| 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 |
| 28.2-a1 |
28.2-a |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{36} \cdot 7^{2} \) |
$0.54385$ |
$(a), (-a+1), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$1$ |
$0.875417135$ |
0.330876576 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -171\) , \( -874\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-171{x}-874$ |
| 28.2-a2 |
28.2-a |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{15} \cdot 7^{3} \) |
$0.54385$ |
$(a), (-a+1), (-2a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$2.626251405$ |
0.330876576 |
\( -\frac{10538337875}{200704} a - \frac{13018580375}{100352} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -10 a + 15\) , \( -5 a - 16\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-10a+15\right){x}-5a-16$ |
| 28.2-a3 |
28.2-a |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{15} \cdot 7^{3} \) |
$0.54385$ |
$(a), (-a+1), (-2a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$2.626251405$ |
0.330876576 |
\( \frac{10538337875}{200704} a - \frac{36575498625}{200704} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( 9 a + 6\) , \( 4 a - 20\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(9a+6\right){x}+4a-20$ |
| 28.2-a4 |
28.2-a |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{5} \cdot 7 \) |
$0.54385$ |
$(a), (-a+1), (-2a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$7.878754216$ |
0.330876576 |
\( -\frac{831875}{112} a - \frac{166375}{112} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -a + 1\) , \( -a + 1\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a+1\right){x}-a+1$ |
| 28.2-a5 |
28.2-a |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{5} \cdot 7 \) |
$0.54385$ |
$(a), (-a+1), (-2a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$7.878754216$ |
0.330876576 |
\( \frac{831875}{112} a - \frac{499125}{56} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}$ |
| 28.2-a6 |
28.2-a |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$0.54385$ |
$(a), (-a+1), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{3} \) |
$1$ |
$7.878754216$ |
0.330876576 |
\( -\frac{15625}{28} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}$ |
| 28.2-a7 |
28.2-a |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{12} \cdot 7^{6} \) |
$0.54385$ |
$(a), (-a+1), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2Cs, 3Cs.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$2.626251405$ |
0.330876576 |
\( \frac{9938375}{21952} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 4\) , \( -6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+4{x}-6$ |
| 28.2-a8 |
28.2-a |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{45} \cdot 7 \) |
$0.54385$ |
$(a), (-a+1), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$0.875417135$ |
0.330876576 |
\( -\frac{70135314719125}{481036337152} a + \frac{179276652423375}{240518168576} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 30 a - 40\) , \( -30 a - 154\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(30a-40\right){x}-30a-154$ |
| 28.2-a9 |
28.2-a |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{45} \cdot 7 \) |
$0.54385$ |
$(a), (-a+1), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$0.875417135$ |
0.330876576 |
\( \frac{70135314719125}{481036337152} a + \frac{288417990127625}{481036337152} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -31 a - 9\) , \( 29 a - 183\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-31a-9\right){x}+29a-183$ |
| 28.2-a10 |
28.2-a |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{6} \cdot 7^{12} \) |
$0.54385$ |
$(a), (-a+1), (-2a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$1.313125702$ |
0.330876576 |
\( \frac{4956477625}{941192} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -36\) , \( -70\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-36{x}-70$ |
| 28.2-a11 |
28.2-a |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{2} \cdot 7^{4} \) |
$0.54385$ |
$(a), (-a+1), (-2a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$3.939377108$ |
0.330876576 |
\( \frac{128787625}{98} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -11\) , \( 12\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-11{x}+12$ |
| 28.2-a12 |
28.2-a |
$12$ |
$36$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{18} \cdot 7^{4} \) |
$0.54385$ |
$(a), (-a+1), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$0.437708567$ |
0.330876576 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -2731\) , \( -55146\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-2731{x}-55146$ |
*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.
