| 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 |
| 512.1-a1 |
512.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( - 2^{17} \) |
$1.47247$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$18.70900640$ |
1.350206235 |
\( -2002968 a + 3470040 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 3064 a - 5306\) , \( -119716 a + 207354\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(3064a-5306\right){x}-119716a+207354$ |
| 512.1-a2 |
512.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( 2^{10} \) |
$1.47247$ |
$(a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$37.41801281$ |
1.350206235 |
\( 3456 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 209 a - 361\) , \( -1410 a + 2442\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(209a-361\right){x}-1410a+2442$ |
| 512.1-a3 |
512.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( 2^{20} \) |
$1.47247$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$9.354503203$ |
1.350206235 |
\( 23328 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 6 a - 11\) , \( 15 a - 26\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(6a-11\right){x}+15a-26$ |
| 512.1-a4 |
512.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( - 2^{17} \) |
$1.47247$ |
$(a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$18.70900640$ |
1.350206235 |
\( 2002968 a + 3470040 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -556 a + 964\) , \( -10464 a + 18124\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-556a+964\right){x}-10464a+18124$ |
| 512.1-b1 |
512.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( - 2^{17} \) |
$1.47247$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.648148600$ |
$5.952999858$ |
2.227664748 |
\( -2002968 a + 3470040 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 220 a - 380\) , \( 2464 a - 4268\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(220a-380\right){x}+2464a-4268$ |
| 512.1-b2 |
512.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( 2^{10} \) |
$1.47247$ |
$(a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1.296297200$ |
$23.81199943$ |
2.227664748 |
\( 3456 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 15 a - 25\) , \( 38 a - 66\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(15a-25\right){x}+38a-66$ |
| 512.1-b3 |
512.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( 2^{20} \) |
$1.47247$ |
$(a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.648148600$ |
$23.81199943$ |
2.227664748 |
\( 23328 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -6 a - 11\) , \( 15 a + 26\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-6a-11\right){x}+15a+26$ |
| 512.1-b4 |
512.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( - 2^{17} \) |
$1.47247$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$2.592594401$ |
$5.952999858$ |
2.227664748 |
\( 2002968 a + 3470040 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -40 a + 70\) , \( 172 a - 298\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-40a+70\right){x}+172a-298$ |
| 512.1-c1 |
512.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( 2^{10} \) |
$1.47247$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$15.68304578$ |
2.263652675 |
\( 128 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -5 a + 9\) , \( 30 a - 52\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-5a+9\right){x}+30a-52$ |
| 512.1-c2 |
512.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( 2^{20} \) |
$1.47247$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$15.68304578$ |
2.263652675 |
\( 10976 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -4 a - 8\) , \( 12 a + 20\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-4a-8\right){x}+12a+20$ |
| 512.1-d1 |
512.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( 2^{10} \) |
$1.47247$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$15.68304578$ |
2.263652675 |
\( 128 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -69 a + 122\) , \( -1490 a + 2580\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-69a+122\right){x}-1490a+2580$ |
| 512.1-d2 |
512.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( 2^{20} \) |
$1.47247$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$15.68304578$ |
2.263652675 |
\( 10976 \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 4 a - 8\) , \( -12 a + 20\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(4a-8\right){x}-12a+20$ |
| 512.1-e1 |
512.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( 2^{10} \) |
$1.47247$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.363017536$ |
$15.68304578$ |
1.643491233 |
\( 128 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -5 a + 9\) , \( -30 a + 52\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-5a+9\right){x}-30a+52$ |
| 512.1-e2 |
512.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( 2^{20} \) |
$1.47247$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.181508768$ |
$15.68304578$ |
1.643491233 |
\( 10976 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -4 a - 8\) , \( -12 a - 20\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-4a-8\right){x}-12a-20$ |
| 512.1-f1 |
512.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( 2^{10} \) |
$1.47247$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.363017536$ |
$15.68304578$ |
1.643491233 |
\( 128 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -69 a + 122\) , \( 1490 a - 2580\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-69a+122\right){x}+1490a-2580$ |
| 512.1-f2 |
512.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( 2^{20} \) |
$1.47247$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.181508768$ |
$15.68304578$ |
1.643491233 |
\( 10976 \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 4 a - 8\) , \( 12 a - 20\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(4a-8\right){x}+12a-20$ |
| 512.1-g1 |
512.1-g |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( - 2^{17} \) |
$1.47247$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$2.592594401$ |
$5.952999858$ |
2.227664748 |
\( -2002968 a + 3470040 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 3064 a - 5306\) , \( 119716 a - 207354\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(3064a-5306\right){x}+119716a-207354$ |
| 512.1-g2 |
512.1-g |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( 2^{10} \) |
$1.47247$ |
$(a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1.296297200$ |
$23.81199943$ |
2.227664748 |
\( 3456 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 209 a - 361\) , \( 1410 a - 2442\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(209a-361\right){x}+1410a-2442$ |
| 512.1-g3 |
512.1-g |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( 2^{20} \) |
$1.47247$ |
$(a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.648148600$ |
$23.81199943$ |
2.227664748 |
\( 23328 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 6 a - 11\) , \( -15 a + 26\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(6a-11\right){x}-15a+26$ |
| 512.1-g4 |
512.1-g |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( - 2^{17} \) |
$1.47247$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.648148600$ |
$5.952999858$ |
2.227664748 |
\( 2002968 a + 3470040 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -556 a + 964\) , \( 10464 a - 18124\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-556a+964\right){x}+10464a-18124$ |
| 512.1-h1 |
512.1-h |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( - 2^{17} \) |
$1.47247$ |
$(a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$18.70900640$ |
1.350206235 |
\( -2002968 a + 3470040 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 220 a - 380\) , \( -2464 a + 4268\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(220a-380\right){x}-2464a+4268$ |
| 512.1-h2 |
512.1-h |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( 2^{10} \) |
$1.47247$ |
$(a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$37.41801281$ |
1.350206235 |
\( 3456 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( 15 a - 25\) , \( -38 a + 66\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(15a-25\right){x}-38a+66$ |
| 512.1-h3 |
512.1-h |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( 2^{20} \) |
$1.47247$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$9.354503203$ |
1.350206235 |
\( 23328 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -6 a - 11\) , \( -15 a - 26\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-6a-11\right){x}-15a-26$ |
| 512.1-h4 |
512.1-h |
$4$ |
$4$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
512.1 |
\( 2^{9} \) |
\( - 2^{17} \) |
$1.47247$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$18.70900640$ |
1.350206235 |
\( 2002968 a + 3470040 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -40 a + 70\) , \( -172 a + 298\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-40a+70\right){x}-172a+298$ |
*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.
