| 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 |
| 18.2-a1 |
18.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.2 |
\( 2 \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{12} \) |
$0.97395$ |
$(a+3), (-a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$10.54590844$ |
1.992989364 |
\( -\frac{275587}{1458} a + \frac{289048}{729} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( -6 a - 8\) , \( 34 a + 93\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-6a-8\right){x}+34a+93$ |
| 18.2-a2 |
18.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.2 |
\( 2 \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{8} \) |
$0.97395$ |
$(a+3), (-a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$3.515302815$ |
1.992989364 |
\( \frac{1500083}{72} a - \frac{495212}{9} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( 74 a - 199\) , \( 691 a - 1830\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(74a-199\right){x}+691a-1830$ |
| 18.2-a3 |
18.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.2 |
\( 2 \cdot 3^{2} \) |
\( 2^{3} \cdot 3^{10} \) |
$0.97395$ |
$(a+3), (-a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$3.515302815$ |
1.992989364 |
\( -\frac{752904308551}{324} a + \frac{1992001566899}{324} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( 1204 a - 3189\) , \( 38811 a - 102686\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(1204a-3189\right){x}+38811a-102686$ |
| 18.2-a4 |
18.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.2 |
\( 2 \cdot 3^{2} \) |
\( 2 \cdot 3^{18} \) |
$0.97395$ |
$(a+3), (-a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$10.54590844$ |
1.992989364 |
\( \frac{62446099201}{1062882} a + \frac{168555918709}{1062882} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( -121 a - 313\) , \( 1188 a + 3145\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-121a-313\right){x}+1188a+3145$ |
| 18.2-b1 |
18.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.2 |
\( 2 \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{12} \) |
$0.97395$ |
$(a+3), (-a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$6.453608505$ |
1.219617368 |
\( -\frac{275587}{1458} a + \frac{289048}{729} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -4 a - 9\) , \( -39 a - 104\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-4a-9\right){x}-39a-104$ |
| 18.2-b2 |
18.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.2 |
\( 2 \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{8} \) |
$0.97395$ |
$(a+3), (-a-2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$19.36082551$ |
1.219617368 |
\( \frac{1500083}{72} a - \frac{495212}{9} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 76 a - 196\) , \( -616 a + 1632\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(76a-196\right){x}-616a+1632$ |
| 18.2-b3 |
18.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.2 |
\( 2 \cdot 3^{2} \) |
\( 2^{3} \cdot 3^{10} \) |
$0.97395$ |
$(a+3), (-a-2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$19.36082551$ |
1.219617368 |
\( -\frac{752904308551}{324} a + \frac{1992001566899}{324} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 1206 a - 3186\) , \( -37606 a + 99498\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(1206a-3186\right){x}-37606a+99498$ |
| 18.2-b4 |
18.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.2 |
\( 2 \cdot 3^{2} \) |
\( 2 \cdot 3^{18} \) |
$0.97395$ |
$(a+3), (-a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$6.453608505$ |
1.219617368 |
\( \frac{62446099201}{1062882} a + \frac{168555918709}{1062882} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -119 a - 314\) , \( -1308 a - 3461\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-119a-314\right){x}-1308a-3461$ |
*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.
