 6\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-6a^{2}+1\right){x}+14a^{2}-6$ |
| 16.4-b6 |
16.4-b |
$8$ |
$16$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{8} \) |
$12.15283$ |
$(a)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.490780576$ |
$2729.263528$ |
1.741080855 |
\( -1995 a^{2} + 11006 \) |
\( \bigl[a\) , \( -a^{2} + 2\) , \( a\) , \( -a^{2} + 1\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-a^{2}+1\right){x}$ |
| 16.4-b7 |
16.4-b |
$8$ |
$16$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{11} \) |
$12.15283$ |
$(a)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1.963122304$ |
$682.3158820$ |
1.741080855 |
\( 1409277718922882 a^{3} - 933097728375808 a^{2} - 6428663656193677 a + 4256735329251048 \) |
\( \bigl[a\) , \( -a^{2} + 2\) , \( a\) , \( 15 a^{3} - 46 a^{2} + 20 a + 1\) , \( -166 a^{3} + 392 a^{2} + 22 a - 138\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(15a^{3}-46a^{2}+20a+1\right){x}-166a^{3}+392a^{2}+22a-138$ |
| 16.4-b8 |
16.4-b |
$8$ |
$16$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{10} \) |
$12.15283$ |
$(a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.245390288$ |
$682.3158820$ |
1.741080855 |
\( 343 a^{2} \) |
\( \bigl[a\) , \( -a^{2} + 4\) , \( a\) , \( -2 a^{2} + 7\) , \( -2 a^{2} + 8\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-2a^{2}+7\right){x}-2a^{2}+8$ |
*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.
