| 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 |
| 49.4-a1 |
49.4-a |
$2$ |
$7$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
49.4 |
\( 7^{2} \) |
\( - 7^{2} \) |
$9.42532$ |
$(-a^3+2a^2+3a-3)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$7$ |
7B.1.3 |
|
\( 1 \) |
$1$ |
$3.842943563$ |
2.266172452 |
\( -156521478494 a^{3} + 257550722470 a^{2} + 616417035615 a - 241680788762 \) |
\( \bigl[a^{2} - a - 2\) , \( -2 a^{3} + 3 a^{2} + 9 a - 1\) , \( a\) , \( -24 a^{2} - 24 a + 1\) , \( -50 a^{3} - 134 a^{2} - 61 a\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+a{y}={x}^{3}+\left(-2a^{3}+3a^{2}+9a-1\right){x}^{2}+\left(-24a^{2}-24a+1\right){x}-50a^{3}-134a^{2}-61a$ |
| 49.4-a2 |
49.4-a |
$2$ |
$7$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
49.4 |
\( 7^{2} \) |
\( - 7^{2} \) |
$9.42532$ |
$(-a^3+2a^2+3a-3)$ |
$0 \le r \le 1$ |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$7$ |
7B.1.1 |
|
\( 1 \) |
$1$ |
$1318.129642$ |
2.266172452 |
\( -236043 a^{3} + 152940 a^{2} + 1233230 a + 670892 \) |
\( \bigl[-a^{3} + 2 a^{2} + 4 a - 1\) , \( a - 1\) , \( -a^{3} + 2 a^{2} + 4 a\) , \( 5 a^{3} - 8 a^{2} - 20 a + 7\) , \( -8 a^{3} + 13 a^{2} + 32 a - 13\bigr] \) |
${y}^2+\left(-a^{3}+2a^{2}+4a-1\right){x}{y}+\left(-a^{3}+2a^{2}+4a\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(5a^{3}-8a^{2}-20a+7\right){x}-8a^{3}+13a^{2}+32a-13$ |
| 49.4-b1 |
49.4-b |
$4$ |
$15$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
49.4 |
\( 7^{2} \) |
\( 7^{6} \) |
$9.42532$ |
$(-a^3+2a^2+3a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3, 5$ |
3B, 5B.4.1 |
$1$ |
\( 2 \) |
$0.272017818$ |
$53.18635999$ |
1.784861860 |
\( -1407628760845 a^{3} + 1407628760845 a^{2} + 8445772565070 a - 3086342051803 \) |
\( \bigl[a^{3} - a^{2} - 5 a - 1\) , \( 2 a^{3} - 3 a^{2} - 8 a + 1\) , \( a^{3} - a^{2} - 5 a\) , \( -18 a^{3} + 32 a^{2} + 67 a - 43\) , \( -155 a^{3} + 223 a^{2} + 667 a - 71\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-1\right){x}{y}+\left(a^{3}-a^{2}-5a\right){y}={x}^{3}+\left(2a^{3}-3a^{2}-8a+1\right){x}^{2}+\left(-18a^{3}+32a^{2}+67a-43\right){x}-155a^{3}+223a^{2}+667a-71$ |
| 49.4-b2 |
49.4-b |
$4$ |
$15$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
49.4 |
\( 7^{2} \) |
\( 7^{6} \) |
$9.42532$ |
$(-a^3+2a^2+3a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3, 5$ |
3B, 5B.4.1 |
$1$ |
\( 2 \) |
$0.090672606$ |
$159.5590799$ |
1.784861860 |
\( -3515 a^{3} + 3515 a^{2} + 21090 a - 7688 \) |
\( \bigl[a^{3} - a^{2} - 5 a - 1\) , \( 2 a^{3} - 3 a^{2} - 8 a + 1\) , \( a^{3} - a^{2} - 5 a\) , \( 2 a^{3} - 3 a^{2} - 8 a + 2\) , \( 2 a^{3} - 2 a^{2} - 10 a - 4\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a-1\right){x}{y}+\left(a^{3}-a^{2}-5a\right){y}={x}^{3}+\left(2a^{3}-3a^{2}-8a+1\right){x}^{2}+\left(2a^{3}-3a^{2}-8a+2\right){x}+2a^{3}-2a^{2}-10a-4$ |
