| 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 |
| 18.1-a7 |
18.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{20} \) |
$0.97395$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{4} \) |
$1$ |
$2.125470044$ |
1.606704330 |
\( \frac{15840177853915205}{86093442} a + \frac{20954710660855016}{43046721} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( 2273 a - 6016\) , \( 59627 a - 157762\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(2273a-6016\right){x}+59627a-157762$ |
| 18.1-b7 |
18.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{20} \) |
$0.97395$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$8.501880177$ |
1.606704330 |
\( \frac{15840177853915205}{86093442} a + \frac{20954710660855016}{43046721} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 2273 a - 6013\) , \( -59628 a + 157758\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(2273a-6013\right){x}-59628a+157758$ |
| 162.1-a7 |
162.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{2} \cdot 3^{32} \) |
$1.68693$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$2.833960059$ |
1.071136220 |
\( \frac{15840177853915205}{86093442} a + \frac{20954710660855016}{43046721} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 20457 a - 54135\) , \( -1643612 a + 4348600\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(20457a-54135\right){x}-1643612a+4348600$ |
| 162.1-b7 |
162.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
162.1 |
\( 2 \cdot 3^{4} \) |
\( 2^{2} \cdot 3^{32} \) |
$1.68693$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{5} \) |
$1$ |
$0.708490014$ |
1.071136220 |
\( \frac{15840177853915205}{86093442} a + \frac{20954710660855016}{43046721} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 20457 a - 54135\) , \( 1643612 a - 4348600\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(20457a-54135\right){x}+1643612a-4348600$ |
| 432.1-b7 |
432.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
432.1 |
\( 2^{4} \cdot 3^{3} \) |
\( 2^{14} \cdot 3^{26} \) |
$2.15570$ |
$(a+3), (-a+2), (-a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1.942786018$ |
$2.021068929$ |
2.968158376 |
\( \frac{15840177853915205}{86093442} a + \frac{20954710660855016}{43046721} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -24972 a - 66084\) , \( 3497216 a + 9252736\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-24972a-66084\right){x}+3497216a+9252736$ |
| 432.1-i7 |
432.1-i |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
432.1 |
\( 2^{4} \cdot 3^{3} \) |
\( 2^{14} \cdot 3^{26} \) |
$2.15570$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{6} \) |
$1$ |
$0.745088047$ |
2.252934490 |
\( \frac{15840177853915205}{86093442} a + \frac{20954710660855016}{43046721} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -24972 a - 66087\) , \( -3522188 a - 9318822\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-24972a-66087\right){x}-3522188a-9318822$ |
| 432.2-b7 |
432.2-b |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
432.2 |
\( 2^{4} \cdot 3^{3} \) |
\( 2^{14} \cdot 3^{26} \) |
$2.15570$ |
$(a+3), (-a+2), (-a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1.942786018$ |
$2.021068929$ |
2.968158376 |
\( \frac{15840177853915205}{86093442} a + \frac{20954710660855016}{43046721} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 3772 a - 10084\) , \( 131200 a - 346784\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(3772a-10084\right){x}+131200a-346784$ |
| 432.2-i7 |
432.2-i |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
432.2 |
\( 2^{4} \cdot 3^{3} \) |
\( 2^{14} \cdot 3^{26} \) |
$2.15570$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{8} \) |
$1$ |
$0.745088047$ |
2.252934490 |
\( \frac{15840177853915205}{86093442} a + \frac{20954710660855016}{43046721} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 3770 a - 10087\) , \( -127429 a + 336698\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(3770a-10087\right){x}-127429a+336698$ |
*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.
