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 11
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 |
| 36864.3-CMb1 |
36864.3-CMb |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{6} \) |
$4.10663$ |
$(a-1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{yes}$ |
$-11$ |
$\mathrm{U}(1)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$3.328893478$ |
2.007398297 |
\( -32768 \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -3 a - 6\) , \( 6 a - 1\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-3a-6\right){x}+6a-1$ |
| 36864.3-CMa1 |
36864.3-CMa |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{6} \) |
$4.10663$ |
$(a-1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{yes}$ |
$-11$ |
$\mathrm{U}(1)$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$3.328893478$ |
2.007398297 |
\( -32768 \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -3 a - 6\) , \( -6 a + 1\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-3a-6\right){x}-6a+1$ |
| 36864.3-a1 |
36864.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{6} \) |
$4.10663$ |
$(a-1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2Cn, 3Nn |
$1$ |
\( 1 \) |
$1.135323379$ |
$3.868333389$ |
5.296721352 |
\( 1024 a + 1536 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -2 a + 2\) , \( -2\bigr] \) |
${y}^2={x}^{3}+\left(-2a+2\right){x}-2$ |
| 36864.3-b1 |
36864.3-b |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{6} \) |
$4.10663$ |
$(a-1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2Cn, 3Nn |
$1$ |
\( 1 \) |
$0.655366408$ |
$3.868333389$ |
3.057537011 |
\( -1024 a + 2560 \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( a - 4\) , \( -1\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(a-4\right){x}-1$ |
| 36864.3-c1 |
36864.3-c |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{28} \cdot 3^{6} \) |
$4.10663$ |
$(a-1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cn |
$1$ |
\( 1 \) |
$1$ |
$1.582969524$ |
0.954566539 |
\( 1024 \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 5 a + 10\) , \( -6 a - 17\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(5a+10\right){x}-6a-17$ |
| 36864.3-d1 |
36864.3-d |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{24} \cdot 3^{9} \) |
$4.10663$ |
$(a-1), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$0.696305545$ |
$1.508042231$ |
2.532835602 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -8 a + 20\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(-8a+20\right){x}$ |
| 36864.3-d2 |
36864.3-d |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{9} \) |
$4.10663$ |
$(a-1), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1.392611090$ |
$3.016084463$ |
2.532835602 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 2 a - 5\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(2a-5\right){x}$ |
| 36864.3-e1 |
36864.3-e |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{3} \) |
$4.10663$ |
$(a-1), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$5.224011530$ |
1.575098740 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( a - 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}$ |
| 36864.3-e2 |
36864.3-e |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{24} \cdot 3^{3} \) |
$4.10663$ |
$(a-1), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$2.612005765$ |
1.575098740 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -4 a + 4\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(-4a+4\right){x}$ |
| 36864.3-f1 |
36864.3-f |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{6} \) |
$4.10663$ |
$(a-1), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$3.969390382$ |
1.196816231 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -a - 2\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(-a-2\right){x}$ |
| 36864.3-f2 |
36864.3-f |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{24} \cdot 3^{6} \) |
$4.10663$ |
$(a-1), (2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$1.984695191$ |
1.196816231 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 4 a + 8\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(4a+8\right){x}$ |
| 36864.3-f3 |
36864.3-f |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{30} \cdot 3^{6} \) |
$4.10663$ |
$(a-1), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$1$ |
$0.992347595$ |
1.196816231 |
\( 287496 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 44 a + 88\) , \( 224 a - 560\bigr] \) |
${y}^2={x}^{3}+\left(44a+88\right){x}+224a-560$ |
| 36864.3-f4 |
36864.3-f |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{30} \cdot 3^{6} \) |
$4.10663$ |
$(a-1), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$1$ |
$0.992347595$ |
1.196816231 |
\( 287496 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 44 a + 88\) , \( -224 a + 560\bigr] \) |
${y}^2={x}^{3}+\left(44a+88\right){x}-224a+560$ |
| 36864.3-g1 |
36864.3-g |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{24} \cdot 3^{3} \) |
$4.10663$ |
$(a-1), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$2.612005765$ |
1.575098740 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 4 a - 4\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(4a-4\right){x}$ |
| 36864.3-g2 |
36864.3-g |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{3} \) |
$4.10663$ |
$(a-1), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$5.224011530$ |
1.575098740 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -a + 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}$ |
| 36864.3-h1 |
36864.3-h |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{9} \) |
$4.10663$ |
$(a-1), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$4.868571633$ |
$3.016084463$ |
8.854799193 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -2 a + 5\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(-2a+5\right){x}$ |
| 36864.3-h2 |
36864.3-h |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{24} \cdot 3^{9} \) |
$4.10663$ |
$(a-1), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{3} \) |
$2.434285816$ |
$1.508042231$ |
8.854799193 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 8 a - 20\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(8a-20\right){x}$ |
| 36864.3-i1 |
36864.3-i |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{6} \) |
$4.10663$ |
$(a-1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2Cn, 3Nn |
$1$ |
\( 1 \) |
$0.643837466$ |
$3.868333389$ |
3.003750049 |
\( -1024 a + 2560 \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( a - 4\) , \( 1\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(a-4\right){x}+1$ |
| 36864.3-j1 |
36864.3-j |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{28} \cdot 3^{6} \) |
$4.10663$ |
$(a-1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cn |
$1$ |
\( 1 \) |
$1$ |
$1.582969524$ |
0.954566539 |
\( 1024 \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 5 a + 10\) , \( 6 a + 17\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(5a+10\right){x}+6a+17$ |
| 36864.3-k1 |
36864.3-k |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
36864.3 |
\( 2^{12} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{6} \) |
$4.10663$ |
$(a-1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2Cn, 3Nn |
$1$ |
\( 1 \) |
$2.864824586$ |
$3.868333389$ |
13.36551138 |
\( 1024 a + 1536 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -2 a + 2\) , \( 2\bigr] \) |
${y}^2={x}^{3}+\left(-2a+2\right){x}+2$ |
*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.
