The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 1000 over imaginary quadratic fields with absolute discriminant 15
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 |
| 605.1-a1 |
605.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
605.1 |
\( 5 \cdot 11^{2} \) |
\( 2^{12} \cdot 5^{2} \cdot 11^{2} \) |
$1.71642$ |
$(5,a+2), (11)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.786066235$ |
$7.082488443$ |
1.437471977 |
\( \frac{59319}{55} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -a - 3\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-a-3\right){x}$ |
| 605.1-a2 |
605.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
605.1 |
\( 5 \cdot 11^{2} \) |
\( 2^{12} \cdot 5^{4} \cdot 11^{4} \) |
$1.71642$ |
$(5,a+2), (11)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.786066235$ |
$3.541244221$ |
1.437471977 |
\( \frac{8120601}{3025} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 4 a + 12\) , \( 9 a - 21\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(4a+12\right){x}+9a-21$ |
| 605.1-a3 |
605.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
605.1 |
\( 5 \cdot 11^{2} \) |
\( 2^{12} \cdot 5^{2} \cdot 11^{8} \) |
$1.71642$ |
$(5,a+2), (11)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.196516558$ |
$1.770622110$ |
1.437471977 |
\( \frac{2749884201}{73205} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 29 a + 87\) , \( -156 a + 364\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(29a+87\right){x}-156a+364$ |
| 605.1-a4 |
605.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
605.1 |
\( 5 \cdot 11^{2} \) |
\( 2^{12} \cdot 5^{8} \cdot 11^{2} \) |
$1.71642$ |
$(5,a+2), (11)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.786066235$ |
$1.770622110$ |
1.437471977 |
\( \frac{22930509321}{6875} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 59 a + 177\) , \( 570 a - 1330\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(59a+177\right){x}+570a-1330$ |
| 605.1-b1 |
605.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
605.1 |
\( 5 \cdot 11^{2} \) |
\( 5^{2} \cdot 11^{2} \) |
$1.71642$ |
$(5,a+2), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1.078747509$ |
$7.082488443$ |
3.945390974 |
\( \frac{59319}{55} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2+{x}$ |
| 605.1-b2 |
605.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
605.1 |
\( 5 \cdot 11^{2} \) |
\( 5^{4} \cdot 11^{4} \) |
$1.71642$ |
$(5,a+2), (11)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$2.157495018$ |
$3.541244221$ |
3.945390974 |
\( \frac{8120601}{3025} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -4\) , \( 3\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-4{x}+3$ |
| 605.1-b3 |
605.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
605.1 |
\( 5 \cdot 11^{2} \) |
\( 5^{2} \cdot 11^{8} \) |
$1.71642$ |
$(5,a+2), (11)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.078747509$ |
$1.770622110$ |
3.945390974 |
\( \frac{2749884201}{73205} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -29\) , \( -52\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-29{x}-52$ |
| 605.1-b4 |
605.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-15}) \) |
$2$ |
$[0, 1]$ |
605.1 |
\( 5 \cdot 11^{2} \) |
\( 5^{8} \cdot 11^{2} \) |
$1.71642$ |
$(5,a+2), (11)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$4.314990037$ |
$1.770622110$ |
3.945390974 |
\( \frac{22930509321}{6875} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -59\) , \( 190\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-59{x}+190$ |
*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.
