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 |
| 18496.2-a1 |
18496.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
18496.2 |
\( 2^{6} \cdot 17^{2} \) |
\( 2^{12} \cdot 17^{3} \) |
$2.08419$ |
$(a+1), (a+4), (a-4)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.136412351$ |
$3.130759034$ |
3.416593617 |
\( -\frac{1392128}{289} a + \frac{1681088}{289} \) |
\( \bigl[0\) , \( i + 1\) , \( 0\) , \( -6 i - 1\) , \( 3 i - 1\bigr] \) |
${y}^2={x}^{3}+\left(i+1\right){x}^{2}+\left(-6i-1\right){x}+3i-1$ |
| 18496.2-a2 |
18496.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
18496.2 |
\( 2^{6} \cdot 17^{2} \) |
\( 2^{12} \cdot 17^{3} \) |
$2.08419$ |
$(a+1), (a+4), (a-4)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.136412351$ |
$3.130759034$ |
3.416593617 |
\( \frac{1392128}{289} a + \frac{1681088}{289} \) |
\( \bigl[0\) , \( -i + 1\) , \( 0\) , \( 6 i - 1\) , \( -3 i - 1\bigr] \) |
${y}^2={x}^{3}+\left(-i+1\right){x}^{2}+\left(6i-1\right){x}-3i-1$ |
| 18496.2-b1 |
18496.2-b |
$4$ |
$10$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
18496.2 |
\( 2^{6} \cdot 17^{2} \) |
\( 2^{12} \cdot 17^{7} \) |
$2.08419$ |
$(a+1), (a+4), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{3} \) |
$1$ |
$0.447086619$ |
0.894173239 |
\( -\frac{98634893187584}{1419857} a - \frac{475746252574016}{1419857} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -540 i - 1494\) , \( 12296 i + 21652\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-540i-1494\right){x}+12296i+21652$ |
| 18496.2-b2 |
18496.2-b |
$4$ |
$10$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
18496.2 |
\( 2^{6} \cdot 17^{2} \) |
\( 2^{12} \cdot 17^{7} \) |
$2.08419$ |
$(a+1), (a+4), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{3} \) |
$1$ |
$0.447086619$ |
0.894173239 |
\( \frac{98634893187584}{1419857} a - \frac{475746252574016}{1419857} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 540 i - 1494\) , \( -12296 i + 21652\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(540i-1494\right){x}-12296i+21652$ |
| 18496.2-b3 |
18496.2-b |
$4$ |
$10$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
18496.2 |
\( 2^{6} \cdot 17^{2} \) |
\( 2^{12} \cdot 17^{11} \) |
$2.08419$ |
$(a+1), (a+4), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{3} \) |
$1$ |
$0.447086619$ |
0.894173239 |
\( -\frac{37076742277432832}{2015993900449} a + \frac{5307209474080448}{2015993900449} \) |
\( \bigl[0\) , \( -i\) , \( 0\) , \( 140 i + 372\) , \( 2554 i - 1392\bigr] \) |
${y}^2={x}^{3}-i{x}^{2}+\left(140i+372\right){x}+2554i-1392$ |
| 18496.2-b4 |
18496.2-b |
$4$ |
$10$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
18496.2 |
\( 2^{6} \cdot 17^{2} \) |
\( 2^{12} \cdot 17^{11} \) |
$2.08419$ |
$(a+1), (a+4), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{3} \) |
$1$ |
$0.447086619$ |
0.894173239 |
\( \frac{37076742277432832}{2015993900449} a + \frac{5307209474080448}{2015993900449} \) |
\( \bigl[0\) , \( i\) , \( 0\) , \( -140 i + 372\) , \( -2554 i - 1392\bigr] \) |
${y}^2={x}^{3}+i{x}^{2}+\left(-140i+372\right){x}-2554i-1392$ |
| 18496.2-c1 |
18496.2-c |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
18496.2 |
\( 2^{6} \cdot 17^{2} \) |
\( 2^{6} \cdot 17^{4} \) |
$2.08419$ |
$(a+1), (a+4), (a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.399144030$ |
$3.845689929$ |
3.069968357 |
\( \frac{27000}{289} \) |
\( \bigl[i + 1\) , \( i\) , \( i + 1\) , \( -i - 2\) , \( i\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-i-2\right){x}+i$ |
| 18496.2-c2 |
18496.2-c |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
18496.2 |
\( 2^{6} \cdot 17^{2} \) |
\( 2^{12} \cdot 17^{2} \) |
$2.08419$ |
$(a+1), (a+4), (a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.798288061$ |
$3.845689929$ |
3.069968357 |
\( \frac{216000}{17} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -5\) , \( 4\bigr] \) |
${y}^2={x}^{3}-5{x}+4$ |
| 18496.2-d1 |
18496.2-d |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
18496.2 |
\( 2^{6} \cdot 17^{2} \) |
\( 2^{6} \cdot 17^{5} \) |
$2.08419$ |
$(a+1), (a+4), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.675241266$ |
2.675241266 |
\( -\frac{914075552}{83521} a - \frac{401280248}{83521} \) |
\( \bigl[i + 1\) , \( i + 1\) , \( 0\) , \( -9 i + 1\) , \( 5 i - 5\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(i+1\right){x}^{2}+\left(-9i+1\right){x}+5i-5$ |
| 18496.2-d2 |
18496.2-d |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
18496.2 |
\( 2^{6} \cdot 17^{2} \) |
\( 2^{6} \cdot 17^{5} \) |
$2.08419$ |
$(a+1), (a+4), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.675241266$ |
2.675241266 |
\( \frac{914075552}{83521} a - \frac{401280248}{83521} \) |
\( \bigl[i + 1\) , \( i + 1\) , \( i + 1\) , \( 10 i + 1\) , \( -4 i - 15\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(i+1\right){x}^{2}+\left(10i+1\right){x}-4i-15$ |
| 18496.2-d3 |
18496.2-d |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
18496.2 |
\( 2^{6} \cdot 17^{2} \) |
\( 2^{12} \cdot 17^{4} \) |
$2.08419$ |
$(a+1), (a+4), (a-4)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$2.675241266$ |
2.675241266 |
\( -\frac{140608}{289} \) |
\( \bigl[0\) , \( -i\) , \( 0\) , \( 4\) , \( 6 i\bigr] \) |
${y}^2={x}^{3}-i{x}^{2}+4{x}+6i$ |
| 18496.2-d4 |
18496.2-d |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
18496.2 |
\( 2^{6} \cdot 17^{2} \) |
\( 2^{12} \cdot 17^{2} \) |
$2.08419$ |
$(a+1), (a+4), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.675241266$ |
2.675241266 |
\( \frac{19248832}{17} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -22\) , \( -48\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-22{x}-48$ |
| 18496.2-e1 |
18496.2-e |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
18496.2 |
\( 2^{6} \cdot 17^{2} \) |
\( 2^{6} \cdot 17^{4} \) |
$2.08419$ |
$(a+1), (a+4), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.667735283$ |
3.667735283 |
\( \frac{941192}{289} \) |
\( \bigl[i + 1\) , \( -i\) , \( i + 1\) , \( -i + 4\) , \( i\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}-i{x}^{2}+\left(-i+4\right){x}+i$ |
| 18496.2-e2 |
18496.2-e |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
18496.2 |
\( 2^{6} \cdot 17^{2} \) |
\( 2^{12} \cdot 17^{2} \) |
$2.08419$ |
$(a+1), (a+4), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.667735283$ |
3.667735283 |
\( \frac{438976}{17} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -6\) , \( -8\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-6{x}-8$ |
*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.
