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 |
| 1024.1-CMb1 |
1024.1-CMb |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{18} \) |
$1.01098$ |
$(a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{yes}$ |
$-4$ |
$\mathrm{U}(1)$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$4.861490513$ |
1.215372628 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 2 i\) , \( 0\bigr] \) |
${y}^2={x}^{3}+2i{x}$ |
| 1024.1-CMb2 |
1024.1-CMb |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{24} \) |
$1.01098$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{yes}$ |
$-16$ |
$\mathrm{U}(1)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$2.430745256$ |
1.215372628 |
\( 287496 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 22 i\) , \( -28 i - 28\bigr] \) |
${y}^2={x}^{3}+22i{x}-28i-28$ |
| 1024.1-CMa1 |
1024.1-CMa |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{18} \) |
$1.01098$ |
$(a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{yes}$ |
$-4$ |
$\mathrm{U}(1)$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$4.861490513$ |
1.215372628 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -2 i\) , \( 0\bigr] \) |
${y}^2={x}^{3}-2i{x}$ |
| 1024.1-CMa2 |
1024.1-CMa |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{24} \) |
$1.01098$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{yes}$ |
$-16$ |
$\mathrm{U}(1)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$2.430745256$ |
1.215372628 |
\( 287496 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -22 i\) , \( 28 i - 28\bigr] \) |
${y}^2={x}^{3}-22i{x}+28i-28$ |
| 1024.1-a1 |
1024.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{23} \) |
$1.01098$ |
$(a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.864662329$ |
$2.772397005$ |
1.198593625 |
\( -118792 a - 11528 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 10 i + 11\) , \( 6 i - 23\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(10i+11\right){x}+6i-23$ |
| 1024.1-a2 |
1024.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{23} \) |
$1.01098$ |
$(a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.864662329$ |
$2.772397005$ |
1.198593625 |
\( 118792 a - 11528 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -10 i + 11\) , \( 6 i + 23\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-10i+11\right){x}+6i+23$ |
| 1024.1-a3 |
1024.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{16} \) |
$1.01098$ |
$(a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.432331164$ |
$5.544794010$ |
1.198593625 |
\( 128 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 1\) , \( 1\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+{x}+1$ |
| 1024.1-a4 |
1024.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{14} \) |
$1.01098$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.216165582$ |
$5.544794010$ |
1.198593625 |
\( 10976 \) |
\( \bigl[0\) , \( i\) , \( 0\) , \( 2\) , \( 2 i\bigr] \) |
${y}^2={x}^{3}+i{x}^{2}+2{x}+2i$ |
| 1024.1-b1 |
1024.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{23} \) |
$1.01098$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$2.772397005$ |
1.386198502 |
\( -118792 a - 11528 \) |
\( \bigl[0\) , \( -i\) , \( 0\) , \( -10 i - 11\) , \( 23 i + 6\bigr] \) |
${y}^2={x}^{3}-i{x}^{2}+\left(-10i-11\right){x}+23i+6$ |
| 1024.1-b2 |
1024.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{23} \) |
$1.01098$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$2.772397005$ |
1.386198502 |
\( 118792 a - 11528 \) |
\( \bigl[0\) , \( i\) , \( 0\) , \( 10 i - 11\) , \( -23 i + 6\bigr] \) |
${y}^2={x}^{3}+i{x}^{2}+\left(10i-11\right){x}-23i+6$ |
| 1024.1-b3 |
1024.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{16} \) |
$1.01098$ |
$(a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$5.544794010$ |
1.386198502 |
\( 128 \) |
\( \bigl[0\) , \( i\) , \( 0\) , \( -1\) , \( -i\bigr] \) |
${y}^2={x}^{3}+i{x}^{2}-{x}-i$ |
| 1024.1-b4 |
1024.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{14} \) |
$1.01098$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$5.544794010$ |
1.386198502 |
\( 10976 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -2\) , \( 2\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-2{x}+2$ |
*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.