| 49.4-b3 |
49.4-b |
$4$ |
$15$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
49.4 |
\( 7^{2} \) |
\( 7^{6} \) |
$9.42532$ |
$(-a^3+2a^2+3a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3, 5$ |
3B, 5B.4.2 |
$1$ |
\( 2 \) |
$0.054403563$ |
$265.9317999$ |
1.784861860 |
\( 1407628760845 a^{3} - 1407628760845 a^{2} - 8445772565070 a - 4493970812648 \) |
\( \bigl[-a^{3} + 2 a^{2} + 4 a - 1\) , \( -a + 1\) , \( a + 1\) , \( -17 a^{3} + 30 a^{2} + 61 a - 40\) , \( -661 a^{3} + 1087 a^{2} + 2599 a - 1002\bigr] \) |
${y}^2+\left(-a^{3}+2a^{2}+4a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-17a^{3}+30a^{2}+61a-40\right){x}-661a^{3}+1087a^{2}+2599a-1002$ |
| 49.4-b4 |
49.4-b |
$4$ |
$15$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
49.4 |
\( 7^{2} \) |
\( 7^{6} \) |
$9.42532$ |
$(-a^3+2a^2+3a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3, 5$ |
3B, 5B.4.2 |
$1$ |
\( 2 \) |
$0.018134521$ |
$797.7953998$ |
1.784861860 |
\( 3515 a^{3} - 3515 a^{2} - 21090 a - 11203 \) |
\( \bigl[-a^{3} + 2 a^{2} + 4 a - 1\) , \( -a + 1\) , \( a + 1\) , \( 3 a^{3} - 5 a^{2} - 14 a + 5\) , \( 26 a^{3} - 43 a^{2} - 103 a + 40\bigr] \) |
${y}^2+\left(-a^{3}+2a^{2}+4a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(3a^{3}-5a^{2}-14a+5\right){x}+26a^{3}-43a^{2}-103a+40$ |
| 49.4-c1 |
49.4-c |
$2$ |
$7$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
49.4 |
\( 7^{2} \) |
\( - 7^{8} \) |
$9.42532$ |
$(-a^3+2a^2+3a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$7$ |
7B.6.3 |
$1$ |
\( 1 \) |
$1$ |
$85.38104876$ |
1.316674682 |
\( -156521478494 a^{3} + 257550722470 a^{2} + 616417035615 a - 241680788762 \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( -2 a^{3} + 3 a^{2} + 8 a + 1\) , \( a + 1\) , \( 66 a^{3} - 37 a^{2} - 364 a - 233\) , \( -457 a^{3} + 318 a^{2} + 2384 a + 1238\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-2a^{3}+3a^{2}+8a+1\right){x}^{2}+\left(66a^{3}-37a^{2}-364a-233\right){x}-457a^{3}+318a^{2}+2384a+1238$ |
| 49.4-c2 |
49.4-c |
$2$ |
$7$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
49.4 |
\( 7^{2} \) |
\( - 7^{8} \) |
$9.42532$ |
$(-a^3+2a^2+3a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$7$ |
7B.6.1 |
$1$ |
\( 1 \) |
$1$ |
$85.38104876$ |
1.316674682 |
\( -236043 a^{3} + 152940 a^{2} + 1233230 a + 670892 \) |
\( \bigl[a\) , \( 2 a^{3} - 3 a^{2} - 8 a\) , \( a^{3} - a^{2} - 4 a\) , \( 34 a^{3} - 57 a^{2} - 134 a + 56\) , \( -82 a^{3} + 135 a^{2} + 322 a - 128\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-a^{2}-4a\right){y}={x}^{3}+\left(2a^{3}-3a^{2}-8a\right){x}^{2}+\left(34a^{3}-57a^{2}-134a+56\right){x}-82a^{3}+135a^{2}+322a-128$ |
| 49.4-d1 |
49.4-d |
$4$ |
$10$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
49.4 |
\( 7^{2} \) |
\( - 7^{7} \) |
$9.42532$ |
