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 |
| 12800.3-a1 |
12800.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
12800.3 |
\( 2^{9} \cdot 5^{2} \) |
\( 2^{17} \cdot 5^{6} \) |
$1.90095$ |
$(a+1), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.381642496$ |
$1.753417822$ |
2.676715022 |
\( -118792 a - 11528 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -36 i - 6\) , \( 86 i - 40\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-36i-6\right){x}+86i-40$ |
| 12800.3-a2 |
12800.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
12800.3 |
\( 2^{9} \cdot 5^{2} \) |
\( 2^{17} \cdot 5^{6} \) |
$1.90095$ |
$(a+1), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1.526569986$ |
$1.753417822$ |
2.676715022 |
\( 118792 a - 11528 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 4 i - 36\) , \( 12 i - 72\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(4i-36\right){x}+12i-72$ |
| 12800.3-a3 |
12800.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
12800.3 |
\( 2^{9} \cdot 5^{2} \) |
\( 2^{10} \cdot 5^{6} \) |
$1.90095$ |
$(a+1), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.763284993$ |
$3.506835645$ |
2.676715022 |
\( 128 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -i - 1\) , \( 2 i - 2\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-i-1\right){x}+2i-2$ |
| 12800.3-a4 |
12800.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
12800.3 |
\( 2^{9} \cdot 5^{2} \) |
\( 2^{20} \cdot 5^{6} \) |
$1.90095$ |
$(a+1), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.381642496$ |
$1.753417822$ |
2.676715022 |
\( 10976 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 14 i + 19\) , \( -18 i + 39\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(14i+19\right){x}-18i+39$ |
| 12800.3-b1 |
12800.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
12800.3 |
\( 2^{9} \cdot 5^{2} \) |
\( 2^{17} \cdot 5^{7} \) |
$1.90095$ |
$(a+1), (2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.320233139$ |
1.320233139 |
\( -\frac{2150376}{5} a - \frac{2556152}{5} \) |
\( \bigl[0\) , \( i - 1\) , \( 0\) , \( 70 i + 47\) , \( 21 i - 339\bigr] \) |
${y}^2={x}^{3}+\left(i-1\right){x}^{2}+\left(70i+47\right){x}+21i-339$ |
| 12800.3-b2 |
12800.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
12800.3 |
\( 2^{9} \cdot 5^{2} \) |
\( 2^{20} \cdot 5^{7} \) |
$1.90095$ |
$(a+1), (2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.320233139$ |
1.320233139 |
\( \frac{389056}{5} a - \frac{211168}{5} \) |
\( \bigl[0\) , \( -i + 1\) , \( 0\) , \( 60 i - 8\) , \( -132 i - 84\bigr] \) |
${y}^2={x}^{3}+\left(-i+1\right){x}^{2}+\left(60i-8\right){x}-132i-84$ |
| 12800.3-b3 |
12800.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
12800.3 |
\( 2^{9} \cdot 5^{2} \) |
\( 2^{10} \cdot 5^{8} \) |
$1.90095$ |
$(a+1), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$2.640466279$ |
1.320233139 |
\( -\frac{13056}{25} a + \frac{27008}{25} \) |
\( \bigl[0\) , \( i - 1\) , \( 0\) , \( 5 i + 2\) , \( 2 i - 6\bigr] \) |
${y}^2={x}^{3}+\left(i-1\right){x}^{2}+\left(5i+2\right){x}+2i-6$ |
| 12800.3-b4 |
12800.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
12800.3 |
\( 2^{9} \cdot 5^{2} \) |
\( 2^{17} \cdot 5^{10} \) |
$1.90095$ |
$(a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.320233139$ |
1.320233139 |
\( \frac{615304}{625} a + \frac{1440328}{625} \) |
\( \bigl[0\) , \( i - 1\) , \( 0\) , \( -30 i - 3\) , \( 33 i - 23\bigr] \) |
${y}^2={x}^{3}+\left(i-1\right){x}^{2}+\left(-30i-3\right){x}+33i-23$ |
