The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 1000 over imaginary quadratic fields with absolute discriminant 104
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 |
| 637.2-a1 |
637.2-a |
$3$ |
$9$ |
\(\Q(\sqrt{-26}) \) |
$2$ |
$[0, 1]$ |
637.2 |
\( 7^{2} \cdot 13 \) |
\( 2^{12} \cdot 7^{2} \cdot 13^{2} \) |
$4.57815$ |
$(7,a+3), (7,a+4), (13,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.556657989$ |
$4.379860585$ |
1.912590750 |
\( -\frac{43614208}{91} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -29\) , \( -71\bigr] \) |
${y}^2={x}^3+{x}^2-29{x}-71$ |
| 637.2-a2 |
637.2-a |
$3$ |
$9$ |
\(\Q(\sqrt{-26}) \) |
$2$ |
$[0, 1]$ |
637.2 |
\( 7^{2} \cdot 13 \) |
\( 2^{12} \cdot 7^{18} \cdot 13^{2} \) |
$4.57815$ |
$(7,a+3), (7,a+4), (13,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$5.009921908$ |
$0.486651176$ |
1.912590750 |
\( -\frac{178643795968}{524596891} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -469\) , \( 9489\bigr] \) |
${y}^2={x}^3+{x}^2-469{x}+9489$ |
| 637.2-a3 |
637.2-a |
$3$ |
$9$ |
\(\Q(\sqrt{-26}) \) |
$2$ |
$[0, 1]$ |
637.2 |
\( 7^{2} \cdot 13 \) |
\( 2^{12} \cdot 7^{6} \cdot 13^{6} \) |
$4.57815$ |
$(7,a+3), (7,a+4), (13,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1.669973969$ |
$1.459953528$ |
1.912590750 |
\( \frac{224755712}{753571} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 51\) , \( -287\bigr] \) |
${y}^2={x}^3+{x}^2+51{x}-287$ |
| 637.2-b1 |
637.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{-26}) \) |
$2$ |
$[0, 1]$ |
637.2 |
\( 7^{2} \cdot 13 \) |
\( 7^{2} \cdot 13^{2} \) |
$4.57815$ |
$(7,a+3), (7,a+4), (13,a)$ |
$2$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$2.973852993$ |
$4.379860585$ |
2.270599754 |
\( -\frac{43614208}{91} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -7\) , \( 5\bigr] \) |
${y}^2+{y}={x}^3+{x}^2-7{x}+5$ |
| 637.2-b2 |
637.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{-26}) \) |
$2$ |
$[0, 1]$ |
637.2 |
\( 7^{2} \cdot 13 \) |
\( 7^{18} \cdot 13^{2} \) |
$4.57815$ |
$(7,a+3), (7,a+4), (13,a)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{4} \) |
$0.036714234$ |
$0.486651176$ |
2.270599754 |
\( -\frac{178643795968}{524596891} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -117\) , \( -1245\bigr] \) |
${y}^2+{y}={x}^3+{x}^2-117{x}-1245$ |
| 637.2-b3 |
637.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{-26}) \) |
$2$ |
$[0, 1]$ |
637.2 |
\( 7^{2} \cdot 13 \) |
\( 7^{6} \cdot 13^{6} \) |
$4.57815$ |
$(7,a+3), (7,a+4), (13,a)$ |
$2$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{3} \) |
$0.330428110$ |
$1.459953528$ |
2.270599754 |
\( \frac{224755712}{753571} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 13\) , \( 42\bigr] \) |
${y}^2+{y}={x}^3+{x}^2+13{x}+42$ |
| 637.2-c1 |
637.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{-26}) \) |
$2$ |
$[0, 1]$ |
637.2 |
\( 7^{2} \cdot 13 \) |
\( 2^{12} \cdot 7^{2} \cdot 13^{2} \) |
$4.57815$ |
$(7,a+3), (7,a+4), (13,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$4$ |
\( 2 \) |
$1$ |
$6.505570680$ |
10.20677902 |
\( \frac{110592}{91} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 4\) , \( -2\bigr] \) |
${y}^2={x}^3+4{x}-2$ |
| 637.2-d1 |
637.2-d |
$1$ |
$1$ |
\(\Q(\sqrt{-26}) \) |
$2$ |
$[0, 1]$ |
637.2 |
\( 7^{2} \cdot 13 \) |
\( 7^{2} \cdot 13^{2} \) |
$4.57815$ |
$(7,a+3), (7,a+4), (13,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.142392150$ |
$6.505570680$ |
0.726682608 |
\( \frac{110592}{91} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{y}={x}^3+{x}$ |
*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.
