| 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 |
| 729.1-b3 |
729.1-b |
$4$ |
$27$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( 3^{6} \) |
$1.03826$ |
$(3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3Cs.1.1 |
$1$ |
\( 1 \) |
$0.405591086$ |
$28.08911226$ |
1.132214990 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{y}={x}^{3}$ |
| 729.1-b4 |
729.1-b |
$4$ |
$27$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( 3^{18} \) |
$1.03826$ |
$(3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$1.216773260$ |
$3.121012474$ |
1.132214990 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( -7\bigr] \) |
${y}^2+{y}={x}^{3}-7$ |
| 729.1-d1 |
729.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( 3^{6} \) |
$1.03826$ |
$(3)$ |
0 |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$28.08911226$ |
1.395759210 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( \phi + 1\) , \( 0\) , \( -\phi\bigr] \) |
${y}^2+\left(\phi+1\right){y}={x}^{3}-\phi$ |
| 729.1-d2 |
729.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( 3^{18} \) |
$1.03826$ |
$(3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$3.121012474$ |
1.395759210 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( \phi + 1\) , \( 0\) , \( 6 \phi - 14\bigr] \) |
${y}^2+\left(\phi+1\right){y}={x}^{3}+6\phi-14$ |
| 729.1-f1 |
729.1-f |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( 3^{6} \) |
$1.03826$ |
$(3)$ |
0 |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$28.08911226$ |
1.395759210 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( \phi\) , \( 0\) , \( 0\bigr] \) |
${y}^2+\phi{y}={x}^{3}$ |
| 729.1-f2 |
729.1-f |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( 3^{18} \) |
$1.03826$ |
$(3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$3.121012474$ |
1.395759210 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( \phi\) , \( 0\) , \( -7 \phi - 7\bigr] \) |
${y}^2+\phi{y}={x}^{3}-7\phi-7$ |
| 1296.1-a1 |
1296.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
1296.1 |
\( 2^{4} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{6} \) |
$1.19888$ |
$(2), (3)$ |
0 |
$\Z/6\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$17.69503190$ |
1.318909807 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( 1\bigr] \) |
${y}^2={x}^{3}+1$ |
| 1296.1-a2 |
1296.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
1296.1 |
\( 2^{4} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{18} \) |
$1.19888$ |
$(2), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$1.966114656$ |
1.318909807 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 0\) , \( -27\bigr] \) |
${y}^2={x}^{3}-27$ |
| 2025.1-c1 |
2025.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2025.1 |
\( 3^{4} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{10} \) |
$1.34039$ |
$(-2a+1), (3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5Cs[2] |
$1$ |
\( 2 \) |
$1$ |
$2.448734813$ |
2.190215000 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( -25 \phi - 19\bigr] \) |
${y}^2+{y}={x}^{3}-25\phi-19$ |
| 2025.1-c2 |
2025.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2025.1 |
\( 3^{4} \cdot 5^{2} \) |
\( 3^{18} \cdot 5^{10} \) |
$1.34039$ |
$(-2a+1), (3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5Cs[2] |
$1$ |
\( 2 \) |
$1$ |
$2.448734813$ |
2.190215000 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 675 \phi + 506\bigr] \) |
${y}^2+{y}={x}^{3}+675\phi+506$ |
| 2025.1-d1 |
2025.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2025.1 |
\( 3^{4} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{4} \) |
$1.34039$ |
$(-2a+1), (3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3, 5$ |
3B.1.1, 5Cs[2] |
$1$ |
\( 2 \cdot 3 \) |
$0.153640035$ |
$16.42661250$ |
1.504894809 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 1\bigr] \) |
${y}^2+{y}={x}^{3}+1$ |
| 2025.1-d2 |
2025.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2025.1 |
\( 3^{4} \cdot 5^{2} \) |
\( 3^{18} \cdot 5^{4} \) |
$1.34039$ |
$(-2a+1), (3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3, 5$ |
3B.1.2, 5Cs[2] |
$1$ |
\( 2 \) |
$0.460920105$ |
$1.825179167$ |
1.504894809 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( -34\bigr] \) |
${y}^2+{y}={x}^{3}-34$ |
*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.
