| 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 |
| 588.1-a1 |
588.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 7^{12} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$5.711517702$ |
1.648773141 |
\( -\frac{10061824000}{352947} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -113\) , \( 516\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-113{x}+516$ |
| 588.1-a2 |
588.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{6} \cdot 7^{4} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$5.711517702$ |
1.648773141 |
\( \frac{2048000}{1323} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 7\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+7{x}$ |
| 588.1-a3 |
588.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{4} \cdot 3^{12} \cdot 7^{2} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$11.42303540$ |
1.648773141 |
\( \frac{9826000}{5103} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 29 a - 48\) , \( -24 a + 42\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(29a-48\right){x}-24a+42$ |
| 588.1-a4 |
588.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{4} \cdot 3^{4} \cdot 7^{6} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$11.42303540$ |
1.648773141 |
\( \frac{2640279346000}{3087} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 1829 a - 3198\) , \( -55734 a + 96576\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(1829a-3198\right){x}-55734a+96576$ |
| 588.1-b1 |
588.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 7^{4} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.782729173$ |
1.091979853 |
\( -\frac{16384}{147} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -1\) , \( -2\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-{x}-2$ |
| 588.1-b2 |
588.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{4} \cdot 3^{4} \cdot 7^{2} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$7.565458346$ |
1.091979853 |
\( \frac{20720464}{63} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 36 a - 64\) , \( 171 a - 298\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(36a-64\right){x}+171a-298$ |
| 588.1-c1 |
588.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( - 2^{4} \cdot 3 \cdot 7^{4} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$13.50509355$ |
1.949292349 |
\( -\frac{2886368}{147} a + \frac{238064}{7} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 3 a - 4\) , \( -3 a + 5\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(3a-4\right){x}-3a+5$ |
| 588.1-c2 |
588.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 7^{2} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$13.50509355$ |
1.949292349 |
\( \frac{274432}{21} a + \frac{520192}{21} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -8 a - 14\) , \( -26 a - 45\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-8a-14\right){x}-26a-45$ |
| 588.1-d1 |
588.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 7^{2} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$13.50509355$ |
1.949292349 |
\( -\frac{274432}{21} a + \frac{520192}{21} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 8 a - 14\) , \( 26 a - 45\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(8a-14\right){x}+26a-45$ |
| 588.1-d2 |
588.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( - 2^{4} \cdot 3 \cdot 7^{4} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$13.50509355$ |
1.949292349 |
\( \frac{2886368}{147} a + \frac{238064}{7} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -2 a + 4\) , \( 359 a - 622\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-2a+4\right){x}+359a-622$ |
| 588.1-e1 |
588.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 7^{2} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.090861332$ |
$26.12114948$ |
2.055426808 |
\( -\frac{274432}{21} a + \frac{520192}{21} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 8 a - 14\) , \( -26 a + 45\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(8a-14\right){x}-26a+45$ |
| 588.1-e2 |
588.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( - 2^{4} \cdot 3 \cdot 7^{4} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.181722664$ |
$13.06057474$ |
2.055426808 |
\( \frac{2886368}{147} a + \frac{238064}{7} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -a + 5\) , \( -357 a + 620\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-a+5\right){x}-357a+620$ |
| 588.1-f1 |
588.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( - 2^{4} \cdot 3 \cdot 7^{4} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.181722664$ |
$13.06057474$ |
2.055426808 |
\( -\frac{2886368}{147} a + \frac{238064}{7} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( a - 5\) , \( 5 a - 10\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(a-5\right){x}+5a-10$ |
| 588.1-f2 |
588.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 7^{2} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.090861332$ |
$26.12114948$ |
2.055426808 |
\( \frac{274432}{21} a + \frac{520192}{21} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -8 a - 14\) , \( 26 a + 45\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-8a-14\right){x}+26a+45$ |
| 588.1-g1 |
588.1-g |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 7^{4} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.097774690$ |
$15.45742511$ |
2.617726238 |
\( -\frac{16384}{147} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -1\) , \( 2\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-{x}+2$ |
| 588.1-g2 |
588.1-g |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{4} \cdot 3^{4} \cdot 7^{2} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.048887345$ |
$30.91485022$ |
2.617726238 |
\( \frac{20720464}{63} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 36 a - 63\) , \( -135 a + 234\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(36a-63\right){x}-135a+234$ |
| 588.1-h1 |
588.1-h |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 7^{12} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$2.839682332$ |
$0.533054337$ |
2.621813941 |
\( -\frac{10061824000}{352947} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -113\) , \( -516\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-113{x}-516$ |
| 588.1-h2 |
588.1-h |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{6} \cdot 7^{4} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$0.946560777$ |
$4.797489038$ |
2.621813941 |
\( \frac{2048000}{1323} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 7\) , \( 0\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+7{x}$ |
| 588.1-h3 |
588.1-h |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{4} \cdot 3^{12} \cdot 7^{2} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$0.473280388$ |
$9.594978077$ |
2.621813941 |
\( \frac{9826000}{5103} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 27 a - 51\) , \( 52 a - 92\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(27a-51\right){x}+52a-92$ |
| 588.1-h4 |
588.1-h |
$4$ |
$6$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
588.1 |
\( 2^{2} \cdot 3 \cdot 7^{2} \) |
\( 2^{4} \cdot 3^{4} \cdot 7^{6} \) |
$1.52431$ |
$(a+1), (a), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.419841166$ |
$1.066108675$ |
2.621813941 |
\( \frac{2640279346000}{3087} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 1827 a - 3201\) , \( 57562 a - 99776\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(1827a-3201\right){x}+57562a-99776$ |
*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.
