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 |
3136.2-a1 |
3136.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3136.2 |
\( 2^{6} \cdot 7^{2} \) |
\( - 2^{6} \cdot 7^{6} \) |
$1.89137$ |
$(a), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-32$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$0.434172091$ |
$9.592728448$ |
2.945025478 |
\( -18473000 a + 26125000 \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 10 a - 3\) , \( -25 a - 8\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(10a-3\right){x}-25a-8$ |
3136.2-a2 |
3136.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3136.2 |
\( 2^{6} \cdot 7^{2} \) |
\( - 2^{6} \cdot 7^{6} \) |
$1.89137$ |
$(a), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-32$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$0.868344182$ |
$4.796364224$ |
2.945025478 |
\( -18473000 a + 26125000 \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 10 a - 3\) , \( 25 a + 7\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(10a-3\right){x}+25a+7$ |
3136.2-a3 |
3136.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3136.2 |
\( 2^{6} \cdot 7^{2} \) |
\( 2^{12} \cdot 7^{6} \) |
$1.89137$ |
$(a), (-2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-8$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.868344182$ |
$9.592728448$ |
2.945025478 |
\( 8000 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -10 a - 18\) , \( -20 a - 24\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-10a-18\right){x}-20a-24$ |
3136.2-a4 |
3136.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3136.2 |
\( 2^{6} \cdot 7^{2} \) |
\( 2^{12} \cdot 7^{6} \) |
$1.89137$ |
$(a), (-2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-8$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.434172091$ |
$9.592728448$ |
2.945025478 |
\( 8000 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -10 a - 18\) , \( 20 a + 24\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-10a-18\right){x}+20a+24$ |
3136.2-a5 |
3136.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3136.2 |
\( 2^{6} \cdot 7^{2} \) |
\( - 2^{6} \cdot 7^{6} \) |
$1.89137$ |
$(a), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-32$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.868344182$ |
$9.592728448$ |
2.945025478 |
\( 18473000 a + 26125000 \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -21 a + 26\) , \( 115 a - 160\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-21a+26\right){x}+115a-160$ |
3136.2-a6 |
3136.2-a |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3136.2 |
\( 2^{6} \cdot 7^{2} \) |
\( - 2^{6} \cdot 7^{6} \) |
$1.89137$ |
$(a), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-32$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1.736688365$ |
$4.796364224$ |
2.945025478 |
\( 18473000 a + 26125000 \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -19 a + 25\) , \( -135 a + 185\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-19a+25\right){x}-135a+185$ |
3136.2-b1 |
3136.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3136.2 |
\( 2^{6} \cdot 7^{2} \) |
\( - 2^{6} \cdot 7^{10} \) |
$1.89137$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.142705457$ |
1.464667560 |
\( -\frac{8738600}{2401} a + \frac{16398328}{2401} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( 34 a + 46\) , \( -29 a - 45\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(34a+46\right){x}-29a-45$ |
3136.2-b2 |
3136.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3136.2 |
\( 2^{6} \cdot 7^{2} \) |
\( 2^{12} \cdot 7^{8} \) |
$1.89137$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$8.285410914$ |
1.464667560 |
\( -\frac{65280}{49} a + \frac{179264}{49} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -32 a - 48\) , \( -48 a - 66\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-32a-48\right){x}-48a-66$ |
3136.2-b3 |
3136.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3136.2 |
\( 2^{6} \cdot 7^{2} \) |
\( 2^{6} \cdot 7^{7} \) |
$1.89137$ |
$(a), (-2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$16.57082182$ |
1.464667560 |
\( -\frac{55566312}{7} a + \frac{79770920}{7} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -81 a - 125\) , \( 523 a + 746\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-81a-125\right){x}+523a+746$ |
3136.2-b4 |
3136.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3136.2 |
\( 2^{6} \cdot 7^{2} \) |
\( - 2^{12} \cdot 7^{7} \) |
$1.89137$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.142705457$ |
1.464667560 |
\( \frac{1233152}{7} a + \frac{1746880}{7} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -20 a + 20\) , \( -92 a + 122\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-20a+20\right){x}-92a+122$ |
3136.2-c1 |
3136.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3136.2 |
\( 2^{6} \cdot 7^{2} \) |
\( - 2^{6} \cdot 7^{10} \) |
$1.89137$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$6.418299503$ |
2.269211551 |
\( -\frac{8738600}{2401} a + \frac{16398328}{2401} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 34 a + 46\) , \( 29 a + 45\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(34a+46\right){x}+29a+45$ |
3136.2-c2 |
3136.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3136.2 |
\( 2^{6} \cdot 7^{2} \) |
\( 2^{12} \cdot 7^{8} \) |
$1.89137$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$6.418299503$ |
2.269211551 |
\( -\frac{65280}{49} a + \frac{179264}{49} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -32 a - 48\) , \( 48 a + 66\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-32a-48\right){x}+48a+66$ |
3136.2-c3 |
3136.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3136.2 |
\( 2^{6} \cdot 7^{2} \) |
\( 2^{6} \cdot 7^{7} \) |
$1.89137$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$3.209149751$ |
2.269211551 |
\( -\frac{55566312}{7} a + \frac{79770920}{7} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( -81 a - 125\) , \( -523 a - 747\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-81a-125\right){x}-523a-747$ |
3136.2-c4 |
3136.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3136.2 |
\( 2^{6} \cdot 7^{2} \) |
\( - 2^{12} \cdot 7^{7} \) |
$1.89137$ |
$(a), (-2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$6.418299503$ |
2.269211551 |
\( \frac{1233152}{7} a + \frac{1746880}{7} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -20 a + 20\) , \( 92 a - 122\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-20a+20\right){x}+92a-122$ |
*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.