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 |
108.1-a1 |
108.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{6} \cdot 3^{11} \) |
$0.95541$ |
$(-a), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$0.392228249$ |
$0.681207479$ |
0.966725513 |
\( \frac{116453655937}{8} a - 37004774076 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 443 a - 974\) , \( 6754 a - 9107\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(443a-974\right){x}+6754a-9107$ |
108.1-a2 |
108.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{18} \cdot 3^{9} \) |
$0.95541$ |
$(-a), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{3} \) |
$0.130742749$ |
$2.043622437$ |
0.966725513 |
\( -\frac{488881}{256} a + \frac{1381533}{512} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 3 a - 14\) , \( 2 a - 11\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(3a-14\right){x}+2a-11$ |
108.1-a3 |
108.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{6} \cdot 3^{3} \) |
$0.95541$ |
$(-a), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$0.392228249$ |
$6.130867313$ |
0.966725513 |
\( -\frac{21349}{4} a + \frac{328857}{8} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -2 a + 1\) , \( a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2a+1\right){x}+a-1$ |
108.1-a4 |
108.1-a |
$4$ |
$27$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{2} \cdot 3^{5} \) |
$0.95541$ |
$(-a), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1.176684747$ |
$6.130867313$ |
0.966725513 |
\( \frac{21493}{2} a + 66744 \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -2 a + 1\) , \( 3\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2a+1\right){x}+3$ |
108.1-b1 |
108.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{2} \cdot 3^{9} \) |
$0.95541$ |
$(-a), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$2.453117830$ |
1.479285710 |
\( -13361111 a - \frac{6886077}{2} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( -21 a + 30\) , \( 6 a - 81\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-21a+30\right){x}+6a-81$ |
108.1-b2 |
108.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{2} \cdot 3^{5} \) |
$0.95541$ |
$(-a), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$7.359353490$ |
1.479285710 |
\( -\frac{3637}{2} a - 1296 \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}$ |
108.1-b3 |
108.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
108.1 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{6} \cdot 3^{3} \) |
$0.95541$ |
$(-a), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$1$ |
$7.359353490$ |
1.479285710 |
\( \frac{1261}{4} a + \frac{11127}{8} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( -a\) , \( 1\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}-a{x}+1$ |
*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.