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 |
27.2-a1 |
27.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 3^{11} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( \frac{27256702}{81} a - \frac{57582961}{81} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( -2 a + 13\) , \( -11 a - 20\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(-a-1\right){x}^2+\left(-2a+13\right){x}-11a-20$ |
27.2-a2 |
27.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{11} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( -\frac{27256702}{81} a - \frac{10108753}{27} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -10 a + 60\) , \( 93 a + 58\bigr] \) |
${y}^2+a{x}{y}={x}^3+a{x}^2+\left(-10a+60\right){x}+93a+58$ |
27.2-a3 |
27.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{14} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( -\frac{2924207}{81} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -16 a + 13\) , \( 34 a - 88\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(-16a+13\right){x}+34a-88$ |
27.2-a4 |
27.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 3^{16} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( \frac{1950520}{6561} a + \frac{58507}{2187} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 1\) , \( a - 2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(-a-1\right){x}^2+{x}+a-2$ |
27.2-a5 |
27.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{16} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( -\frac{1950520}{6561} a + \frac{2126041}{6561} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -2 a + 12\) , \( -a - 26\bigr] \) |
${y}^2+a{x}{y}={x}^3+a{x}^2+\left(-2a+12\right){x}-a-26$ |
27.2-a6 |
27.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{23} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.589586966$ |
0.662903589 |
\( \frac{67902559538}{43046721} a + \frac{66394340881}{43046721} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -17 a - 78\) , \( -130 a - 80\bigr] \) |
${y}^2+a{x}{y}={x}^3+a{x}^2+\left(-17a-78\right){x}-130a-80$ |
27.2-a7 |
27.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 3^{23} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.589586966$ |
0.662903589 |
\( -\frac{67902559538}{43046721} a + \frac{44765633473}{14348907} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 10 a - 14\) , \( 13 a + 34\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+\left(-a-1\right){x}^2+\left(10a-14\right){x}+13a+34$ |
27.2-a8 |
27.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.2 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{10} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.589586966$ |
0.662903589 |
\( \frac{12214672127}{9} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -241 a + 148\) , \( 1654 a - 4948\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(-241a+148\right){x}+1654a-4948$ |
*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.