| 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-a1 |
729.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( - 3^{6} \) |
$1.03826$ |
$(3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$0.280968537$ |
$43.64077547$ |
1.218576037 |
\( 20439 a - 13095 \) |
\( \bigl[1\) , \( -1\) , \( \phi\) , \( \phi - 3\) , \( -2 \phi + 3\bigr] \) |
${y}^2+{x}{y}+\phi{y}={x}^{3}-{x}^{2}+\left(\phi-3\right){x}-2\phi+3$ |
| 729.1-a2 |
729.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( - 3^{18} \) |
$1.03826$ |
$(3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$0.093656179$ |
$4.848975052$ |
1.218576037 |
\( 72927 a + 46656 \) |
\( \bigl[1\) , \( -1\) , \( \phi\) , \( -14 \phi + 12\) , \( -7 \phi + 1\bigr] \) |
${y}^2+{x}{y}+\phi{y}={x}^{3}-{x}^{2}+\left(-14\phi+12\right){x}-7\phi+1$ |
| 729.1-b1 |
729.1-b |
$4$ |
$27$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( 3^{10} \) |
$1.03826$ |
$(3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-27$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$0.135197028$ |
$28.08911226$ |
1.132214990 |
\( -12288000 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -30\) , \( 63\bigr] \) |
${y}^2+{y}={x}^{3}-30{x}+63$ |
| 729.1-b2 |
729.1-b |
$4$ |
$27$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( 3^{22} \) |
$1.03826$ |
$(3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-27$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$3.650319782$ |
$0.346779163$ |
1.132214990 |
\( -12288000 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -270\) , \( -1708\bigr] \) |
${y}^2+{y}={x}^{3}-270{x}-1708$ |
| 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-c1 |
729.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( - 3^{18} \) |
$1.03826$ |
$(3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$2.858635467$ |
1.278420645 |
\( -20439 a + 7344 \) |
\( \bigl[1\) , \( -1\) , \( \phi\) , \( -14 \phi - 15\) , \( -34 \phi - 26\bigr] \) |
${y}^2+{x}{y}+\phi{y}={x}^{3}-{x}^{2}+\left(-14\phi-15\right){x}-34\phi-26$ |
| 729.1-c2 |
729.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( - 3^{6} \) |
$1.03826$ |
$(3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$25.72771921$ |
1.278420645 |
\( -72927 a + 119583 \) |
\( \bigl[1\) , \( -1\) , \( \phi\) , \( \phi\) , \( -\phi\bigr] \) |
${y}^2+{x}{y}+\phi{y}={x}^{3}-{x}^{2}+\phi{x}-\phi$ |
| 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-e1 |
729.1-e |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( - 3^{18} \) |
$1.03826$ |
$(3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$2.858635467$ |
1.278420645 |
\( 20439 a - 13095 \) |
\( \bigl[1\) , \( -1\) , \( \phi + 1\) , \( 13 \phi - 29\) , \( 33 \phi - 60\bigr] \) |
${y}^2+{x}{y}+\left(\phi+1\right){y}={x}^{3}-{x}^{2}+\left(13\phi-29\right){x}+33\phi-60$ |
| 729.1-e2 |
729.1-e |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( - 3^{6} \) |
$1.03826$ |
$(3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$25.72771921$ |
1.278420645 |
\( 72927 a + 46656 \) |
\( \bigl[1\) , \( -1\) , \( \phi + 1\) , \( -2 \phi + 1\) , \( -1\bigr] \) |
${y}^2+{x}{y}+\left(\phi+1\right){y}={x}^{3}-{x}^{2}+\left(-2\phi+1\right){x}-1$ |
| 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 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( \phi\) , \( 0\) , \( -7 \phi - 7\bigr] \) |
${y}^2+\phi{y}={x}^{3}-7\phi-7$ |
| 729.1-g1 |
729.1-g |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( - 3^{6} \) |
$1.03826$ |
$(3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$0.280968537$ |
$43.64077547$ |
1.218576037 |
\( -20439 a + 7344 \) |
\( \bigl[1\) , \( -1\) , \( \phi + 1\) , \( -2 \phi - 2\) , \( \phi + 1\bigr] \) |
${y}^2+{x}{y}+\left(\phi+1\right){y}={x}^{3}-{x}^{2}+\left(-2\phi-2\right){x}+\phi+1$ |
| 729.1-g2 |
729.1-g |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
729.1 |
\( 3^{6} \) |
\( - 3^{18} \) |
$1.03826$ |
$(3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$0.093656179$ |
$4.848975052$ |
1.218576037 |
\( -72927 a + 119583 \) |
\( \bigl[1\) , \( -1\) , \( \phi + 1\) , \( 13 \phi - 2\) , \( 6 \phi - 6\bigr] \) |
${y}^2+{x}{y}+\left(\phi+1\right){y}={x}^{3}-{x}^{2}+\left(13\phi-2\right){x}+6\phi-6$ |
*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.
