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 |
16.4-a1 |
16.4-a |
$2$ |
$11$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{12} \) |
$2.58370$ |
$(11a+74)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-11$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.404753223$ |
$38.24607426$ |
2.141578668 |
\( -32768 \) |
\( \bigl[0\) , \( a - 1\) , \( a\) , \( -317 a - 2113\) , \( 7753 a + 52137\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-317a-2113\right){x}+7753a+52137$ |
16.4-a2 |
16.4-a |
$2$ |
$11$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{12} \) |
$2.58370$ |
$(11a+74)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-11$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$4.452285453$ |
$3.476915842$ |
2.141578668 |
\( -32768 \) |
\( \bigl[0\) , \( -a\) , \( a\) , \( 931337 a - 7197740\) , \( 1351685914 a - 10446391276\bigr] \) |
${y}^2+a{y}={x}^{3}-a{x}^{2}+\left(931337a-7197740\right){x}+1351685914a-10446391276$ |
16.5-a1 |
16.5-a |
$2$ |
$11$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
16.5 |
\( 2^{4} \) |
\( 2^{12} \) |
$2.58370$ |
$(11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-11$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$4.452285453$ |
$3.476915842$ |
2.141578668 |
\( -32768 \) |
\( \bigl[0\) , \( a - 1\) , \( a + 1\) , \( -931337 a - 6266403\) , \( -1351685915 a - 9094705362\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-931337a-6266403\right){x}-1351685915a-9094705362$ |
16.5-a2 |
16.5-a |
$2$ |
$11$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
16.5 |
\( 2^{4} \) |
\( 2^{12} \) |
$2.58370$ |
$(11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-11$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.404753223$ |
$38.24607426$ |
2.141578668 |
\( -32768 \) |
\( \bigl[0\) , \( -a\) , \( a + 1\) , \( 317 a - 2430\) , \( -7754 a + 59890\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(317a-2430\right){x}-7754a+59890$ |
64.6-f1 |
64.6-f |
$2$ |
$11$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
64.6 |
\( 2^{6} \) |
\( 2^{6} \) |
$3.65390$ |
$(11a-85)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-11$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$16.30818086$ |
1.128060458 |
\( -32768 \) |
\( \bigl[0\) , \( a\) , \( a + 1\) , \( 108556933 a - 838973134\) , \( 1698119186795 a - 13123771743439\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(108556933a-838973134\right){x}+1698119186795a-13123771743439$ |
64.6-f2 |
64.6-f |
$2$ |
$11$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
64.6 |
\( 2^{6} \) |
\( 2^{6} \) |
$3.65390$ |
$(11a-85)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-11$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$16.30818086$ |
1.128060458 |
\( -32768 \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( -a + 13\) , \( -2 a - 5\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a+13\right){x}-2a-5$ |
64.7-f1 |
64.7-f |
$2$ |
$11$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{6} \) |
$3.65390$ |
$(11a+74)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-11$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$16.30818086$ |
1.128060458 |
\( -32768 \) |
\( \bigl[0\) , \( a\) , \( a\) , \( a + 12\) , \( a - 6\bigr] \) |
${y}^2+a{y}={x}^{3}+a{x}^{2}+\left(a+12\right){x}+a-6$ |
64.7-f2 |
64.7-f |
$2$ |
$11$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{6} \) |
$3.65390$ |
$(11a+74)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-11$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$16.30818086$ |
1.128060458 |
\( -32768 \) |
\( \bigl[0\) , \( -a + 1\) , \( a\) , \( -108556933 a - 730416201\) , \( -1698119186796 a - 11425652556643\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-108556933a-730416201\right){x}-1698119186796a-11425652556643$ |
*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.