The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 150000 over imaginary quadratic fields with absolute discriminant 3
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 |
| 8281.5-a1 |
8281.5-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
8281.5 |
\( 7^{2} \cdot 13^{2} \) |
\( 7^{2} \cdot 13^{2} \) |
$1.47645$ |
$(-3a+1), (3a-2), (-4a+1), (4a-3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1[2] |
$1$ |
\( 1 \) |
$1.059245086$ |
$4.379860585$ |
1.190456688 |
\( -\frac{43614208}{91} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( -7 a + 7\) , \( 5\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(-7a+7\right){x}+5$ |
| 8281.5-a2 |
8281.5-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
8281.5 |
\( 7^{2} \cdot 13^{2} \) |
\( 7^{18} \cdot 13^{2} \) |
$1.47645$ |
$(-3a+1), (3a-2), (-4a+1), (4a-3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1[2] |
$1$ |
\( 3^{4} \) |
$0.117693898$ |
$0.486651176$ |
1.190456688 |
\( -\frac{178643795968}{524596891} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( -117 a + 117\) , \( -1245\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(-117a+117\right){x}-1245$ |
| 8281.5-a3 |
8281.5-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
8281.5 |
\( 7^{2} \cdot 13^{2} \) |
\( 7^{6} \cdot 13^{6} \) |
$1.47645$ |
$(-3a+1), (3a-2), (-4a+1), (4a-3)$ |
$1$ |
$\Z/3\Z\oplus\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1[2] |
$1$ |
\( 3^{4} \) |
$0.353081695$ |
$1.459953528$ |
1.190456688 |
\( \frac{224755712}{753571} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 13 a - 13\) , \( 42\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(13a-13\right){x}+42$ |
| 8281.5-a4 |
8281.5-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
8281.5 |
\( 7^{2} \cdot 13^{2} \) |
\( 7^{2} \cdot 13^{10} \) |
$1.47645$ |
$(-3a+1), (3a-2), (-4a+1), (4a-3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.1[2] |
$1$ |
\( 3^{2} \) |
$1.059245086$ |
$0.486651176$ |
1.190456688 |
\( -\frac{1548384163323379712}{74231495611} a + \frac{4883537544409743360}{74231495611} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 503 a - 1273\) , \( -9086 a + 16555\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(503a-1273\right){x}-9086a+16555$ |
| 8281.5-a5 |
8281.5-a |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
8281.5 |
\( 7^{2} \cdot 13^{2} \) |
\( 7^{2} \cdot 13^{10} \) |
$1.47645$ |
$(-3a+1), (3a-2), (-4a+1), (4a-3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.1[2] |
$1$ |
\( 3^{2} \) |
$1.059245086$ |
$0.486651176$ |
1.190456688 |
\( \frac{1548384163323379712}{74231495611} a + \frac{476450483012337664}{10604499373} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 1273 a - 503\) , \( 9086 a + 7469\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(1273a-503\right){x}+9086a+7469$ |
| 8281.5-b1 |
8281.5-b |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
8281.5 |
\( 7^{2} \cdot 13^{2} \) |
\( 7^{2} \cdot 13^{2} \) |
$1.47645$ |
$(-3a+1), (3a-2), (-4a+1), (4a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.142392150$ |
$6.505570680$ |
2.139295675 |
\( \frac{110592}{91} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( a - 1\) , \( 0\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a-1\right){x}$ |
*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.
