| 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 |
| 40000.3-CMc1 |
40000.3-CMc |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
40000.3 |
\( 2^{6} \cdot 5^{4} \) |
\( 2^{12} \cdot 5^{12} \) |
$2.52746$ |
$(a+1), (-a-2), (2a+1)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{yes}$ |
$-4$ |
$\mathrm{U}(1)$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$0.225502032$ |
$1.375037163$ |
4.961178807 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -25\) , \( 0\bigr] \) |
${y}^2={x}^{3}-25{x}$ |
| 20000.1-g2 |
20000.1-g |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
20000.1 |
\( 2^{5} \cdot 5^{4} \) |
\( 2^{12} \cdot 5^{12} \) |
$3.00567$ |
$(a), (5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.899482172$ |
$1.375037163$ |
3.693725825 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -25\) , \( 0\bigr] \) |
${y}^2={x}^{3}-25{x}$ |
| 64.1-a1 |
64.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-5}) \) |
$2$ |
$[0, 1]$ |
64.1 |
\( 2^{6} \) |
\( 2^{12} \) |
$1.13031$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2, 5$ |
2Cs, 5Ns.2.1 |
$1$ |
\( 2 \) |
$1.899482172$ |
$6.875185818$ |
1.460073332 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -1\) , \( 0\bigr] \) |
${y}^2={x}^3-{x}$ |
| 32.1-b1 |
32.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-10}) \) |
$2$ |
$[0, 1]$ |
32.1 |
\( 2^{5} \) |
\( 2^{24} \) |
$1.34418$ |
$(2,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1.899482172$ |
$6.875185818$ |
2.064855508 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -4\) , \( 0\bigr] \) |
${y}^2={x}^3-4{x}$ |
| 32.1-a2 |
32.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-30}) \) |
$2$ |
$[0, 1]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \cdot 5^{12} \) |
$2.32818$ |
$(2,a)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$4$ |
\( 2 \) |
$1.687929007$ |
$6.875185818$ |
4.237483161 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -25\) , \( 0\bigr] \) |
${y}^2={x}^3-25{x}$ |
| 64.1-a2 |
64.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-65}) \) |
$2$ |
$[0, 1]$ |
64.1 |
\( 2^{6} \) |
\( 2^{12} \cdot 5^{12} \) |
$4.07540$ |
$(2,a+1)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$11.07875206$ |
$6.875185818$ |
9.447537087 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -25\) , \( 0\bigr] \) |
${y}^2={x}^3-25{x}$ |
| 32.1-d1 |
32.1-d |
$4$ |
$4$ |
\(\Q(\sqrt{-70}) \) |
$2$ |
$[0, 1]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \cdot 5^{12} \) |
$3.55635$ |
$(2,a)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$4$ |
\( 2 \) |
$4.053724506$ |
$6.875185818$ |
6.662230381 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -25\) , \( 0\bigr] \) |
${y}^2={x}^3-25{x}$ |
| 64.1-a2 |
64.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-85}) \) |
$2$ |
$[0, 1]$ |
64.1 |
\( 2^{6} \) |
\( 2^{12} \cdot 5^{12} \) |
$4.66040$ |
$(2,a+1)$ |
$1 \le r \le 3$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$16$ |
\( 2 \) |
$1.899482172$ |
$6.875185818$ |
5.665916772 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -25\) , \( 0\bigr] \) |
${y}^2={x}^3-25{x}$ |
| 64.1-a2 |
64.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-105}) \) |
$2$ |
$[0, 1]$ |
64.1 |
\( 2^{6} \) |
\( 2^{12} \cdot 5^{12} \) |
$5.17974$ |
$(2,a+1)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$3.155908371$ |
$6.875185818$ |
8.469819743 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -25\) , \( 0\bigr] \) |
${y}^2={x}^3-25{x}$ |
| 32.1-c1 |
32.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{-110}) \) |
$2$ |
$[0, 1]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \cdot 5^{12} \) |
$4.45813$ |
$(2,a)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$4$ |
\( 2 \) |
$8.078773048$ |
$6.875185818$ |
10.59164708 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -25\) , \( 0\bigr] \) |
${y}^2={x}^3-25{x}$ |
| 32.1-a2 |
32.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-130}) \) |
$2$ |
$[0, 1]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \cdot 5^{12} \) |
$4.84649$ |
$(2,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1.899482172$ |
$13.75037163$ |
2.290751511 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -25\) , \( 0\bigr] \) |
${y}^2={x}^3-25{x}$ |
| 64.1-b1 |
64.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-165}) \) |
$2$ |
$[0, 1]$ |
64.1 |
\( 2^{6} \) |
\( 2^{12} \cdot 5^{12} \) |
$6.49315$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$16$ |
\( 2 \) |
$1.899482172$ |
$13.75037163$ |
4.066658291 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -25\) , \( 0\bigr] \) |
${y}^2={x}^3-25{x}$ |
| 32.1-a1 |
32.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-170}) \) |
$2$ |
$[0, 1]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \cdot 5^{12} \) |
$5.54218$ |
$(2,a)$ |
$3$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$13.48570643$ |
$13.75037163$ |
14.22209833 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -25\) , \( 0\bigr] \) |
${y}^2={x}^3-25{x}$ |
| 64.1-a1 |
64.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-185}) \) |
$2$ |
$[0, 1]$ |
64.1 |
\( 2^{6} \) |
\( 2^{12} \cdot 5^{12} \) |
$6.87542$ |
$(2,a+1)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$25.76727478$ |
$13.75037163$ |
13.02468009 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -25\) , \( 0\bigr] \) |
${y}^2={x}^3-25{x}$ |
| 32.1-c1 |
32.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{-190}) \) |
$2$ |
$[0, 1]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \cdot 5^{12} \) |
$5.85913$ |
$(2,a)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$4$ |
\( 2 \) |
$13.73063350$ |
$13.75037163$ |
13.69708689 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -25\) , \( 0\bigr] \) |
${y}^2={x}^3-25{x}$ |
| 32.1-f1 |
32.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{-210}) \) |
$2$ |
$[0, 1]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \cdot 5^{12} \) |
$6.15979$ |
$(2,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$16$ |
\( 2^{2} \) |
$1.899482172$ |
$13.75037163$ |
7.209414604 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -25\) , \( 0\bigr] \) |
${y}^2={x}^3-25{x}$ |
| 32.1-a1 |
32.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-230}) \) |
$2$ |
$[0, 1]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \cdot 5^{12} \) |
$6.44644$ |
$(2,a)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$4$ |
\( 2 \) |
$8.605816478$ |
$13.75037163$ |
7.802658482 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -25\) , \( 0\bigr] \) |
${y}^2={x}^3-25{x}$ |
| 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 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -1\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}$ |
| 32.1-a2 |
32.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{24} \) |
$1.34418$ |
$(2,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1.899482172$ |
$27.50074327$ |
2.064855508 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -4\) , \( 0\bigr] \) |
${y}^2={x}^{3}-4{x}$ |
| 64.1-b2 |
64.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{15}) \) |
$2$ |
$[2, 0]$ |
64.1 |
\( 2^{6} \) |
\( 2^{24} \) |
$1.95776$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.899482172$ |
$27.50074327$ |
3.371894926 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -32 a - 124\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(-32a-124\right){x}$ |
*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.
