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 56
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 |
| 729.4-a1 |
729.4-a |
$2$ |
$3$ |
\(\Q(\sqrt{-14}) \) |
$2$ |
$[0, 1]$ |
729.4 |
\( 3^{6} \) |
\( 2^{12} \cdot 3^{14} \) |
$3.47468$ |
$(3,a+1), (3,a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$3.813130531$ |
$3.372433224$ |
2.291235632 |
\( -3840 a + 5952 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -6 a - 15\) , \( 14 a + 8\bigr] \) |
${y}^2={x}^3+\left(-6a-15\right){x}+14a+8$ |
| 729.4-a2 |
729.4-a |
$2$ |
$3$ |
\(\Q(\sqrt{-14}) \) |
$2$ |
$[0, 1]$ |
729.4 |
\( 3^{6} \) |
\( 3^{26} \) |
$3.47468$ |
$(3,a+1), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1.271043510$ |
$3.372433224$ |
2.291235632 |
\( 3840 a + 5952 \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( -14 a - 27\) , \( 68 a + 55\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3-{x}^2+\left(-14a-27\right){x}+68a+55$ |
| 729.4-b1 |
729.4-b |
$2$ |
$3$ |
\(\Q(\sqrt{-14}) \) |
$2$ |
$[0, 1]$ |
729.4 |
\( 3^{6} \) |
\( 3^{26} \) |
$3.47468$ |
$(3,a+1), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$2.099971139$ |
$3.372433224$ |
3.785494879 |
\( -3840 a + 5952 \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( -14 a - 27\) , \( -27 a + 20\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3-{x}^2+\left(-14a-27\right){x}-27a+20$ |
| 729.4-b2 |
729.4-b |
$2$ |
$3$ |
\(\Q(\sqrt{-14}) \) |
$2$ |
$[0, 1]$ |
729.4 |
\( 3^{6} \) |
\( 3^{14} \) |
$3.47468$ |
$(3,a+1), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B |
$1$ |
\( 3 \) |
$0.699990379$ |
$3.372433224$ |
3.785494879 |
\( 3840 a + 5952 \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( a + 3\) , \( -4 a + 3\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3-{x}^2+\left(a+3\right){x}-4a+3$ |
| 729.4-c1 |
729.4-c |
$4$ |
$27$ |
\(\Q(\sqrt{-14}) \) |
$2$ |
$[0, 1]$ |
729.4 |
\( 3^{6} \) |
\( 2^{12} \cdot 3^{22} \) |
$3.47468$ |
$(3,a+1), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-27$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3Cs |
$1$ |
\( 1 \) |
$6.791192128$ |
$0.900958696$ |
3.270520505 |
\( -12288000 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -1080\) , \( -13662\bigr] \) |
${y}^2={x}^3-1080{x}-13662$ |
| 729.4-c2 |
729.4-c |
$4$ |
$27$ |
\(\Q(\sqrt{-14}) \) |
$2$ |
$[0, 1]$ |
729.4 |
\( 3^{6} \) |
\( 2^{12} \cdot 3^{10} \) |
$3.47468$ |
$(3,a+1), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-27$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3Cs |
$1$ |
\( 3^{2} \) |
$0.251525634$ |
$2.702876088$ |
3.270520505 |
\( -12288000 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -120\) , \( 506\bigr] \) |
${y}^2={x}^3-120{x}+506$ |
| 729.4-c3 |
729.4-c |
$4$ |
$27$ |
\(\Q(\sqrt{-14}) \) |
$2$ |
$[0, 1]$ |
729.4 |
\( 3^{6} \) |
\( 2^{12} \cdot 3^{6} \) |
$3.47468$ |
$(3,a+1), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3Cs |
$1$ |
\( 1 \) |
$0.754576903$ |
$8.108628264$ |
3.270520505 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( 2\bigr] \) |
${y}^2={x}^3+2$ |
| 729.4-c4 |
729.4-c |
$4$ |
$27$ |
\(\Q(\sqrt{-14}) \) |
$2$ |
$[0, 1]$ |
729.4 |
\( 3^{6} \) |
\( 2^{12} \cdot 3^{18} \) |
$3.47468$ |
$(3,a+1), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3Cs |
$1$ |
\( 1 \) |
$2.263730709$ |
$2.702876088$ |
