| 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 |
| 49.2-a1 |
49.2-a |
$2$ |
$3$ |
6.6.592661.1 |
$6$ |
$[6, 0]$ |
49.2 |
\( 7^{2} \) |
\( 7^{6} \) |
$95.14619$ |
$(a^4-4a^2+a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 2 \) |
$0.013940662$ |
$12990.64016$ |
2.82288 |
\( 24164258357939329603 a^{5} - 32090567672134595043 a^{4} - 110295011211469229354 a^{3} + 132835771594028505313 a^{2} + 77248781181165510499 a - 73667499983669418847 \) |
\( \bigl[a^{5} - a^{4} - 5 a^{3} + 5 a^{2} + 4 a - 3\) , \( a^{5} - a^{4} - 4 a^{3} + 4 a^{2} + 2 a - 2\) , \( a\) , \( 9 a^{5} + 10 a^{4} - 36 a^{3} - 43 a^{2} + 3 a - 6\) , \( -49 a^{5} - 33 a^{4} + 208 a^{3} + 164 a^{2} - 53 a - 19\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-5a^{3}+5a^{2}+4a-3\right){x}{y}+a{y}={x}^{3}+\left(a^{5}-a^{4}-4a^{3}+4a^{2}+2a-2\right){x}^{2}+\left(9a^{5}+10a^{4}-36a^{3}-43a^{2}+3a-6\right){x}-49a^{5}-33a^{4}+208a^{3}+164a^{2}-53a-19$ |
| 49.2-a2 |
49.2-a |
$2$ |
$3$ |
6.6.592661.1 |
$6$ |
$[6, 0]$ |
49.2 |
\( 7^{2} \) |
\( 7^{6} \) |
$95.14619$ |
$(a^4-4a^2+a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 2 \) |
$0.004646887$ |
$38971.92049$ |
2.82288 |
\( 1285592 a^{5} - 1709472 a^{4} - 5867135 a^{3} + 7077626 a^{2} + 4106851 a - 3927865 \) |
\( \bigl[a^{5} - a^{4} - 5 a^{3} + 5 a^{2} + 4 a - 3\) , \( a^{5} - a^{4} - 4 a^{3} + 4 a^{2} + 2 a - 2\) , \( a\) , \( -a^{5} + 4 a^{3} + 2 a^{2} - 2 a - 1\) , \( 0\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-5a^{3}+5a^{2}+4a-3\right){x}{y}+a{y}={x}^{3}+\left(a^{5}-a^{4}-4a^{3}+4a^{2}+2a-2\right){x}^{2}+\left(-a^{5}+4a^{3}+2a^{2}-2a-1\right){x}$ |
| 49.2-b1 |
49.2-b |
$2$ |
$7$ |
6.6.592661.1 |
$6$ |
$[6, 0]$ |
49.2 |
\( 7^{2} \) |
\( - 7^{8} \) |
$95.14619$ |
$(a^4-4a^2+a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$7$ |
7B.6.3 |
$1$ |
\( 1 \) |
$1$ |
$1081.567760$ |
1.40492 |
\( -21437955273398086628814342060 a^{5} - 8296492504833651562385453152 a^{4} + 95682539786756414754531486072 a^{3} + 46959878613528243860092427239 a^{2} - 42056416375390440919728940937 a - 15456346448131269056429661858 \) |
\( \bigl[a^{5} - a^{4} - 5 a^{3} + 5 a^{2} + 4 a - 3\) , \( 2 a^{5} - 3 a^{4} - 9 a^{3} + 11 a^{2} + 5 a - 3\) , \( 2 a^{5} - 2 a^{4} - 9 a^{3} + 9 a^{2} + 5 a - 6\) , \( -150 a^{5} + 288 a^{4} + 503 a^{3} - 1083 a^{2} + 192 a + 167\) , \( 1413 a^{5} - 2764 a^{4} - 4519 a^{3} + 10175 a^{2} - 2425 a - 1163\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-5a^{3}+5a^{2}+4a-3\right){x}{y}+\left(2a^{5}-2a^{4}-9a^{3}+9a^{2}+5a-6\right){y}={x}^{3}+\left(2a^{5}-3a^{4}-9a^{3}+11a^{2}+5a-3\right){x}^{2}+\left(-150a^{5}+288a^{4}+503a^{3}-1083a^{2}+192a+167\right){x}+1413a^{5}-2764a^{4}-4519a^{3}+10175a^{2}-2425a-1163$ |
