The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 50000 over imaginary quadratic fields with absolute discriminant 7
Note: The completeness Only modular elliptic curves are included
| 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 |
| 288.2-a1 |
288.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
288.2 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{4} \) |
$0.97395$ |
$(a), (-a+1), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.131934487$ |
$4.775104742$ |
0.952471978 |
\( -\frac{1771}{36} a + \frac{26425}{18} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( -1\) , \( -a + 1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}-{x}-a+1$ |
| 288.2-a2 |
288.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
288.2 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{15} \cdot 3^{2} \) |
$0.97395$ |
$(a), (-a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.065967243$ |
$4.775104742$ |
0.952471978 |
\( \frac{3191}{48} a + \frac{48539}{24} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -a - 1\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-a-1\right){x}$ |
| 288.2-a3 |
288.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
288.2 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{8} \) |
$0.97395$ |
$(a), (-a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.263868974$ |
$2.387552371$ |
0.952471978 |
\( \frac{8978189}{54} a - \frac{3246631}{81} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( 10 a - 21\) , \( -37 a + 33\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(10a-21\right){x}-37a+33$ |
| 288.2-a4 |
288.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
288.2 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{9} \cdot 3^{2} \) |
$0.97395$ |
$(a), (-a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.263868974$ |
$4.775104742$ |
0.952471978 |
\( -\frac{1410889}{6} a + \frac{497819}{3} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -a - 6\) , \( -a + 6\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-a-6\right){x}-a+6$ |
| 288.2-b1 |
288.2-b |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
288.2 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{20} \cdot 3^{8} \) |
$0.97395$ |
$(a), (-a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$2.125470044$ |
1.606704330 |
\( \frac{4913}{1296} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( a - 1\) , \( 6 a - 14\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(a-1\right){x}+6a-14$ |
| 288.2-b2 |
288.2-b |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
288.2 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{29} \cdot 3^{2} \) |
$0.97395$ |
$(a), (-a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.125470044$ |
1.606704330 |
\( -\frac{43993943}{196608} a + \frac{140725583}{65536} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -3 a - 8\) , \( 2 a - 5\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-3a-8\right){x}+2a-5$ |
| 288.2-b3 |
288.2-b |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
288.2 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{29} \cdot 3^{2} \) |
$0.97395$ |
$(a), (-a+1), (3)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$2.125470044$ |
1.606704330 |
\( \frac{43993943}{196608} a + \frac{189091403}{98304} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -a + 12\) , \( 2 a - 5\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-a+12\right){x}+2a-5$ |
| 288.2-b4 |
288.2-b |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
288.2 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{16} \cdot 3^{16} \) |
$0.97395$ |
$(a), (-a+1), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$1.062735022$ |
1.606704330 |
\( \frac{838561807}{26244} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -59 a + 39\) , \( 106 a - 294\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-59a+39\right){x}+106a-294$ |
| 288.2-b5 |
288.2-b |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
288.2 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{22} \cdot 3^{4} \) |
$0.97395$ |
$(a), (-a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$2.125470044$ |
1.606704330 |
\( -\frac{56620795}{2304} a + \frac{85821697}{1152} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 17 a - 1\) , \( 2 a - 50\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(17a-1\right){x}+2a-50$ |
| 288.2-b6 |
288.2-b |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
288.2 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{22} \cdot 3^{4} \) |
$0.97395$ |
$(a), (-a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$2.125470044$ |
1.606704330 |
\( \frac{56620795}{2304} a + \frac{115022599}{2304} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 14 a - 20\) , \( 28 a - 20\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(14a-20\right){x}+28a-20$ |
| 288.2-b7 |
288.2-b |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
288.2 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{17} \cdot 3^{2} \) |
$0.97395$ |
$(a), (-a+1), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$1.062735022$ |
1.606704330 |
\( -\frac{145011769343}{48} a + \frac{101553555457}{24} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 257 a - 1\) , \( -334 a - 2546\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(257a-1\right){x}-334a-2546$ |
| 288.2-b8 |
288.2-b |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
288.2 |
\( 2^{5} \cdot 3^{2} \) |
\( 2^{17} \cdot 3^{2} \) |
$0.97395$ |
$(a), (-a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$1.062735022$ |
1.606704330 |
\( \frac{145011769343}{48} a + \frac{19365113857}{16} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 224 a - 320\) , \( 1942 a - 1328\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(224a-320\right){x}+1942a-1328$ |
*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.