$(-a^3+2a^2+3a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2^{2} \) |
$0.083197706$ |
$478.3046663$ |
2.454669158 |
\( \frac{2677817}{7} a^{3} - \frac{117781460}{7} a^{2} - \frac{145844186}{7} a + \frac{65824690}{7} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( a^{3} - a^{2} - 6 a\) , \( a^{2} - 2 a - 1\) , \( -2 a^{3} + 3 a^{2} + 6 a - 4\) , \( 3 a^{3} - 6 a^{2} - 5 a + 2\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{2}-2a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a\right){x}^{2}+\left(-2a^{3}+3a^{2}+6a-4\right){x}+3a^{3}-6a^{2}-5a+2$ |
| 49.4-d2 |
49.4-d |
$4$ |
$10$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
49.4 |
\( 7^{2} \) |
\( - 7^{8} \) |
$9.42532$ |
$(-a^3+2a^2+3a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2^{2} \) |
$0.166395413$ |
$239.1523331$ |
2.454669158 |
\( \frac{40225241338678555}{49} a^{3} + \frac{74036012651745535}{49} a^{2} + \frac{9175766779869835}{49} a - \frac{14161143749601521}{49} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( a^{3} - a^{2} - 6 a\) , \( a^{2} - 2 a - 1\) , \( 8 a^{3} - 32 a^{2} - 14 a + 1\) , \( 70 a^{3} - 124 a^{2} - 57 a + 38\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{2}-2a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a\right){x}^{2}+\left(8a^{3}-32a^{2}-14a+1\right){x}+70a^{3}-124a^{2}-57a+38$ |
| 49.4-d3 |
49.4-d |
$4$ |
$10$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
49.4 |
\( 7^{2} \) |
\( - 7^{11} \) |
$9.42532$ |
$(-a^3+2a^2+3a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{2} \) |
$0.415988534$ |
$95.66093327$ |
2.454669158 |
\( \frac{883009024}{16807} a^{3} - \frac{1375719561}{16807} a^{2} - \frac{3570496060}{16807} a + \frac{1021377428}{16807} \) |
\( \bigl[-a^{3} + 2 a^{2} + 3 a - 1\) , \( a^{2} - 2 a - 2\) , \( a^{2} - a - 2\) , \( -a^{3} + 3 a^{2} + 5 a\) , \( 3 a^{2} - 2\bigr] \) |
${y}^2+\left(-a^{3}+2a^{2}+3a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-2a-2\right){x}^{2}+\left(-a^{3}+3a^{2}+5a\right){x}+3a^{2}-2$ |
| 49.4-d4 |
49.4-d |
$4$ |
$10$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
49.4 |
\( 7^{2} \) |
\( - 7^{16} \) |
$9.42532$ |
$(-a^3+2a^2+3a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{2} \) |
$0.831977069$ |
$47.83046663$ |
2.454669158 |
\( -\frac{30631833226798148}{282475249} a^{3} + \frac{19913145044553620}{282475249} a^{2} + \frac{160003945294303955}{282475249} a + \frac{86891270576754902}{282475249} \) |
\( \bigl[-a^{3} + 2 a^{2} + 3 a - 1\) , \( a^{2} - 2 a - 2\) , \( a^{2} - a - 2\) , \( 9 a^{3} - 32 a^{2} - 15 a + 5\) , \( -17 a^{3} + 65 a^{2} + 23 a - 18\bigr] \) |
${y}^2+\left(-a^{3}+2a^{2}+3a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-2a-2\right){x}^{2}+\left(9a^{3}-32a^{2}-15a+5\right){x}-17a^{3}+65a^{2}+23a-18$ |
*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.
