| 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 |
| 392.1-a1 |
392.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( - 2^{10} \cdot 7^{9} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$0.908137285$ |
1.284300066 |
\( -\frac{1159856388322676}{5764801} a + \frac{1640242904389426}{5764801} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( -194 a - 314\) , \( -2200 a - 3186\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-194a-314\right){x}-2200a-3186$ |
| 392.1-a2 |
392.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( - 2^{8} \cdot 7^{3} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$14.53019657$ |
1.284300066 |
\( -\frac{110288896}{49} a + \frac{155981824}{49} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 6 a + 7\) , \( -23 a - 32\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(6a+7\right){x}-23a-32$ |
| 392.1-a3 |
392.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( - 2^{8} \cdot 7^{9} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$7.265098286$ |
1.284300066 |
\( \frac{13282665232}{5764801} a + \frac{23566456972}{5764801} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -15 a + 18\) , \( -30 a + 43\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-15a+18\right){x}-30a+43$ |
| 392.1-a4 |
392.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{4} \cdot 7^{6} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$14.53019657$ |
1.284300066 |
\( \frac{4566144}{2401} a + \frac{14497232}{2401} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 5 a - 7\) , \( -7 a + 9\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(5a-7\right){x}-7a+9$ |
| 392.1-a5 |
392.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{6} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$3.632549143$ |
1.284300066 |
\( \frac{53744933616}{2401} a + \frac{76065896132}{2401} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 25 a - 52\) , \( 84 a - 159\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(25a-52\right){x}+84a-159$ |
| 392.1-a6 |
392.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( - 2^{10} \cdot 7^{3} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$0.908137285$ |
1.284300066 |
\( \frac{69495892205440052}{49} a + \frac{98282033286152638}{49} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -115 a - 122\) , \( -322 a - 1587\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-115a-122\right){x}-322a-1587$ |
| 392.1-b1 |
392.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{4} \cdot 7^{2} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.696738583$ |
$24.47471212$ |
1.507240199 |
\( \frac{432}{7} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 0\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}$ |
| 392.1-b2 |
392.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( - 2^{11} \cdot 7^{10} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.393477166$ |
$1.529669507$ |
1.507240199 |
\( -\frac{29774895462729}{5764801} a + \frac{42111203990760}{5764801} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 90 a - 95\) , \( 458 a - 737\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(90a-95\right){x}+458a-737$ |
| 392.1-b3 |
392.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{10} \cdot 7^{8} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.696738583$ |
$6.118678030$ |
1.507240199 |
\( \frac{11090466}{2401} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -15\) , \( -25\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-15{x}-25$ |
| 392.1-b4 |
392.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{4} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.348369291$ |
$24.47471212$ |
1.507240199 |
\( \frac{740772}{49} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -5\) , \( 1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-5{x}+1$ |
| 392.1-b5 |
392.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( - 2^{11} \cdot 7^{10} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.393477166$ |
$1.529669507$ |
1.507240199 |
\( \frac{29774895462729}{5764801} a + \frac{42111203990760}{5764801} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -90 a - 95\) , \( -458 a - 737\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-90a-95\right){x}-458a-737$ |
| 392.1-b6 |
392.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{10} \cdot 7^{2} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.696738583$ |
$24.47471212$ |
1.507240199 |
\( \frac{1443468546}{7} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -75\) , \( 211\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-75{x}+211$ |
| 392.1-c1 |
392.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{2} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$7.189921948$ |
1.271010641 |
\( -\frac{4}{7} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -1\) , \( -1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}-{x}-1$ |
| 392.1-c2 |
392.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{11} \cdot 7^{5} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.594960974$ |
1.271010641 |
\( -\frac{4347206325605}{2401} a + \frac{6147883179496}{2401} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 50 a - 91\) , \( 274 a - 371\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(50a-91\right){x}+274a-371$ |
| 392.1-c3 |
392.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{10} \cdot 7^{4} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$7.189921948$ |
1.271010641 |
\( \frac{3543122}{49} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -11\) , \( -11\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}-11{x}-11$ |
| 392.1-c4 |
392.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{11} \cdot 7^{5} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.594960974$ |
1.271010641 |
\( \frac{4347206325605}{2401} a + \frac{6147883179496}{2401} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -50 a - 91\) , \( -274 a - 371\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-50a-91\right){x}-274a-371$ |
| 392.1-d1 |
392.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( - 2^{10} \cdot 7^{3} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$0.908137285$ |
1.284300066 |
\( -\frac{69495892205440052}{49} a + \frac{98282033286152638}{49} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( 115 a - 122\) , \( 322 a - 1587\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(115a-122\right){x}+322a-1587$ |
| 392.1-d2 |
392.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( - 2^{8} \cdot 7^{9} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$7.265098286$ |
1.284300066 |
\( -\frac{13282665232}{5764801} a + \frac{23566456972}{5764801} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( 15 a + 18\) , \( 30 a + 43\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(15a+18\right){x}+30a+43$ |
| 392.1-d3 |
392.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{4} \cdot 7^{6} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$14.53019657$ |
1.284300066 |
\( -\frac{4566144}{2401} a + \frac{14497232}{2401} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( -5 a - 7\) , \( 7 a + 9\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(-5a-7\right){x}+7a+9$ |
| 392.1-d4 |
392.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( 2^{8} \cdot 7^{6} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$3.632549143$ |
1.284300066 |
\( -\frac{53744933616}{2401} a + \frac{76065896132}{2401} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( -25 a - 52\) , \( -84 a - 159\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(-25a-52\right){x}-84a-159$ |
| 392.1-d5 |
392.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( - 2^{8} \cdot 7^{3} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$14.53019657$ |
1.284300066 |
\( \frac{110288896}{49} a + \frac{155981824}{49} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -6 a + 7\) , \( 23 a - 32\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-6a+7\right){x}+23a-32$ |
| 392.1-d6 |
392.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
392.1 |
\( 2^{3} \cdot 7^{2} \) |
\( - 2^{10} \cdot 7^{9} \) |
$1.12462$ |
$(a), (-2a+1), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$0.908137285$ |
1.284300066 |
\( \frac{1159856388322676}{5764801} a + \frac{1640242904389426}{5764801} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 194 a - 314\) , \( 2200 a - 3186\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(194a-314\right){x}+2200a-3186$ |
*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.
