| 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 |
| 1.1-a1 |
1.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{497}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$1.99213$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-7$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$71$ |
71Ns.2.1 |
$9$ |
\( 1 \) |
$1$ |
$26.16385905$ |
2.640621315 |
\( -3375 \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 33695 a - 392437\) , \( 13813271 a - 160879692\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(33695a-392437\right){x}+13813271a-160879692$ |
| 1.1-a2 |
1.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{497}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$1.99213$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-7$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$71$ |
71Ns.2.1 |
$9$ |
\( 1 \) |
$1$ |
$26.16385905$ |
2.640621315 |
\( -3375 \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -33695 a - 358742\) , \( -13813271 a - 147066421\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-33695a-358742\right){x}-13813271a-147066421$ |
| 1.1-a3 |
1.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{497}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$1.99213$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-28$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$71$ |
71Ns.2.1 |
$9$ |
\( 1 \) |
$1$ |
$26.16385905$ |
2.640621315 |
\( 16581375 \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 572815 a - 6671432\) , \( 787693397 a - 9174066815\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(572815a-6671432\right){x}+787693397a-9174066815$ |
| 1.1-a4 |
1.1-a |
$4$ |
$14$ |
\(\Q(\sqrt{497}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$1.99213$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-28$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$71$ |
71Ns.2.1 |
$9$ |
\( 1 \) |
$1$ |
$26.16385905$ |
2.640621315 |
\( 16581375 \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -572815 a - 6098617\) , \( -787693397 a - 8386373418\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-572815a-6098617\right){x}-787693397a-8386373418$ |
| 4.1-a1 |
4.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{497}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{18} \) |
$2.81729$ |
$(17a-198), (17a+181)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \cdot 5 \) |
$2.001658716$ |
$16.49789458$ |
4.937636413 |
\( \frac{2663311}{32768} a + \frac{21913717}{8192} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( -4420590 a - 47064874\) , \( -12231406379 a - 130224706396\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-4420590a-47064874\right){x}-12231406379a-130224706396$ |
| 4.1-a2 |
4.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{497}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{14} \) |
$2.81729$ |
$(17a-198), (17a+181)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 5 \) |
$6.004976148$ |
$1.833099397$ |
4.937636413 |
\( \frac{496068661001}{512} a + \frac{5518116702491}{512} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( -329759135 a - 3510862509\) , \( -10883902199851 a - 115878168384348\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-329759135a-3510862509\right){x}-10883902199851a-115878168384348$ |
| 4.1-b1 |
4.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{497}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{18} \) |
$2.81729$ |
$(17a-198), (17a+181)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \cdot 5 \) |
$0.173838151$ |
$22.06412917$ |
5.161488303 |
\( -\frac{2663311}{32768} a + \frac{90318179}{32768} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( -13672 a - 145557\) , \( 611456 a + 6509995\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-13672a-145557\right){x}+611456a+6509995$ |
| 4.1-b2 |
4.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{497}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{14} \) |
$2.81729$ |
$(17a-198), (17a+181)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 5 \) |
$0.521514455$ |
$22.06412917$ |
5.161488303 |
\( -\frac{496068661001}{512} a + \frac{1503546340873}{128} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( -842727 a - 8972297\) , \( 1405587529 a + 14964936759\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-842727a-8972297\right){x}+1405587529a+14964936759$ |
| 4.1-c1 |
4.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{497}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{18} \) |
$2.81729$ |
$(17a-198), (17a+181)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \cdot 5 \) |
$2.001658716$ |
$16.49789458$ |
4.937636413 |
\( -\frac{2663311}{32768} a + \frac{90318179}{32768} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 4420590 a - 51485464\) , \( 12231406379 a - 142456112775\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(4420590a-51485464\right){x}+12231406379a-142456112775$ |
| 4.1-c2 |
4.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{497}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{14} \) |
$2.81729$ |
$(17a-198), (17a+181)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 5 \) |
$6.004976148$ |
$1.833099397$ |
4.937636413 |
\( -\frac{496068661001}{512} a + \frac{1503546340873}{128} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 329759135 a - 3840621644\) , \( 10883902199851 a - 126762070584199\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(329759135a-3840621644\right){x}+10883902199851a-126762070584199$ |
| 4.1-d1 |
4.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{497}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{18} \) |
$2.81729$ |
$(17a-198), (17a+181)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \cdot 5 \) |
$0.173838151$ |
$22.06412917$ |
5.161488303 |
\( \frac{2663311}{32768} a + \frac{21913717}{8192} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( 13671 a - 159228\) , \( -611457 a + 7121452\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(13671a-159228\right){x}-611457a+7121452$ |
| 4.1-d2 |
4.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{497}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{14} \) |
$2.81729$ |
$(17a-198), (17a+181)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 5 \) |
$0.521514455$ |
$22.06412917$ |
5.161488303 |
\( \frac{496068661001}{512} a + \frac{5518116702491}{512} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( 842726 a - 9815023\) , \( -1405587530 a + 16370524289\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(842726a-9815023\right){x}-1405587530a+16370524289$ |
*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.