| 12800.3-c1 |
12800.3-c |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
12800.3 |
\( 2^{9} \cdot 5^{2} \) |
\( 2^{17} \cdot 5^{6} \) |
$1.90095$ |
$(a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.753417822$ |
1.753417822 |
\( -118792 a - 11528 \) |
\( \bigl[0\) , \( i\) , \( 0\) , \( 36 i + 6\) , \( -40 i - 86\bigr] \) |
${y}^2={x}^{3}+i{x}^{2}+\left(36i+6\right){x}-40i-86$ |
| 12800.3-c2 |
12800.3-c |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
12800.3 |
\( 2^{9} \cdot 5^{2} \) |
\( 2^{17} \cdot 5^{6} \) |
$1.90095$ |
$(a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.753417822$ |
1.753417822 |
\( 118792 a - 11528 \) |
\( \bigl[0\) , \( i\) , \( 0\) , \( -4 i + 36\) , \( -72 i - 12\bigr] \) |
${y}^2={x}^{3}+i{x}^{2}+\left(-4i+36\right){x}-72i-12$ |
| 12800.3-c3 |
12800.3-c |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
12800.3 |
\( 2^{9} \cdot 5^{2} \) |
\( 2^{10} \cdot 5^{6} \) |
$1.90095$ |
$(a+1), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$3.506835645$ |
1.753417822 |
\( 128 \) |
\( \bigl[0\) , \( i\) , \( 0\) , \( i + 1\) , \( -2 i - 2\bigr] \) |
${y}^2={x}^{3}+i{x}^{2}+\left(i+1\right){x}-2i-2$ |
| 12800.3-c4 |
12800.3-c |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
12800.3 |
\( 2^{9} \cdot 5^{2} \) |
\( 2^{20} \cdot 5^{6} \) |
$1.90095$ |
$(a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.753417822$ |
1.753417822 |
\( 10976 \) |
\( \bigl[0\) , \( -i\) , \( 0\) , \( -14 i - 19\) , \( 39 i + 18\bigr] \) |
${y}^2={x}^{3}-i{x}^{2}+\left(-14i-19\right){x}+39i+18$ |
| 12800.3-d1 |
12800.3-d |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
12800.3 |
\( 2^{9} \cdot 5^{2} \) |
\( 2^{17} \cdot 5^{7} \) |
$1.90095$ |
$(a+1), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.550722046$ |
$1.320233139$ |
3.367547776 |
\( -\frac{2150376}{5} a - \frac{2556152}{5} \) |
\( \bigl[0\) , \( -i - 1\) , \( 0\) , \( -70 i - 47\) , \( 339 i + 21\bigr] \) |
${y}^2={x}^{3}+\left(-i-1\right){x}^{2}+\left(-70i-47\right){x}+339i+21$ |
| 12800.3-d2 |
12800.3-d |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
12800.3 |
\( 2^{9} \cdot 5^{2} \) |
\( 2^{20} \cdot 5^{7} \) |
$1.90095$ |
$(a+1), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.637680511$ |
$1.320233139$ |
3.367547776 |
\( \frac{389056}{5} a - \frac{211168}{5} \) |
\( \bigl[0\) , \( i + 1\) , \( 0\) , \( -60 i + 8\) , \( 84 i - 132\bigr] \) |
${y}^2={x}^{3}+\left(i+1\right){x}^{2}+\left(-60i+8\right){x}+84i-132$ |
| 12800.3-d3 |
12800.3-d |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
12800.3 |
\( 2^{9} \cdot 5^{2} \) |
\( 2^{10} \cdot 5^{8} \) |
$1.90095$ |
$(a+1), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.275361023$ |
$2.640466279$ |
3.367547776 |
\( -\frac{13056}{25} a + \frac{27008}{25} \) |
\( \bigl[0\) , \( -i - 1\) , \( 0\) , \( -5 i - 2\) , \( 6 i + 2\bigr] \) |
${y}^2={x}^{3}+\left(-i-1\right){x}^{2}+\left(-5i-2\right){x}+6i+2$ |
| 12800.3-d4 |
12800.3-d |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
12800.3 |
\( 2^{9} \cdot 5^{2} \) |
\( 2^{17} \cdot 5^{10} \) |
$1.90095$ |
$(a+1), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.637680511$ |
$1.320233139$ |
3.367547776 |
\( \frac{615304}{625} a + \frac{1440328}{625} \) |
\( \bigl[0\) , \( -i - 1\) , \( 0\) , \( 30 i + 3\) , \( 23 i + 33\bigr] \) |
${y}^2={x}^{3}+\left(-i-1\right){x}^{2}+\left(30i+3\right){x}+23i+33$ |
*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.