| 49.2-b2 |
49.2-b |
$2$ |
$7$ |
6.6.592661.1 |
$6$ |
$[6, 0]$ |
49.2 |
\( 7^{2} \) |
\( - 7^{8} \) |
$95.14619$ |
$(a^4-4a^2+a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$7$ |
7B.6.1 |
$1$ |
\( 1 \) |
$1$ |
$1081.567760$ |
1.40492 |
\( -466893 a^{5} + 96319 a^{4} + 2480115 a^{3} + 52414 a^{2} - 2640289 a - 1001005 \) |
\( \bigl[a^{5} - 4 a^{3} + a^{2} + a - 2\) , \( a^{5} - a^{4} - 4 a^{3} + 4 a^{2} + 2 a - 3\) , \( a^{5} - a^{4} - 5 a^{3} + 4 a^{2} + 5 a - 2\) , \( 6 a^{5} - 6 a^{4} - 24 a^{3} + 27 a^{2} + 16 a - 15\) , \( 6 a^{5} - 5 a^{4} - 22 a^{3} + 21 a^{2} + 14 a - 13\bigr] \) |
${y}^2+\left(a^{5}-4a^{3}+a^{2}+a-2\right){x}{y}+\left(a^{5}-a^{4}-5a^{3}+4a^{2}+5a-2\right){y}={x}^{3}+\left(a^{5}-a^{4}-4a^{3}+4a^{2}+2a-3\right){x}^{2}+\left(6a^{5}-6a^{4}-24a^{3}+27a^{2}+16a-15\right){x}+6a^{5}-5a^{4}-22a^{3}+21a^{2}+14a-13$ |
| 49.2-c1 |
49.2-c |
$2$ |
$7$ |
6.6.592661.1 |
$6$ |
$[6, 0]$ |
49.2 |
\( 7^{2} \) |
\( - 7^{2} \) |
$95.14619$ |
$(a^4-4a^2+a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$7$ |
7B.1.3 |
$49$ |
\( 1 \) |
$1.100846541$ |
$6.680361328$ |
2.80848 |
\( -21437955273398086628814342060 a^{5} - 8296492504833651562385453152 a^{4} + 95682539786756414754531486072 a^{3} + 46959878613528243860092427239 a^{2} - 42056416375390440919728940937 a - 15456346448131269056429661858 \) |
\( \bigl[2 a^{5} - a^{4} - 9 a^{3} + 5 a^{2} + 6 a - 4\) , \( -a^{3} + a^{2} + 2 a - 2\) , \( a^{5} - a^{4} - 5 a^{3} + 5 a^{2} + 4 a - 3\) , \( 23 a^{5} + 6 a^{4} - 108 a^{3} - 42 a^{2} + 62 a + 17\) , \( 85 a^{5} + 42 a^{4} - 373 a^{3} - 245 a^{2} + 97 a + 41\bigr] \) |
${y}^2+\left(2a^{5}-a^{4}-9a^{3}+5a^{2}+6a-4\right){x}{y}+\left(a^{5}-a^{4}-5a^{3}+5a^{2}+4a-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a-2\right){x}^{2}+\left(23a^{5}+6a^{4}-108a^{3}-42a^{2}+62a+17\right){x}+85a^{5}+42a^{4}-373a^{3}-245a^{2}+97a+41$ |
| 49.2-c2 |
49.2-c |
$2$ |
$7$ |
6.6.592661.1 |
$6$ |
$[6, 0]$ |
49.2 |
\( 7^{2} \) |
\( - 7^{2} \) |
$95.14619$ |
$(a^4-4a^2+a+2)$ |
$1$ |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$7$ |
7B.1.1 |
$1$ |
\( 1 \) |
$0.157263791$ |
$112276.8328$ |
2.80848 |
\( -466893 a^{5} + 96319 a^{4} + 2480115 a^{3} + 52414 a^{2} - 2640289 a - 1001005 \) |
