| 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-a1 |
18.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2 \cdot 3^{34} \) |
$0.97395$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$0.531367511$ |
1.606704330 |
\( -\frac{22026189082216125793}{3706040377703682} a - \frac{54626740869226485845}{3706040377703682} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( -6732 a + 17809\) , \( 429164 a - 1135465\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-6732a+17809\right){x}+429164a-1135465$ |
| 18.1-a2 |
18.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{8} \) |
$0.97395$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$8.501880177$ |
1.606704330 |
\( \frac{4913}{1296} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 16 a + 47\) , \( -1313 a - 3476\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(16a+47\right){x}-1313a-3476$ |
| 18.1-a3 |
18.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2 \cdot 3^{34} \) |
$0.97395$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$2.125470044$ |
1.606704330 |
\( \frac{22026189082216125793}{3706040377703682} a - \frac{54626740869226485845}{3706040377703682} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 6731 a + 17812\) , \( 429164 a + 1135461\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(6731a+17812\right){x}+429164a+1135461$ |
| 18.1-a4 |
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 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$8.501880177$ |
1.606704330 |
\( -\frac{15840177853915205}{86093442} a + \frac{20954710660855016}{43046721} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -2274 a - 6013\) , \( 59627 a + 157758\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2274a-6013\right){x}+59627a+157758$ |
| 18.1-a5 |
18.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{16} \) |
$0.97395$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$8.501880177$ |
1.606704330 |
\( \frac{838561807}{26244} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -944 a - 2493\) , \( -25533 a - 67556\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-944a-2493\right){x}-25533a-67556$ |
| 18.1-a6 |
18.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2 \cdot 3^{10} \) |
$0.97395$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$8.501880177$ |
1.606704330 |
\( -\frac{2350503708439404473473}{13122} a + \frac{6218848268321376696725}{13122} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -32559 a - 86158\) , \( 5180690 a + 13706871\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-32559a-86158\right){x}+5180690a+13706871$ |
| 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-a8 |
18.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2 \cdot 3^{10} \) |
$0.97395$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$0.531367511$ |
1.606704330 |
\( \frac{2350503708439404473473}{13122} a + \frac{6218848268321376696725}{13122} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( 32558 a - 86161\) , \( 5180690 a - 13706875\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(32558a-86161\right){x}+5180690a-13706875$ |
| 18.1-b1 |
18.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2 \cdot 3^{34} \) |
$0.97395$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$2.125470044$ |
1.606704330 |
\( -\frac{22026189082216125793}{3706040377703682} a - \frac{54626740869226485845}{3706040377703682} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( -6732 a + 17812\) , \( -429165 a + 1135461\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6732a+17812\right){x}-429165a+1135461$ |
| 18.1-b2 |
18.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{8} \) |
$0.97395$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$8.501880177$ |
1.606704330 |
\( \frac{4913}{1296} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( 16 a + 44\) , \( 1312 a + 3472\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(16a+44\right){x}+1312a+3472$ |
| 18.1-b3 |
18.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2 \cdot 3^{34} \) |
$0.97395$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$0.531367511$ |
1.606704330 |
\( \frac{22026189082216125793}{3706040377703682} a - \frac{54626740869226485845}{3706040377703682} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( 6731 a + 17809\) , \( -429165 a - 1135465\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(6731a+17809\right){x}-429165a-1135465$ |
| 18.1-b4 |
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 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{4} \) |
$1$ |
$2.125470044$ |
1.606704330 |
\( -\frac{15840177853915205}{86093442} a + \frac{20954710660855016}{43046721} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -2274 a - 6016\) , \( -59628 a - 157762\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2274a-6016\right){x}-59628a-157762$ |
| 18.1-b5 |
18.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{16} \) |
$0.97395$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$8.501880177$ |
1.606704330 |
\( \frac{838561807}{26244} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -944 a - 2496\) , \( 25532 a + 67552\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-944a-2496\right){x}+25532a+67552$ |
| 18.1-b6 |
18.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2 \cdot 3^{10} \) |
$0.97395$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$0.531367511$ |
1.606704330 |
\( -\frac{2350503708439404473473}{13122} a + \frac{6218848268321376696725}{13122} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -32559 a - 86161\) , \( -5180691 a - 13706875\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-32559a-86161\right){x}-5180691a-13706875$ |
| 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 |
\( \frac{15840177853915205}{86093442} a + \frac{20954710660855016}{43046721} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 2273 a - 6013\) , \( -59628 a + 157758\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(2273a-6013\right){x}-59628a+157758$ |
| 18.1-b8 |
18.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{7}) \) |
$2$ |
$[2, 0]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2 \cdot 3^{10} \) |
$0.97395$ |
$(a+3), (-a+2), (-a-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$8.501880177$ |
1.606704330 |
\( \frac{2350503708439404473473}{13122} a + \frac{6218848268321376696725}{13122} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 32558 a - 86158\) , \( -5180691 a + 13706871\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(32558a-86158\right){x}-5180691a+13706871$ |
*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.
