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 |
56.1-a1 |
56.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( 2^{8} \cdot 7^{2} \) |
$1.82928$ |
$(-a+4), (-2a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$22.75712104$ |
3.041048216 |
\( -\frac{4}{7} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 4\) , \( 2\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+4{x}+2$ |
56.1-a2 |
56.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( 2^{10} \cdot 7^{4} \) |
$1.82928$ |
$(-a+4), (-2a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$22.75712104$ |
3.041048216 |
\( \frac{3543122}{49} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -6\) , \( 2\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-6{x}+2$ |
56.1-b1 |
56.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( - 2^{11} \cdot 7 \) |
$1.82928$ |
$(-a+4), (-2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$3.866114466$ |
$12.23735606$ |
3.161100443 |
\( -\frac{39775849362076815}{7} a + 21261085797246696 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -8820 a - 33003\) , \( 912600 a + 3414635\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-8820a-33003\right){x}+912600a+3414635$ |
56.1-b2 |
56.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$1.82928$ |
$(-a+4), (-2a+7)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.933057233$ |
$24.47471212$ |
3.161100443 |
\( \frac{432}{7} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -30 a + 112\) , \( -944 a + 3532\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-30a+112\right){x}-944a+3532$ |
56.1-b3 |
56.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( 2^{10} \cdot 7^{8} \) |
$1.82928$ |
$(-a+4), (-2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.933057233$ |
$6.118678030$ |
3.161100443 |
\( \frac{11090466}{2401} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 1770 a - 6623\) , \( 64686 a - 242033\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(1770a-6623\right){x}+64686a-242033$ |
56.1-b4 |
56.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( 2^{8} \cdot 7^{4} \) |
$1.82928$ |
$(-a+4), (-2a+7)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.966528616$ |
$24.47471212$ |
3.161100443 |
\( \frac{740772}{49} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -570 a - 2133\) , \( 12630 a + 47257\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-570a-2133\right){x}+12630a+47257$ |
56.1-b5 |
56.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( 2^{10} \cdot 7^{2} \) |
$1.82928$ |
$(-a+4), (-2a+7)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.933057233$ |
$24.47471212$ |
3.161100443 |
\( \frac{1443468546}{7} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 8970 a - 33563\) , \( -881050 a + 3296587\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(8970a-33563\right){x}-881050a+3296587$ |
56.1-b6 |
56.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( - 2^{11} \cdot 7 \) |
$1.82928$ |
$(-a+4), (-2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$3.866114466$ |
$12.23735606$ |
3.161100443 |
\( \frac{39775849362076815}{7} a + 21261085797246696 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 8820 a - 33003\) , \( -912600 a + 3414635\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(8820a-33003\right){x}-912600a+3414635$ |
56.1-c1 |
56.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( 2^{8} \cdot 7^{2} \) |
$1.82928$ |
$(-a+4), (-2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.844493815$ |
$7.189921948$ |
1.622768733 |
\( -\frac{4}{7} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 7 a - 33\) , \( 1856 a - 6948\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(7a-33\right){x}+1856a-6948$ |
56.1-c2 |
56.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( 2^{10} \cdot 7^{4} \) |
$1.82928$ |
$(-a+4), (-2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.422246907$ |
$7.189921948$ |
1.622768733 |
\( \frac{3543122}{49} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 1207 a - 4523\) , \( 47106 a - 176258\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(1207a-4523\right){x}+47106a-176258$ |
56.1-d1 |
56.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( - 2^{11} \cdot 7 \) |
$1.82928$ |
$(-a+4), (-2a+7)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$32.36335355$ |
$0.659073382$ |
2.850317744 |
\( -\frac{39775849362076815}{7} a + 21261085797246696 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 150 a - 635\) , \( 2318 a - 9041\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(150a-635\right){x}+2318a-9041$ |
56.1-d2 |
56.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$1.82928$ |
$(-a+4), (-2a+7)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$2.022709596$ |
$10.54517411$ |
2.850317744 |
\( \frac{432}{7} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 0\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}$ |
56.1-d3 |
56.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( 2^{10} \cdot 7^{8} \) |
$1.82928$ |
$(-a+4), (-2a+7)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.505677399$ |
$10.54517411$ |
2.850317744 |
\( \frac{11090466}{2401} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -15\) , \( -5\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-15{x}-5$ |
56.1-d4 |
56.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( 2^{8} \cdot 7^{4} \) |
$1.82928$ |
$(-a+4), (-2a+7)$ |
$2$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$2.022709596$ |
$10.54517411$ |
2.850317744 |
\( \frac{740772}{49} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -5\) , \( -11\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-5{x}-11$ |
56.1-d5 |
56.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( 2^{10} \cdot 7^{2} \) |
$1.82928$ |
$(-a+4), (-2a+7)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$8.090838387$ |
$2.636293528$ |
2.850317744 |
\( \frac{1443468546}{7} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -75\) , \( -361\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-75{x}-361$ |
56.1-d6 |
56.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{14}) \) |
$2$ |
$[2, 0]$ |
56.1 |
\( 2^{3} \cdot 7 \) |
\( - 2^{11} \cdot 7 \) |
$1.82928$ |
$(-a+4), (-2a+7)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$32.36335355$ |
$0.659073382$ |
2.850317744 |
\( \frac{39775849362076815}{7} a + 21261085797246696 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -150 a - 635\) , \( -2318 a - 9041\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-150a-635\right){x}-2318a-9041$ |
*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.