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 |
7.1-a1 |
7.1-a |
$2$ |
$3$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{9} \) |
$3.34611$ |
$(a^2+2a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{2} \) |
$1$ |
$3.407152534$ |
1.132614452 |
\( -\frac{16146410084579303}{40353607} a^{2} - \frac{24513786018922996}{40353607} a + \frac{51294295430574624}{40353607} \) |
\( \bigl[a^{2} + a - 4\) , \( a^{2} + a - 6\) , \( a + 1\) , \( 77 a^{2} + 9 a - 494\) , \( 499 a^{2} + 93 a - 3377\bigr] \) |
${y}^2+\left(a^{2}+a-4\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}+a-6\right){x}^{2}+\left(77a^{2}+9a-494\right){x}+499a^{2}+93a-3377$ |
7.1-a2 |
7.1-a |
$2$ |
$3$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{3} \) |
$3.34611$ |
$(a^2+2a-3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$91.99311844$ |
1.132614452 |
\( -\frac{470436}{343} a^{2} + \frac{1143587}{343} a - \frac{168936}{343} \) |
\( \bigl[a^{2} + a - 4\) , \( a^{2} + a - 6\) , \( a + 1\) , \( -3 a^{2} + 4 a + 26\) , \( 11 a^{2} + 4 a - 58\bigr] \) |
${y}^2+\left(a^{2}+a-4\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}+a-6\right){x}^{2}+\left(-3a^{2}+4a+26\right){x}+11a^{2}+4a-58$ |
7.1-b1 |
7.1-b |
$6$ |
$8$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
7.1 |
\( 7 \) |
\( -7 \) |
$3.34611$ |
$(a^2+2a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$64$ |
\( 1 \) |
$1$ |
$1.433724702$ |
0.847293282 |
\( \frac{960431553635305495}{7} a^{2} - \frac{3550132253411453690}{7} a + \frac{2849530108248409337}{7} \) |
\( \bigl[a^{2} - 5\) , \( a^{2} - a - 5\) , \( 1\) , \( -572 a^{2} + 2121 a - 1723\) , \( -25034 a^{2} + 92549 a - 74316\bigr] \) |
${y}^2+\left(a^{2}-5\right){x}{y}+{y}={x}^{3}+\left(a^{2}-a-5\right){x}^{2}+\left(-572a^{2}+2121a-1723\right){x}-25034a^{2}+92549a-74316$ |
7.1-b2 |
7.1-b |
$6$ |
$8$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
7.1 |
\( 7 \) |
\( 7 \) |
$3.34611$ |
$(a^2+2a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$64$ |
\( 1 \) |
$1$ |
$1.433724702$ |
0.847293282 |
\( -\frac{84096891433574753384471}{7} a^{2} - \frac{14985568201611275023622}{7} a + \frac{571022332294137577763767}{7} \) |
\( \bigl[a^{2} - 5\) , \( a^{2} - a - 5\) , \( 1\) , \( 28 a^{2} + 141 a - 533\) , \( -38 a^{2} + 1741 a - 4144\bigr] \) |
${y}^2+\left(a^{2}-5\right){x}{y}+{y}={x}^{3}+\left(a^{2}-a-5\right){x}^{2}+\left(28a^{2}+141a-533\right){x}-38a^{2}+1741a-4144$ |
7.1-b3 |
7.1-b |
$6$ |
$8$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
7.1 |
\( 7 \) |
\( 7^{2} \) |
$3.34611$ |
$(a^2+2a-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$16$ |
\( 2 \) |
$1$ |
$11.46979762$ |
0.847293282 |
\( -\frac{2353024794450}{49} a^{2} - \frac{434940018425}{49} a + \frac{16016561377497}{49} \) |
\( \bigl[a^{2} - 5\) , \( a^{2} - a - 5\) , \( 1\) , \( -32 a^{2} + 131 a - 128\) , \( -442 a^{2} + 1653 a - 1360\bigr] \) |
${y}^2+\left(a^{2}-5\right){x}{y}+{y}={x}^{3}+\left(a^{2}-a-5\right){x}^{2}+\left(-32a^{2}+131a-128\right){x}-442a^{2}+1653a-1360$ |
7.1-b4 |
7.1-b |
$6$ |
$8$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
7.1 |
\( 7 \) |
\( 7^{8} \) |
$3.34611$ |
$(a^2+2a-3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$45.87919048$ |
0.847293282 |
\( \frac{1824928468}{5764801} a^{2} - \frac{1345230953}{5764801} a - \frac{2105854007}{5764801} \) |
\( \bigl[1\) , \( a^{2} - a - 5\) , \( a^{2} + a - 5\) , \( 274158 a^{2} + 416226 a - 870957\) , \( 116637957 a^{2} + 177079592 a - 370544096\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}+a-5\right){y}={x}^{3}+\left(a^{2}-a-5\right){x}^{2}+\left(274158a^{2}+416226a-870957\right){x}+116637957a^{2}+177079592a-370544096$ |
7.1-b5 |
7.1-b |
$6$ |
$8$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
7.1 |
\( 7 \) |
\( 7^{4} \) |
$3.34611$ |
$(a^2+2a-3)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$91.75838097$ |
0.847293282 |
\( -\frac{225573007}{2401} a^{2} - \frac{58067834}{2401} a + \frac{1583231073}{2401} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( 248 a^{2} + 46 a - 1678\) , \( 3590 a^{2} + 641 a - 24373\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(248a^{2}+46a-1678\right){x}+3590a^{2}+641a-24373$ |
7.1-b6 |
7.1-b |
$6$ |
$8$ |
3.3.733.1 |
$3$ |
$[3, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{2} \) |
$3.34611$ |
$(a^2+2a-3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$183.5167619$ |
0.847293282 |
\( \frac{24849000}{49} a^{2} + \frac{37748161}{49} a - \frac{78881609}{49} \) |
\( \bigl[a + 1\) , \( -a^{2} + 4\) , \( a^{2} - 4\) , \( 454041244 a^{2} + 80907460 a - 3082964015\) , \( 1380930983897 a^{2} + 246073726249 a - 9376594279770\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(454041244a^{2}+80907460a-3082964015\right){x}+1380930983897a^{2}+246073726249a-9376594279770$ |
*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.