| 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 |
| 27.3-a1 |
27.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{11} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( \frac{27256702}{81} a - \frac{57582961}{81} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 10 a + 47\) , \( -32 a + 94\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a+1\right){x}^2+\left(10a+47\right){x}-32a+94$ |
| 27.3-a2 |
27.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{11} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( -\frac{27256702}{81} a - \frac{10108753}{27} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 4 a + 11\) , \( 14 a - 20\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a+1\right){x}^2+\left(4a+11\right){x}+14a-20$ |
| 27.3-a3 |
27.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{14} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( -\frac{2924207}{81} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 14 a - 1\) , \( -35 a - 53\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(14a-1\right){x}-35a-53$ |
| 27.3-a4 |
27.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{16} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( \frac{1950520}{6561} a + \frac{58507}{2187} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 2 a + 7\) , \( 14 a - 36\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a+1\right){x}^2+\left(2a+7\right){x}+14a-36$ |
| 27.3-a5 |
27.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{16} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.179173933$ |
0.662903589 |
\( -\frac{1950520}{6561} a + \frac{2126041}{6561} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 2 a + 1\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a+1\right){x}^2+\left(2a+1\right){x}$ |
| 27.3-a6 |
27.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{23} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.589586966$ |
0.662903589 |
\( \frac{67902559538}{43046721} a + \frac{66394340881}{43046721} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( -8 a - 4\) , \( -22 a + 43\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a+1\right){x}^2+\left(-8a-4\right){x}-22a+43$ |
| 27.3-a7 |
27.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{23} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.589586966$ |
0.662903589 |
\( -\frac{67902559538}{43046721} a + \frac{44765633473}{14348907} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 17 a - 98\) , \( 53 a - 309\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a+1\right){x}^2+\left(17a-98\right){x}+53a-309$ |
| 27.3-a8 |
27.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-23}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 2^{12} \cdot 3^{10} \) |
$0.97688$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.589586966$ |
0.662903589 |
\( \frac{12214672127}{9} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 239 a - 91\) , \( -1655 a - 3293\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(239a-91\right){x}-1655a-3293$ |
*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.
