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 55
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 |
| 245.2-a1 |
245.2-a |
$3$ |
$9$ |
\(\Q(\sqrt{-55}) \) |
$2$ |
$[0, 1]$ |
245.2 |
\( 5 \cdot 7^{2} \) |
\( 5^{30} \cdot 7^{2} \) |
$2.62187$ |
$(5,a+2), (7,a), (7,a+6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$0.833160004$ |
$0.774975202$ |
0.696505999 |
\( -\frac{250523582464}{13671875} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -3283\) , \( -74657\bigr] \) |
${y}^2+{y}={x}^3-{x}^2-3283{x}-74657$ |
| 245.2-a2 |
245.2-a |
$3$ |
$9$ |
\(\Q(\sqrt{-55}) \) |
$2$ |
$[0, 1]$ |
245.2 |
\( 5 \cdot 7^{2} \) |
\( 5^{14} \cdot 7^{2} \) |
$2.62187$ |
$(5,a+2), (7,a), (7,a+6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$0.092573333$ |
$6.974776820$ |
0.696505999 |
\( -\frac{262144}{35} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -33\) , \( 93\bigr] \) |
${y}^2+{y}={x}^3-{x}^2-33{x}+93$ |
| 245.2-a3 |
245.2-a |
$3$ |
$9$ |
\(\Q(\sqrt{-55}) \) |
$2$ |
$[0, 1]$ |
245.2 |
\( 5 \cdot 7^{2} \) |
\( 5^{18} \cdot 7^{6} \) |
$2.62187$ |
$(5,a+2), (7,a), (7,a+6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$0.277720001$ |
$2.324925606$ |
0.696505999 |
\( \frac{71991296}{42875} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( 217\) , \( -282\bigr] \) |
${y}^2+{y}={x}^3-{x}^2+217{x}-282$ |
| 245.2-b1 |
245.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{-55}) \) |
$2$ |
$[0, 1]$ |
245.2 |
\( 5 \cdot 7^{2} \) |
\( 5^{18} \cdot 7^{2} \) |
$2.62187$ |
$(5,a+2), (7,a), (7,a+6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \) |
$2.993171204$ |
$0.774975202$ |
2.502234495 |
\( -\frac{250523582464}{13671875} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -131\) , \( -650\bigr] \) |
${y}^2+{y}={x}^3+{x}^2-131{x}-650$ |
| 245.2-b2 |
245.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{-55}) \) |
$2$ |
$[0, 1]$ |
245.2 |
\( 5 \cdot 7^{2} \) |
\( 5^{2} \cdot 7^{2} \) |
$2.62187$ |
$(5,a+2), (7,a), (7,a+6)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \) |
$2.993171204$ |
$6.974776820$ |
2.502234495 |
\( -\frac{262144}{35} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{y}={x}^3+{x}^2-{x}$ |
| 245.2-b3 |
245.2-b |
$3$ |
$9$ |
\(\Q(\sqrt{-55}) \) |
$2$ |
$[0, 1]$ |
245.2 |
\( 5 \cdot 7^{2} \) |
\( 5^{6} \cdot 7^{6} \) |
$2.62187$ |
$(5,a+2), (7,a), (7,a+6)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.997723734$ |
$2.324925606$ |
2.502234495 |
\( \frac{71991296}{42875} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 9\) , \( 1\bigr] \) |
${y}^2+{y}={x}^3+{x}^2+9{x}+1$ |
*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.
