| 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-a1 |
2.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( 2^{12} \) |
$1.53628$ |
$(11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Nn |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.048108188$ |
$34.45290976$ |
2.751585456 |
\( \frac{1714092597}{4096} a - \frac{3313345635}{1024} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -333486 a - 2243755\) , \( 412430147 a + 2775001836\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-333486a-2243755\right){x}+412430147a+2775001836$ |
| 2.2-d1 |
2.2-d |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( 2^{12} \) |
$1.53628$ |
$(11a-85)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Nn |
$1$ |
\( 2 \) |
$1$ |
$2.835776404$ |
0.392309511 |
\( \frac{1714092597}{4096} a - \frac{3313345635}{1024} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 1797 a - 13879\) , \( 122321 a - 945339\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1797a-13879\right){x}+122321a-945339$ |
| 50.3-c1 |
50.3-c |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
50.3 |
\( 2 \cdot 5^{2} \) |
\( 2^{12} \cdot 5^{6} \) |
$3.43522$ |
$(11a-85), (-4a+31)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Nn |
$1$ |
\( 2^{4} \cdot 3 \) |
$0.056309543$ |
$15.40780965$ |
5.761306811 |
\( \frac{1714092597}{4096} a - \frac{3313345635}{1024} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( 6048290 a - 46743723\) , \( -21737532752 a + 167996699115\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(6048290a-46743723\right){x}-21737532752a+167996699115$ |
| 50.3-h1 |
50.3-h |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
50.3 |
\( 2 \cdot 5^{2} \) |
\( 2^{12} \cdot 5^{6} \) |
$3.43522$ |
$(11a-85), (-4a+31)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Nn |
$1$ |
\( 2^{3} \) |
$1$ |
$1.268197761$ |
0.701784587 |
\( \frac{1714092597}{4096} a - \frac{3313345635}{1024} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( -2492 a - 16667\) , \( -271006 a - 1823258\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2492a-16667\right){x}-271006a-1823258$ |
| 50.4-c1 |
50.4-c |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
50.4 |
\( 2 \cdot 5^{2} \) |
\( 2^{12} \cdot 5^{6} \) |
$3.43522$ |
$(11a-85), (-4a-27)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Nn |
$1$ |
\( 2 \) |
$1$ |
$15.40780965$ |
2.131560958 |
\( \frac{1714092597}{4096} a - \frac{3313345635}{1024} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 7 a - 47\) , \( -8 a + 68\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(7a-47\right){x}-8a+68$ |
| 50.4-j1 |
50.4-j |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
50.4 |
\( 2 \cdot 5^{2} \) |
\( 2^{12} \cdot 5^{6} \) |
$3.43522$ |
$(11a-85), (-4a-27)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Nn |
$1$ |
\( 2^{2} \cdot 3 \) |
$3.426968061$ |
$1.268197761$ |
7.214980105 |
\( \frac{1714092597}{4096} a - \frac{3313345635}{1024} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -1118486716 a - 7525644003\) , \( -80247521088440 a - 539938716680145\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1118486716a-7525644003\right){x}-80247521088440a-539938716680145$ |
| 64.6-a1 |
64.6-a |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
64.6 |
\( 2^{6} \) |
\( 2^{30} \) |
$3.65390$ |
$(11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Nn |
$1$ |
\( 2 \) |
$2.272387693$ |
$4.983618103$ |
3.133386959 |
\( \frac{1714092597}{4096} a - \frac{3313345635}{1024} \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 3363116 a - 25991560\) , \( -9027332820 a + 69766984715\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(3363116a-25991560\right){x}-9027332820a+69766984715$ |
| 64.6-h1 |
64.6-h |
$1$ |
$1$ |
\(\Q(\sqrt{209}) \) |
$2$ |
$[2, 0]$ |
64.6 |
\( 2^{6} \) |
\( 2^{30} \) |
$3.65390$ |
$(11a-85)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Nn |
$1$ |
\( 2 \) |
$1.991622570$ |
$2.450547637$ |
1.350383233 |
\( \frac{1714092597}{4096} a - \frac{3313345635}{1024} \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( -11444 a - 77000\) , \( 2626271 a + 17670636\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-11444a-77000\right){x}+2626271a+17670636$ |
*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.
