The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 100000 over imaginary quadratic fields with absolute discriminant 4
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 |
| 8450.6-a1 |
8450.6-a |
$2$ |
$3$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
8450.6 |
\( 2 \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{3} \cdot 5^{9} \cdot 13^{4} \) |
$1.71349$ |
$(a+1), (-a-2), (2a+1), (2a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$0.129578811$ |
$1.219656637$ |
1.896499887 |
\( \frac{2412409957}{62500} a - \frac{352209201}{62500} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -25 i + 57\) , \( 144 i + 130\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(-25i+57\right){x}+144i+130$ |
| 8450.6-a2 |
8450.6-a |
$2$ |
$3$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
8450.6 |
\( 2 \cdot 5^{2} \cdot 13^{2} \) |
\( 2 \cdot 5^{3} \cdot 13^{4} \) |
$1.71349$ |
$(a+1), (-a-2), (2a+1), (2a+3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.388736433$ |
$3.658969912$ |
1.896499887 |
\( \frac{40729}{50} a + \frac{80613}{50} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -3\) , \( i\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}-3{x}+i$ |
| 8450.6-b1 |
8450.6-b |
$2$ |
$3$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
8450.6 |
\( 2 \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{27} \cdot 5^{5} \cdot 13^{8} \) |
$1.71349$ |
$(a+1), (-a-2), (2a+1), (2a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$0.179846587$ |
1.079079527 |
\( \frac{577233446569}{2048000} a - \frac{853138583973}{2048000} \) |
\( \bigl[i\) , \( 0\) , \( i\) , \( -936 i - 4280\) , \( 35656 i + 106340\bigr] \) |
${y}^2+i{x}{y}+i{y}={x}^{3}+\left(-936i-4280\right){x}+35656i+106340$ |
| 8450.6-b2 |
8450.6-b |
$2$ |
$3$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
8450.6 |
\( 2 \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{9} \cdot 5^{7} \cdot 13^{8} \) |
$1.71349$ |
$(a+1), (-a-2), (2a+1), (2a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$0.539539763$ |
1.079079527 |
\( -\frac{2944910839}{500000} a + \frac{24766677}{500000} \) |
\( \bigl[i\) , \( 0\) , \( i\) , \( 104 i - 185\) , \( 952 i - 732\bigr] \) |
${y}^2+i{x}{y}+i{y}={x}^{3}+\left(104i-185\right){x}+952i-732$ |
| 8450.6-c1 |
8450.6-c |
$1$ |
$1$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
8450.6 |
\( 2 \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{7} \cdot 5^{7} \cdot 13^{2} \) |
$1.71349$ |
$(a+1), (-a-2), (2a+1), (2a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 5 \cdot 7 \) |
$0.011180466$ |
$2.341190353$ |
3.664584158 |
\( -\frac{2255889}{50000} a + \frac{83040173}{50000} \) |
\( \bigl[1\) , \( 0\) , \( i + 1\) , \( 8 i - 1\) , \( i - 2\bigr] \) |
${y}^2+{x}{y}+\left(i+1\right){y}={x}^{3}+\left(8i-1\right){x}+i-2$ |
*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.
