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-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-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-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-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 |
\( 128 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -69 a + 122\) , \( 1490 a - 2580\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-69a+122\right){x}+1490a-2580$ |
1024.1-e1 |
1024.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{16} \) |
$1.75107$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$16.29302268$ |
2.351695258 |
\( 128 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -2 a + 6\) , \( -12 a + 20\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2a+6\right){x}-12a+20$ |
1024.1-f1 |
1024.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{16} \) |
$1.75107$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1.632943382$ |
$7.547952572$ |
3.558030500 |
\( 128 \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -38 a + 66\) , \( 612 a - 1060\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-38a+66\right){x}+612a-1060$ |
1024.1-o1 |
1024.1-o |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{16} \) |
$1.75107$ |
$(a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$7.547952572$ |
1.089453112 |
\( 128 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -2 a + 6\) , \( 12 a - 20\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-2a+6\right){x}+12a-20$ |
1024.1-p1 |
1024.1-p |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
1024.1 |
\( 2^{10} \) |
\( 2^{16} \) |
$1.75107$ |
$(a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.432331164$ |
$16.29302268$ |
2.033422299 |
\( 128 \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -38 a + 66\) , \( -612 a + 1060\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-38a+66\right){x}-612a+1060$ |
4608.1-h1 |
4608.1-h |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
4608.1 |
\( 2^{9} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{6} \) |
$2.55039$ |
$(a+1), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$6.162877468$ |
1.779069482 |
\( 128 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -15 a + 27\) , \( 156 a - 270\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-15a+27\right){x}+156a-270$ |
4608.1-i1 |
4608.1-i |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
4608.1 |
\( 2^{9} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{6} \) |
$2.55039$ |
$(a+1), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$6.162877468$ |
1.779069482 |
\( 128 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -209 a + 363\) , \( 7532 a - 13046\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-209a+363\right){x}+7532a-13046$ |
4608.1-v1 |
4608.1-v |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
4608.1 |
\( 2^{9} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{6} \) |
$2.55039$ |
$(a+1), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.512963232$ |
$13.30319731$ |
3.939867736 |
\( 128 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -209 a + 363\) , \( -7532 a + 13046\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-209a+363\right){x}-7532a+13046$ |
4608.1-ba1 |
4608.1-ba |
$2$ |
$2$ |
\(\Q(\sqrt{3}) \) |
$2$ |
$[2, 0]$ |
4608.1 |
\( 2^{9} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{6} \) |
$2.55039$ |
$(a+1), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.512963232$ |
$13.30319731$ |
3.939867736 |
\( 128 \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -15 a + 27\) , \( -156 a + 270\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(-15a+27\right){x}-156a+270$ |
*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.