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 |
2.2-b2 |
2.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( 2^{10} \) |
$1.53628$ |
$(11a-85)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \) |
$1$ |
$18.51817441$ |
2.561857817 |
\( \frac{102557}{1024} a + \frac{242501}{256} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( 6449267 a + 43393366\) , \( 1546707281 a + 10406890251\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(6449267a+43393366\right){x}+1546707281a+10406890251$ |
2.2-c2 |
2.2-c |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( 2^{10} \) |
$1.53628$ |
$(11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B |
$1$ |
\( 2 \cdot 5 \) |
$0.105892522$ |
$13.03498084$ |
1.909556628 |
\( \frac{102557}{1024} a + \frac{242501}{256} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -10 a + 66\) , \( -20 a + 156\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-10a+66\right){x}-20a+156$ |
50.3-a2 |
50.3-a |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
50.3 |
\( 2 \cdot 5^{2} \) |
\( 2^{10} \cdot 5^{6} \) |
$3.43522$ |
$(11a-85), (-4a+31)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B |
$1$ |
\( 2^{3} \) |
$0.129561378$ |
$8.281579361$ |
2.375010653 |
\( \frac{102557}{1024} a + \frac{242501}{256} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -7329 a + 56779\) , \( -2133698 a + 16490518\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-7329a+56779\right){x}-2133698a+16490518$ |
50.3-g2 |
50.3-g |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
50.3 |
\( 2 \cdot 5^{2} \) |
\( 2^{10} \cdot 5^{6} \) |
$3.43522$ |
$(11a-85), (-4a+31)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B |
$1$ |
\( 2^{3} \cdot 5 \) |
$0.357204483$ |
$5.829420649$ |
11.52282962 |
\( \frac{102557}{1024} a + \frac{242501}{256} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( 48081 a + 323494\) , \( -615506 a - 4141387\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(48081a+323494\right){x}-615506a-4141387$ |
50.4-b2 |
50.4-b |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
50.4 |
\( 2 \cdot 5^{2} \) |
\( 2^{10} \cdot 5^{6} \) |
$3.43522$ |
$(11a-85), (-4a-27)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B |
$1$ |
\( 2 \cdot 5 \) |
$1.422389310$ |
$8.281579361$ |
16.29628084 |
\( \frac{102557}{1024} a + \frac{242501}{256} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( 4 a + 66\) , \( 16 a + 149\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(4a+66\right){x}+16a+149$ |
50.4-k2 |
50.4-k |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
50.4 |
\( 2 \cdot 5^{2} \) |
\( 2^{10} \cdot 5^{6} \) |
$3.43522$ |
$(11a-85), (-4a-27)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B |
$1$ |
\( 2 \) |
$1$ |
$5.829420649$ |
0.806458915 |
\( \frac{102557}{1024} a + \frac{242501}{256} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( -141381207 a + 1092652854\) , \( 5674189094061 a - 43852494617814\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-141381207a+1092652854\right){x}+5674189094061a-43852494617814$ |
64.6-c2 |
64.6-c |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
64.6 |
\( 2^{6} \) |
\( 2^{28} \) |
$3.65390$ |
$(11a-85)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B |
|
\( 2 \) |
$1$ |
$6.440367677$ |
7.117173427 |
\( \frac{102557}{1024} a + \frac{242501}{256} \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( -4081 a + 31540\) , \( -878697 a + 6790917\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-4081a+31540\right){x}-878697a+6790917$ |
64.6-g2 |
64.6-g |
$2$ |
$5$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
64.6 |
\( 2^{6} \) |
\( 2^{28} \) |
$3.65390$ |
$(11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B |
$1$ |
\( 2 \) |
$1.356515814$ |
$4.684981914$ |
1.758407907 |
\( \frac{102557}{1024} a + \frac{242501}{256} \) |
\( \bigl[0\) , \( -a - 1\) , \( a + 1\) , \( 221320 a + 1489144\) , \( 8865401 a + 59650106\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(221320a+1489144\right){x}+8865401a+59650106$ |
*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.