\( \bigl[a^{5} - a^{4} - 4 a^{3} + 5 a^{2} + 2 a - 4\) , \( a^{4} + a^{3} - 4 a^{2} - 4 a + 2\) , \( 2 a^{5} - a^{4} - 9 a^{3} + 5 a^{2} + 5 a - 5\) , \( 2 a^{5} - 2 a^{4} - 7 a^{3} + 11 a^{2} + 3 a - 6\) , \( 5 a^{5} - a^{4} - 20 a^{3} + 6 a^{2} + 10 a - 5\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-4a^{3}+5a^{2}+2a-4\right){x}{y}+\left(2a^{5}-a^{4}-9a^{3}+5a^{2}+5a-5\right){y}={x}^{3}+\left(a^{4}+a^{3}-4a^{2}-4a+2\right){x}^{2}+\left(2a^{5}-2a^{4}-7a^{3}+11a^{2}+3a-6\right){x}+5a^{5}-a^{4}-20a^{3}+6a^{2}+10a-5$ |
| 49.2-d1 |
49.2-d |
$2$ |
$2$ |
6.6.592661.1 |
$6$ |
$[6, 0]$ |
49.2 |
\( 7^{2} \) |
\( - 7^{16} \) |
$95.14619$ |
$(a^4-4a^2+a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.140992452$ |
$2368.483818$ |
2.60264 |
\( -\frac{6378555847697446312}{282475249} a^{5} - \frac{2474411833285999287}{282475249} a^{4} + \frac{28471855515534194816}{282475249} a^{3} + \frac{13996124332632398039}{282475249} a^{2} - \frac{12521297730192365783}{282475249} a - \frac{4603543349670581566}{282475249} \) |
\( \bigl[a^{5} - 4 a^{3} + a^{2} + 2 a - 3\) , \( -a^{5} + 5 a^{3} + a^{2} - 3 a - 2\) , \( 2 a^{5} - a^{4} - 9 a^{3} + 5 a^{2} + 5 a - 5\) , \( 27 a^{5} - 81 a^{4} - 85 a^{3} + 376 a^{2} - 22 a - 297\) , \( 242 a^{5} - 536 a^{4} - 965 a^{3} + 2432 a^{2} + 294 a - 1763\bigr] \) |
${y}^2+\left(a^{5}-4a^{3}+a^{2}+2a-3\right){x}{y}+\left(2a^{5}-a^{4}-9a^{3}+5a^{2}+5a-5\right){y}={x}^{3}+\left(-a^{5}+5a^{3}+a^{2}-3a-2\right){x}^{2}+\left(27a^{5}-81a^{4}-85a^{3}+376a^{2}-22a-297\right){x}+242a^{5}-536a^{4}-965a^{3}+2432a^{2}+294a-1763$ |
| 49.2-d2 |
49.2-d |
$2$ |
$2$ |
6.6.592661.1 |
$6$ |
$[6, 0]$ |
49.2 |
\( 7^{2} \) |
\( - 7^{11} \) |
$95.14619$ |
$(a^4-4a^2+a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.070496226$ |
$4736.967637$ |
2.60264 |
\( \frac{1519176145}{16807} a^{5} + \frac{970371936}{16807} a^{4} - \frac{5971587107}{16807} a^{3} - \frac{3315408956}{16807} a^{2} + \frac{2656017515}{16807} a + \frac{1004044424}{16807} \) |
\( \bigl[a^{5} - 5 a^{3} + 4 a - 1\) , \( a^{5} - 2 a^{4} - 5 a^{3} + 9 a^{2} + 3 a - 4\) , \( a^{5} - a^{4} - 4 a^{3} + 5 a^{2} + a - 4\) , \( -a^{5} + 2 a^{4} - 8 a^{2} + 5 a + 1\) , \( -4 a^{4} - a^{3} + 15 a^{2} - 4 a - 7\bigr] \) |
${y}^2+\left(a^{5}-5a^{3}+4a-1\right){x}{y}+\left(a^{5}-a^{4}-4a^{3}+5a^{2}+a-4\right){y}={x}^{3}+\left(a^{5}-2a^{4}-5a^{3}+9a^{2}+3a-4\right){x}^{2}+\left(-a^{5}+2a^{4}-8a^{2}+5a+1\right){x}-4a^{4}-a^{3}+15a^{2}-4a-7$ |
*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.
