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 |
| 27075.2-a1 |
27075.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
27075.2 |
\( 3 \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{2} \cdot 5^{6} \cdot 19^{4} \) |
$1.98537$ |
$(-2a+1), (-5a+3), (-5a+2), (5)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.047706362$ |
$1.959065350$ |
2.590036220 |
\( \frac{357911}{135375} \) |
\( \bigl[a + 1\) , \( a\) , \( 1\) , \( 2 a - 3\) , \( -19\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(2a-3\right){x}-19$ |
| 27075.2-a2 |
27075.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
27075.2 |
\( 3 \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{4} \cdot 5^{12} \cdot 19^{2} \) |
$1.98537$ |
$(-2a+1), (-5a+3), (-5a+2), (5)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.190825450$ |
$0.979532675$ |
2.590036220 |
\( \frac{90458382169}{2671875} \) |
\( \bigl[a + 1\) , \( a\) , \( 1\) , \( -93 a + 92\) , \( -285\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-93a+92\right){x}-285$ |
| 27075.2-b1 |
27075.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
27075.2 |
\( 3 \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{16} \cdot 5^{6} \cdot 19^{2} \) |
$1.98537$ |
$(-2a+1), (-5a+3), (-5a+2), (5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$0.885163107$ |
1.533147475 |
\( \frac{1256216039}{15582375} \) |
\( \bigl[a + 1\) , \( a\) , \( 1\) , \( 23 a - 24\) , \( -199\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(23a-24\right){x}-199$ |
| 27075.2-b2 |
27075.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
27075.2 |
\( 3 \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{4} \cdot 5^{24} \cdot 19^{2} \) |
$1.98537$ |
$(-2a+1), (-5a+3), (-5a+2), (5)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$0.221290776$ |
1.533147475 |
\( \frac{209595169258201}{41748046875} \) |
\( \bigl[a + 1\) , \( a\) , \( 1\) , \( -1237 a + 1236\) , \( 14291\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-1237a+1236\right){x}+14291$ |
| 27075.2-b3 |
27075.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
27075.2 |
\( 3 \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{8} \cdot 5^{12} \cdot 19^{4} \) |
$1.98537$ |
$(-2a+1), (-5a+3), (-5a+2), (5)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$0.442581553$ |
1.533147475 |
\( \frac{6189976379881}{456890625} \) |
\( \bigl[a + 1\) , \( a\) , \( 1\) , \( -382 a + 381\) , \( -2467\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-382a+381\right){x}-2467$ |
| 27075.2-b4 |
27075.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
27075.2 |
\( 3 \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{4} \cdot 5^{6} \cdot 19^{8} \) |
$1.98537$ |
$(-2a+1), (-5a+3), (-5a+2), (5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$0.221290776$ |
1.533147475 |
\( \frac{23977812996389881}{146611125} \) |
\( \bigl[a + 1\) , \( a\) , \( 1\) , \( -6007 a + 6006\) , \( -175717\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-6007a+6006\right){x}-175717$ |
| 27075.2-c1 |
27075.2-c |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
27075.2 |
\( 3 \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{10} \cdot 5^{2} \cdot 19^{4} \) |
$1.98537$ |
$(-2a+1), (-5a+3), (-5a+2), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 5 \) |
$0.067413588$ |
$1.577532480$ |
2.455981678 |
\( \frac{756058031}{438615} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -19 a\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-19a{x}$ |
| 27075.2-c2 |
27075.2-c |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
27075.2 |
\( 3 \cdot 5^{2} \cdot 19^{2} \) |
\( 3^{20} \cdot 5^{4} \cdot 19^{2} \) |
$1.98537$ |
$(-2a+1), (-5a+3), (-5a+2), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 5 \) |
$0.134827177$ |
$0.788766240$ |
2.455981678 |
\( \frac{48587168449}{28048275} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 76 a\) , \( -19\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+76a{x}-19$ |
*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.
