The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 50000 over imaginary quadratic fields with absolute discriminant 8
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 |
| 23328.4-a1 |
23328.4-a |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{12} \cdot 3^{6} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$3$ |
3Nn |
$1$ |
\( 2 \) |
$0.360874900$ |
$3.519633048$ |
3.592518868 |
\( -13824 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -6\) , \( -6\bigr] \) |
${y}^2={x}^{3}-6{x}-6$ |
| 23328.4-b1 |
23328.4-b |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{12} \cdot 3^{18} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$3$ |
3Nn |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$0.055027577$ |
$1.173211016$ |
6.573611317 |
\( -13824 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -54\) , \( -162\bigr] \) |
${y}^2={x}^{3}-54{x}-162$ |
| 23328.4-c1 |
23328.4-c |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{12} \cdot 3^{22} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$0.930919368$ |
1.316518796 |
\( 192 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -27\) , \( -108 a\bigr] \) |
${y}^2={x}^{3}-27{x}-108a$ |
| 23328.4-d1 |
23328.4-d |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{12} \cdot 3^{10} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \cdot 3 \) |
$0.215161280$ |
$2.792758105$ |
5.098749669 |
\( 192 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -3\) , \( -4 a\bigr] \) |
${y}^2={x}^{3}-3{x}-4a$ |
| 23328.4-e1 |
23328.4-e |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{6} \cdot 3^{18} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$3$ |
3Nn |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.069203095$ |
$1.889072622$ |
6.655668365 |
\( -216 \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -5\) , \( 24\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}-5{x}+24$ |
| 23328.4-f1 |
23328.4-f |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{6} \cdot 3^{6} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$3$ |
3Nn |
$1$ |
\( 2 \) |
$0.292341859$ |
$5.667217867$ |
4.686039087 |
\( -216 \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 0\) , \( 1\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+1$ |
| 23328.4-g1 |
23328.4-g |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{12} \cdot 3^{18} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3Nn |
$1$ |
\( 2 \cdot 3 \) |
$0.870627779$ |
$0.613621303$ |
4.533140687 |
\( -21024576 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 621\) , \( -4212 a\bigr] \) |
${y}^2={x}^{3}+621{x}-4212a$ |
| 23328.4-h1 |
23328.4-h |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{12} \cdot 3^{6} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3Nn |
$1$ |
\( 2 \) |
$1$ |
$1.840863911$ |
2.603374709 |
\( -21024576 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 69\) , \( -156 a\bigr] \) |
${y}^2={x}^{3}+69{x}-156a$ |
| 23328.4-i1 |
23328.4-i |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{12} \cdot 3^{10} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$0.023973090$ |
$2.755509096$ |
6.726255610 |
\( 1536 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 6\) , \( 2\bigr] \) |
${y}^2={x}^{3}+6{x}+2$ |
| 23328.4-j1 |
23328.4-j |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{12} \cdot 3^{22} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$0.918503032$ |
2.597918890 |
\( 1536 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 54\) , \( 54\bigr] \) |
${y}^2={x}^{3}+54{x}+54$ |
| 23328.4-k1 |
23328.4-k |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{12} \cdot 3^{10} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.622549773$ |
$2.755509096$ |
4.852001448 |
\( 1536 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 6\) , \( -2\bigr] \) |
${y}^2={x}^{3}+6{x}-2$ |
| 23328.4-l1 |
23328.4-l |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{12} \cdot 3^{22} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$2.021940563$ |
$0.918503032$ |
5.252837584 |
\( 1536 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 54\) , \( -54\bigr] \) |
${y}^2={x}^{3}+54{x}-54$ |
| 23328.4-m1 |
23328.4-m |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{12} \cdot 3^{18} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3Nn |
$1$ |
\( 2 \cdot 3 \) |
$0.870627779$ |
$0.613621303$ |
4.533140687 |
\( -21024576 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 621\) , \( 4212 a\bigr] \) |
${y}^2={x}^{3}+621{x}+4212a$ |
| 23328.4-n1 |
23328.4-n |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{12} \cdot 3^{6} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3Nn |
$1$ |
\( 2 \) |
$1$ |
$1.840863911$ |
2.603374709 |
\( -21024576 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 69\) , \( 156 a\bigr] \) |
${y}^2={x}^{3}+69{x}+156a$ |
| 23328.4-o1 |
23328.4-o |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{6} \cdot 3^{18} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$3$ |
3Nn |
$1$ |
\( 2 \) |
$1$ |
$1.889072622$ |
2.671552123 |
\( -216 \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( -6\) , \( -17\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}-6{x}-17$ |
| 23328.4-p1 |
23328.4-p |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{6} \cdot 3^{6} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$3$ |
3Nn |
$1$ |
\( 2 \) |
$0.317388851$ |
$5.667217867$ |
5.087525156 |
\( -216 \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 1\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+{x}$ |
| 23328.4-q1 |
23328.4-q |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{12} \cdot 3^{22} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$0.930919368$ |
1.316518796 |
\( 192 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -27\) , \( 108 a\bigr] \) |
${y}^2={x}^{3}-27{x}+108a$ |
| 23328.4-r1 |
23328.4-r |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{12} \cdot 3^{10} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \cdot 3 \) |
$0.215161280$ |
$2.792758105$ |
5.098749669 |
\( 192 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -3\) , \( 4 a\bigr] \) |
${y}^2={x}^{3}-3{x}+4a$ |
| 23328.4-s1 |
23328.4-s |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{12} \cdot 3^{6} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$3$ |
3Nn |
$1$ |
\( 2 \) |
$0.555032222$ |
$3.519633048$ |
5.525359971 |
\( -13824 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -6\) , \( 6\bigr] \) |
${y}^2={x}^{3}-6{x}+6$ |
| 23328.4-t1 |
23328.4-t |
$1$ |
$1$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
23328.4 |
\( 2^{5} \cdot 3^{6} \) |
\( 2^{12} \cdot 3^{18} \) |
$3.12358$ |
$(a), (-a-1), (a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$3$ |
3Nn |
$1$ |
\( 2^{2} \) |
$1$ |
$1.173211016$ |
3.318341860 |
\( -13824 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -54\) , \( 162\bigr] \) |
${y}^2={x}^{3}-54{x}+162$ |
*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.
