| 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 |
| 729.1-a1 |
729.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( 3^{6} \) |
$2.27471$ |
$(a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1.096515940$ |
$6.389222357$ |
2.860140232 |
\( -3386880 a - 8296128 \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( -2 a - 6\) , \( -3 a - 7\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-2a-6\right){x}-3a-7$ |
| 729.1-a2 |
729.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( 3^{6} \) |
$2.27471$ |
$(a+3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$3.289547821$ |
$19.16766707$ |
2.860140232 |
\( 3386880 a - 8296128 \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( -2 a - 6\) , \( -157 a - 385\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-2a-6\right){x}-157a-385$ |
| 729.1-b1 |
729.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( 3^{6} \) |
$2.27471$ |
$(a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$0.781767729$ |
$9.363037422$ |
2.988263384 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 0\) , \( -a - 3\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-a-3$ |
| 729.1-b2 |
729.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( 3^{6} \) |
$2.27471$ |
$(a+3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$2.345303189$ |
$28.08911226$ |
2.988263384 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( -5 a + 12\bigr] \) |
${y}^2+{y}={x}^{3}-5a+12$ |
| 729.1-c1 |
729.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( 3^{6} \) |
$2.27471$ |
$(a+3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$3.289547821$ |
$19.16766707$ |
2.860140232 |
\( -3386880 a - 8296128 \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( a - 6\) , \( 156 a - 385\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-6\right){x}+156a-385$ |
| 729.1-c2 |
729.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( 3^{6} \) |
$2.27471$ |
$(a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1.096515940$ |
$6.389222357$ |
2.860140232 |
\( 3386880 a - 8296128 \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( a - 6\) , \( 2 a - 7\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-6\right){x}+2a-7$ |
| 729.1-d1 |
729.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( 3^{6} \) |
$2.27471$ |
$(a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$0.781767729$ |
$9.363037422$ |
2.988263384 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 0\) , \( -3\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-3$ |
| 729.1-d2 |
729.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( 3^{6} \) |
$2.27471$ |
$(a+3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$2.345303189$ |
$28.08911226$ |
2.988263384 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 5 a + 12\bigr] \) |
${y}^2+{y}={x}^{3}+5a+12$ |
*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.
