| 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-a2 |
1.1-a |
$2$ |
$3$ |
4.4.14013.1 |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$10.57801$ |
$\textsf{none}$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$1046.592732$ |
0.982357678 |
\( -211822760 a^{3} + 290798826 a^{2} + 1162516086 a - 1704375723 \) |
\( \bigl[a^{3} + a^{2} - 5 a - 1\) , \( a - 1\) , \( 1\) , \( -a^{2} + 3 a + 1\) , \( -a^{3} + a^{2} + a\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-5a-1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-a^{2}+3a+1\right){x}-a^{3}+a^{2}+a$ |
| 1.1-b2 |
1.1-b |
$2$ |
$3$ |
4.4.14013.1 |
$4$ |
$[4, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$10.57801$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$199.7789155$ |
1.687656631 |
\( -211822760 a^{3} + 290798826 a^{2} + 1162516086 a - 1704375723 \) |
\( \bigl[a^{3} - 5 a + 2\) , \( -a^{2} + 2\) , \( 1\) , \( -a^{2} - a + 7\) , \( -a^{2} + 4\bigr] \) |
${y}^2+\left(a^{3}-5a+2\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-a^{2}-a+7\right){x}-a^{2}+4$ |
| 49.3-c2 |
49.3-c |
$2$ |
$3$ |
4.4.14013.1 |
$4$ |
$[4, 0]$ |
49.3 |
\( 7^{2} \) |
\( 7^{6} \) |
$17.20595$ |
$(2a^3+2a^2-7a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 2 \) |
$2.736485742$ |
$39.18604650$ |
7.246846547 |
\( -211822760 a^{3} + 290798826 a^{2} + 1162516086 a - 1704375723 \) |
\( \bigl[a^{3} - 5 a + 1\) , \( -a^{3} + a^{2} + 5 a - 4\) , \( a^{2} - 3\) , \( a^{3} - 11 a^{2} + 13 a + 14\) , \( 12 a^{3} - 46 a^{2} + 35 a + 17\bigr] \) |
${y}^2+\left(a^{3}-5a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-4\right){x}^{2}+\left(a^{3}-11a^{2}+13a+14\right){x}+12a^{3}-46a^{2}+35a+17$ |
| 49.3-j1 |
49.3-j |
$2$ |
$3$ |
4.4.14013.1 |
$4$ |
$[4, 0]$ |
49.3 |
\( 7^{2} \) |
\( 7^{6} \) |
$17.20595$ |
$(2a^3+2a^2-7a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 2 \) |
$0.045869643$ |
$348.9883410$ |
1.081834198 |
\( -211822760 a^{3} + 290798826 a^{2} + 1162516086 a - 1704375723 \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} - a + 3\) , \( a^{2} + a - 3\) , \( -33 a^{3} - 18 a^{2} + 171 a + 65\) , \( -221 a^{3} - 119 a^{2} + 1143 a + 430\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(-33a^{3}-18a^{2}+171a+65\right){x}-221a^{3}-119a^{2}+1143a+430$ |
| 81.3-c2 |
81.3-c |
$2$ |
$3$ |
4.4.14013.1 |
$4$ |
$[4, 0]$ |
81.3 |
\( 3^{4} \) |
\( 3^{12} \) |
$18.32166$ |
$(-a^3+5a), (-a^3-a^2+4a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$236.0548787$ |
1.994102233 |
\( -211822760 a^{3} + 290798826 a^{2} + 1162516086 a - 1704375723 \) |
\( \bigl[a\) , \( a^{3} + a^{2} - 6 a - 3\) , \( a^{3} + a^{2} - 4 a - 2\) , \( -2 a^{3} - 12 a^{2} + 37 a - 9\) , \( 33 a^{3} - 96 a^{2} + 17 a + 73\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}+a^{2}-4a-2\right){y}={x}^{3}+\left(a^{3}+a^{2}-6a-3\right){x}^{2}+\left(-2a^{3}-12a^{2}+37a-9\right){x}+33a^{3}-96a^{2}+17a+73$ |
| 81.3-h2 |
81.3-h |
$2$ |
$3$ |
4.4.14013.1 |
$4$ |
$[4, 0]$ |
81.3 |
\( 3^{4} \) |
\( 3^{12} \) |
$18.32166$ |
$(-a^3+5a), (-a^3-a^2+4a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$45.05934947$ |
3.425798902 |
\( -211822760 a^{3} + 290798826 a^{2} + 1162516086 a - 1704375723 \) |
\( \bigl[a^{3} - 5 a + 1\) , \( a - 1\) , \( a^{3} - 5 a + 2\) , \( -11 a^{3} + 15 a^{2} + 60 a - 88\) , \( -43 a^{3} + 59 a^{2} + 236 a - 346\bigr] \) |
${y}^2+\left(a^{3}-5a+1\right){x}{y}+\left(a^{3}-5a+2\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-11a^{3}+15a^{2}+60a-88\right){x}-43a^{3}+59a^{2}+236a-346$ |
*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.