3.270520505 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( -54\bigr] \) |
${y}^2={x}^3-54$ |
| 729.4-d1 |
729.4-d |
$4$ |
$27$ |
\(\Q(\sqrt{-14}) \) |
$2$ |
$[0, 1]$ |
729.4 |
\( 3^{6} \) |
\( 3^{10} \) |
$3.47468$ |
$(3,a+1), (3,a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-27$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3Cs.1.1 |
$1$ |
\( 1 \) |
$23.73408739$ |
$2.702876088$ |
3.809975138 |
\( -12288000 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -30\) , \( 63\bigr] \) |
${y}^2+{y}={x}^3-30{x}+63$ |
| 729.4-d2 |
729.4-d |
$4$ |
$27$ |
\(\Q(\sqrt{-14}) \) |
$2$ |
$[0, 1]$ |
729.4 |
\( 3^{6} \) |
\( 3^{22} \) |
$3.47468$ |
$(3,a+1), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-27$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3Cs.1.1 |
$1$ |
\( 1 \) |
$7.911362464$ |
$0.900958696$ |
3.809975138 |
\( -12288000 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -270\) , \( -1708\bigr] \) |
${y}^2+{y}={x}^3-270{x}-1708$ |
| 729.4-d3 |
729.4-d |
$4$ |
$27$ |
\(\Q(\sqrt{-14}) \) |
$2$ |
$[0, 1]$ |
729.4 |
\( 3^{6} \) |
\( 3^{18} \) |
$3.47468$ |
$(3,a+1), (3,a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$2.637120821$ |
$2.702876088$ |
3.809975138 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( -7\bigr] \) |
${y}^2+{y}={x}^3-7$ |
| 729.4-d4 |
729.4-d |
$4$ |
$27$ |
\(\Q(\sqrt{-14}) \) |
$2$ |
$[0, 1]$ |
729.4 |
\( 3^{6} \) |
\( 3^{6} \) |
$3.47468$ |
$(3,a+1), (3,a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3Cs.1.1 |
$1$ |
\( 1 \) |
$7.911362464$ |
$8.108628264$ |
3.809975138 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{y}={x}^3$ |
| 729.4-e1 |
729.4-e |
$2$ |
$3$ |
\(\Q(\sqrt{-14}) \) |
$2$ |
$[0, 1]$ |
729.4 |
\( 3^{6} \) |
\( 3^{26} \) |
$3.47468$ |
$(3,a+1), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1.271043510$ |
$3.372433224$ |
2.291235632 |
\( -3840 a + 5952 \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( 13 a - 27\) , \( -68 a + 55\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3-{x}^2+\left(13a-27\right){x}-68a+55$ |
| 729.4-e2 |
729.4-e |
$2$ |
$3$ |
\(\Q(\sqrt{-14}) \) |
$2$ |
$[0, 1]$ |
729.4 |
\( 3^{6} \) |
\( 2^{12} \cdot 3^{14} \) |
$3.47468$ |
$(3,a+1), (3,a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$3.813130531$ |
$3.372433224$ |
2.291235632 |
\( 3840 a + 5952 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 6 a - 15\) , \( -14 a + 8\bigr] \) |
${y}^2={x}^3+\left(6a-15\right){x}-14a+8$ |
| 729.4-f1 |
729.4-f |
$2$ |
$3$ |
\(\Q(\sqrt{-14}) \) |
$2$ |
$[0, 1]$ |
729.4 |
\( 3^{6} \) |
\( 3^{14} \) |
$3.47468$ |
$(3,a+1), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B |
$1$ |
\( 3 \) |
$0.699990379$ |
$3.372433224$ |
3.785494879 |
\( -3840 a + 5952 \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( -2 a + 3\) , \( 4 a + 3\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3-{x}^2+\left(-2a+3\right){x}+4a+3$ |
| 729.4-f2 |
729.4-f |
$2$ |
$3$ |
\(\Q(\sqrt{-14}) \) |
$2$ |
$[0, 1]$ |
729.4 |
\( 3^{6} \) |
\( 3^{26} \) |
$3.47468$ |
$(3,a+1), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$2.099971139$ |
$3.372433224$ |
3.785494879 |
\( 3840 a + 5952 \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( 13 a - 27\) , \( 27 a + 20\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3-{x}^2+\left(13a-27\right){x}+27a+20$ |
*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.
