| 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 |
| 1024.1-a1 |
1024.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{12} \) |
$1.13031$ |
$(2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.474870543$ |
$27.50074327$ |
1.460073332 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -1\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}$ |
| 1024.1-a2 |
1024.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{24} \) |
$1.13031$ |
$(2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$0.949741086$ |
$6.875185818$ |
1.460073332 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 4\) , \( 0\bigr] \) |
${y}^2={x}^{3}+4{x}$ |
| 1024.1-j1 |
1024.1-j |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{12} \) |
$1.13031$ |
$(2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$27.50074327$ |
1.537338284 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -\phi - 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(-\phi-1\right){x}$ |
| 1024.1-j2 |
1024.1-j |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{24} \) |
$1.13031$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$6.875185818$ |
1.537338284 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 4 \phi + 4\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(4\phi+4\right){x}$ |
| 4096.1-c1 |
4096.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
4096.1 |
\( 2^{12} \) |
\( - 2^{12} \) |
$1.59850$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5Ns.2.1[2] |
$1$ |
\( 1 \) |
$1$ |
$19.44596205$ |
2.174124652 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( \phi - 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(\phi-1\right){x}$ |
| 4096.1-c2 |
4096.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
4096.1 |
\( 2^{12} \) |
\( - 2^{24} \) |
$1.59850$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5Ns.2.1[2] |
$1$ |
\( 2 \) |
$1$ |
$9.722981027$ |
2.174124652 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -4 \phi + 4\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(-4\phi+4\right){x}$ |
| 4096.1-d1 |
4096.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
4096.1 |
\( 2^{12} \) |
\( 2^{12} \) |
$1.59850$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$13.75037163$ |
1.537338284 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+{x}$ |
| 4096.1-d2 |
4096.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
4096.1 |
\( 2^{12} \) |
\( 2^{24} \) |
$1.59850$ |
$(2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$13.75037163$ |
1.537338284 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -4\) , \( 0\bigr] \) |
${y}^2={x}^{3}-4{x}$ |
| 4096.1-j1 |
4096.1-j |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
4096.1 |
\( 2^{12} \) |
\( - 2^{12} \) |
$1.59850$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5Ns.2.1[2] |
$1$ |
\( 1 \) |
$1$ |
$19.44596205$ |
2.174124652 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -\phi\) , \( 0\bigr] \) |
${y}^2={x}^{3}-\phi{x}$ |
| 4096.1-j2 |
4096.1-j |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
4096.1 |
\( 2^{12} \) |
\( - 2^{24} \) |
$1.59850$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5Ns.2.1[2] |
$1$ |
\( 2 \) |
$1$ |
$9.722981027$ |
2.174124652 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 4 \phi\) , \( 0\bigr] \) |
${y}^2={x}^{3}+4\phi{x}$ |
| 4096.1-v1 |
4096.1-v |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
4096.1 |
\( 2^{12} \) |
\( 2^{12} \) |
$1.59850$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$13.75037163$ |
1.537338284 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( \phi + 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(\phi+1\right){x}$ |
| 4096.1-v2 |
4096.1-v |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
4096.1 |
\( 2^{12} \) |
\( 2^{24} \) |
$1.59850$ |
$(2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$13.75037163$ |
1.537338284 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -4 \phi - 4\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(-4\phi-4\right){x}$ |
| 4096.1-w1 |
4096.1-w |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
4096.1 |
\( 2^{12} \) |
\( - 2^{12} \) |
$1.59850$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5Ns.2.1[2] |
$1$ |
\( 1 \) |
$0.468297444$ |
$19.44596205$ |
2.036274038 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( \phi\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\phi{x}$ |
| 4096.1-w2 |
4096.1-w |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
4096.1 |
\( 2^{12} \) |
\( - 2^{24} \) |
$1.59850$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5Ns.2.1[2] |
$1$ |
\( 2^{2} \) |
$0.234148722$ |
$9.722981027$ |
2.036274038 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -4 \phi\) , \( 0\bigr] \) |
${y}^2={x}^{3}-4\phi{x}$ |
| 4096.1-ba1 |
4096.1-ba |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
4096.1 |
\( 2^{12} \) |
\( - 2^{12} \) |
$1.59850$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5Ns.2.1[2] |
$1$ |
\( 1 \) |
$0.468297444$ |
$19.44596205$ |
2.036274038 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -\phi + 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(-\phi+1\right){x}$ |
| 4096.1-ba2 |
4096.1-ba |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
4096.1 |
\( 2^{12} \) |
\( - 2^{24} \) |
$1.59850$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5$ |
5Ns.2.1[2] |
$1$ |
\( 2^{2} \) |
$0.234148722$ |
$9.722981027$ |
2.036274038 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 4 \phi - 4\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(4\phi-4\right){x}$ |
*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.
