| 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 |
| 108.4-a1 |
108.4-a |
$4$ |
$27$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
108.4 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{6} \cdot 3^{11} \) |
$0.95541$ |
$(a-1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$0.392228249$ |
$0.681207479$ |
0.966725513 |
\( -\frac{116453655937}{8} a - \frac{179584536671}{8} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( -445 a - 529\) , \( -7728 a - 1018\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-445a-529\right){x}-7728a-1018$ |
| 108.4-a2 |
108.4-a |
$4$ |
$27$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
108.4 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{18} \cdot 3^{9} \) |
$0.95541$ |
$(a-1), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{3} \) |
$0.130742749$ |
$2.043622437$ |
0.966725513 |
\( \frac{488881}{256} a + \frac{403771}{512} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( -5 a - 9\) , \( -16 a + 6\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-5a-9\right){x}-16a+6$ |
| 108.4-a3 |
108.4-a |
$4$ |
$27$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
108.4 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{6} \cdot 3^{3} \) |
$0.95541$ |
$(a-1), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$0.392228249$ |
$6.130867313$ |
0.966725513 |
\( \frac{21349}{4} a + \frac{286159}{8} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( 1\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+{x}$ |
| 108.4-a4 |
108.4-a |
$4$ |
$27$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
108.4 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{2} \cdot 3^{5} \) |
$0.95541$ |
$(a-1), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1.176684747$ |
$6.130867313$ |
0.966725513 |
\( -\frac{21493}{2} a + \frac{154981}{2} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( a\) , \( -a + 4\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+a{x}-a+4$ |
| 108.4-b1 |
108.4-b |
$3$ |
$9$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
108.4 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{2} \cdot 3^{9} \) |
$0.95541$ |
$(a-1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$2.453117830$ |
1.479285710 |
\( 13361111 a - \frac{33608299}{2} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( 20 a + 9\) , \( -7 a - 75\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(20a+9\right){x}-7a-75$ |
| 108.4-b2 |
108.4-b |
$3$ |
$9$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
108.4 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{2} \cdot 3^{5} \) |
$0.95541$ |
$(a-1), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$7.359353490$ |
1.479285710 |
\( \frac{3637}{2} a - \frac{6229}{2} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( -a\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}-a{x}$ |
| 108.4-b3 |
108.4-b |
$3$ |
$9$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
108.4 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{6} \cdot 3^{3} \) |
$0.95541$ |
$(a-1), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$1$ |
$7.359353490$ |
1.479285710 |
\( -\frac{1261}{4} a + \frac{13649}{8} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -1\) , \( -a + 1\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}-{x}-a+1$ |
*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.
