| 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 |
$8$ |
$20$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{18} \) |
$0.67558$ |
$(-a), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{3} \) |
$1$ |
$2.289634401$ |
0.690350747 |
\( -\frac{349209575}{59049} a - \frac{298597801}{19683} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( 5 a + 11\) , \( -4 a + 22\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(5a+11\right){x}-4a+22$ |
| 27.3-a2 |
27.3-a |
$8$ |
$20$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{18} \) |
$0.67558$ |
$(-a), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{3} \) |
$1$ |
$2.289634401$ |
0.690350747 |
\( \frac{349209575}{59049} a - \frac{1245002978}{59049} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 6 a + 7\) , \( 16 a - 34\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(6a+7\right){x}+16a-34$ |
| 27.3-a3 |
27.3-a |
$8$ |
$20$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{12} \) |
$0.67558$ |
$(-a), (a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.579268803$ |
0.690350747 |
\( -\frac{77935}{243} a - \frac{11594}{81} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( 1\) , \( 3\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+{x}+3$ |
| 27.3-a4 |
27.3-a |
$8$ |
$20$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{12} \) |
$0.67558$ |
$(-a), (a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.579268803$ |
0.690350747 |
\( \frac{77935}{243} a - \frac{112717}{243} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( a - 3\) , \( -3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a-3\right){x}-3$ |
| 27.3-a5 |
27.3-a |
$8$ |
$20$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{27} \) |
$0.67558$ |
$(-a), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.144817200$ |
0.690350747 |
\( -\frac{2927543402641}{3486784401} a + \frac{1635099303025}{3486784401} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -4 a + 32\) , \( -23 a - 31\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-4a+32\right){x}-23a-31$ |
| 27.3-a6 |
27.3-a |
$8$ |
$20$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{27} \) |
$0.67558$ |
$(-a), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.144817200$ |
0.690350747 |
\( \frac{2927543402641}{3486784401} a - \frac{430814699872}{1162261467} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( 20 a - 4\) , \( -31 a - 50\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(20a-4\right){x}-31a-50$ |
| 27.3-a7 |
27.3-a |
$8$ |
$20$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{15} \) |
$0.67558$ |
$(-a), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.144817200$ |
0.690350747 |
\( -\frac{54238838797}{243} a + \frac{90191354077}{243} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 96 a + 142\) , \( 619 a - 1681\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(96a+142\right){x}+619a-1681$ |
| 27.3-a8 |
27.3-a |
$8$ |
$20$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27.3 |
\( 3^{3} \) |
\( 3^{15} \) |
$0.67558$ |
$(-a), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.144817200$ |
0.690350747 |
\( \frac{54238838797}{243} a + \frac{1331574640}{9} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( 70 a + 186\) , \( -573 a + 1350\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(70a+186\right){x}-573a+1350$ |
*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.
