The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 100 over imaginary quadratic fields with absolute discriminant 228
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 |
| 19.1-a1 |
19.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 3^{12} \cdot 19^{2} \) |
$2.81705$ |
$(19,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$11.02710220$ |
$0.935309008$ |
5.464357194 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( -6924\) , \( -221746\bigr] \) |
${y}^2+a{y}={x}^3-6924{x}-221746$ |
| 19.1-a2 |
19.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 3^{12} \cdot 19^{6} \) |
$2.81705$ |
$(19,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$3.675700734$ |
$2.805927025$ |
5.464357194 |
\( -\frac{89915392}{6859} \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( -84\) , \( -301\bigr] \) |
${y}^2+a{y}={x}^3-84{x}-301$ |
| 19.1-a3 |
19.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 3^{12} \cdot 19^{2} \) |
$2.81705$ |
$(19,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$1.225233578$ |
$8.417781075$ |
5.464357194 |
\( \frac{32768}{19} \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( 6\) , \( 14\bigr] \) |
${y}^2+a{y}={x}^3+6{x}+14$ |
| 19.1-b1 |
19.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 3^{12} \cdot 19^{2} \) |
$2.81705$ |
$(19,a)$ |
$2$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \) |
$33.64532080$ |
$0.935309008$ |
3.705013890 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -6924\) , \( 221760\bigr] \) |
${y}^2+{y}={x}^3-6924{x}+221760$ |
| 19.1-b2 |
19.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 3^{12} \cdot 19^{6} \) |
$2.81705$ |
$(19,a)$ |
$2$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$3.738368977$ |
$2.805927025$ |
3.705013890 |
\( -\frac{89915392}{6859} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -84\) , \( 315\bigr] \) |
${y}^2+{y}={x}^3-84{x}+315$ |
| 19.1-b3 |
19.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 3^{12} \cdot 19^{2} \) |
$2.81705$ |
$(19,a)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \) |
$0.415374330$ |
$8.417781075$ |
3.705013890 |
\( \frac{32768}{19} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 6\) , \( 0\bigr] \) |
${y}^2+{y}={x}^3+6{x}$ |
| 19.1-c1 |
19.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$2.81705$ |
$(19,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$9.202414482$ |
$0.935309008$ |
4.560153597 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -769\) , \( 8484\bigr] \) |
${y}^2+a{y}={x}^3-{x}^2-769{x}+8484$ |
| 19.1-c2 |
19.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 19^{6} \) |
$2.81705$ |
$(19,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$3.067471494$ |
$2.805927025$ |
4.560153597 |
\( -\frac{89915392}{6859} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -9\) , \( 29\bigr] \) |
${y}^2+a{y}={x}^3-{x}^2-9{x}+29$ |
| 19.1-c3 |
19.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$2.81705$ |
$(19,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$1.022490498$ |
$8.417781075$ |
4.560153597 |
\( \frac{32768}{19} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( 1\) , \( 14\bigr] \) |
${y}^2+a{y}={x}^3-{x}^2+{x}+14$ |
| 19.1-d1 |
19.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$2.81705$ |
$(19,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$4$ |
\( 2 \) |
$1$ |
$0.935309008$ |
0.991077636 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -769\) , \( -8470\bigr] \) |
${y}^2+{y}={x}^3+{x}^2-769{x}-8470$ |
| 19.1-d2 |
19.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 19^{6} \) |
$2.81705$ |
$(19,a)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$4$ |
\( 2 \cdot 3 \) |
$1$ |
$2.805927025$ |
0.991077636 |
\( -\frac{89915392}{6859} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -9\) , \( -15\bigr] \) |
${y}^2+{y}={x}^3+{x}^2-9{x}-15$ |
| 19.1-d3 |
19.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$2.81705$ |
$(19,a)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$4$ |
\( 2 \) |
$1$ |
$8.417781075$ |
0.991077636 |
\( \frac{32768}{19} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{y}={x}^3+{x}^2+{x}$ |
*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.
