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 |
3969.1-a1 |
3969.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( - 3^{12} \cdot 7^{3} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$9$ |
\( 2^{2} \) |
$1$ |
$5.120490302$ |
1.810366707 |
\( -\frac{551719468601289472}{49} a + \frac{780249146376863552}{49} \) |
\( \bigl[a\) , \( 1\) , \( 1\) , \( 3514 a - 8283\) , \( -179933 a + 326221\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+{x}^{2}+\left(3514a-8283\right){x}-179933a+326221$ |
3969.1-a2 |
3969.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( - 3^{12} \cdot 7^{9} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$5.120490302$ |
1.810366707 |
\( -\frac{13219923200}{117649} a + \frac{18683377472}{117649} \) |
\( \bigl[a\) , \( 1\) , \( 1\) , \( 49 a - 93\) , \( -194 a + 448\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+{x}^{2}+\left(49a-93\right){x}-194a+448$ |
3969.1-a3 |
3969.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( - 3^{12} \cdot 7^{3} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$5.120490302$ |
1.810366707 |
\( -\frac{210688}{49} a + \frac{302912}{49} \) |
\( \bigl[a\) , \( 1\) , \( 1\) , \( 4 a - 3\) , \( 4 a - 11\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+{x}^{2}+\left(4a-3\right){x}+4a-11$ |
3969.1-a4 |
3969.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( - 3^{12} \cdot 7^{3} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$5.120490302$ |
1.810366707 |
\( \frac{210688}{49} a + \frac{302912}{49} \) |
\( \bigl[a\) , \( 1\) , \( 1\) , \( -5 a - 3\) , \( -4 a - 11\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-5a-3\right){x}-4a-11$ |
3969.1-a5 |
3969.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( - 3^{12} \cdot 7^{9} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$5.120490302$ |
1.810366707 |
\( \frac{13219923200}{117649} a + \frac{18683377472}{117649} \) |
\( \bigl[a\) , \( 1\) , \( 1\) , \( -50 a - 93\) , \( 194 a + 448\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-50a-93\right){x}+194a+448$ |
3969.1-a6 |
3969.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( - 3^{12} \cdot 7^{3} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$9$ |
\( 2^{2} \) |
$1$ |
$5.120490302$ |
1.810366707 |
\( \frac{551719468601289472}{49} a + \frac{780249146376863552}{49} \) |
\( \bigl[a\) , \( 1\) , \( 1\) , \( -3515 a - 8283\) , \( 179933 a + 326221\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-3515a-8283\right){x}+179933a+326221$ |
3969.1-b1 |
3969.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( 3^{14} \cdot 7^{16} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.217293980$ |
1.721513656 |
\( -\frac{4354703137}{17294403} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -306\) , \( 5859\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-306{x}+5859$ |
3969.1-b2 |
3969.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( 3^{16} \cdot 7^{2} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.869175922$ |
1.721513656 |
\( \frac{103823}{63} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 9\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+9{x}$ |
3969.1-b3 |
3969.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( 3^{20} \cdot 7^{4} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$4.869175922$ |
1.721513656 |
\( \frac{7189057}{3969} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -36\) , \( 27\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-36{x}+27$ |
3969.1-b4 |
3969.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( 3^{28} \cdot 7^{2} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$1.217293980$ |
1.721513656 |
\( \frac{6570725617}{45927} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -351\) , \( -2430\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-351{x}-2430$ |
3969.1-b5 |
3969.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( 3^{16} \cdot 7^{8} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$4.869175922$ |
1.721513656 |
\( \frac{13027640977}{21609} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -441\) , \( 3672\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-441{x}+3672$ |
3969.1-b6 |
3969.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( 3^{14} \cdot 7^{4} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$4.869175922$ |
1.721513656 |
\( \frac{53297461115137}{147} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -7056\) , \( 229905\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-7056{x}+229905$ |
3969.1-c1 |
3969.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( - 3^{18} \cdot 7^{7} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$1.350443682$ |
1.432361828 |
\( -\frac{211597056}{117649} a - \frac{25299648}{117649} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( 148 a - 223\) , \( 688 a - 1003\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(148a-223\right){x}+688a-1003$ |
3969.1-c2 |
3969.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( - 3^{6} \cdot 7^{5} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$12.15399314$ |
1.432361828 |
\( -\frac{1441034496}{343} a + \frac{2038364352}{343} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( 16 a + 17\) , \( 27 a + 41\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(16a+17\right){x}+27a+41$ |
3969.1-c3 |
3969.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( - 3^{6} \cdot 7^{7} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$12.15399314$ |
1.432361828 |
\( \frac{211597056}{117649} a - \frac{25299648}{117649} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( -17 a - 25\) , \( 14 a + 20\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-17a-25\right){x}+14a+20$ |
3969.1-c4 |
3969.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( - 3^{18} \cdot 7^{5} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$1.350443682$ |
1.432361828 |
\( \frac{1441034496}{343} a + \frac{2038364352}{343} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( -149 a + 155\) , \( 445 a - 814\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-149a+155\right){x}+445a-814$ |
3969.1-d1 |
3969.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( - 3^{6} \cdot 7^{7} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$12.15399314$ |
1.432361828 |
\( -\frac{211597056}{117649} a - \frac{25299648}{117649} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( 16 a - 25\) , \( -15 a + 20\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(16a-25\right){x}-15a+20$ |
3969.1-d2 |
3969.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( - 3^{18} \cdot 7^{5} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$1.350443682$ |
1.432361828 |
\( -\frac{1441034496}{343} a + \frac{2038364352}{343} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( 148 a + 155\) , \( -446 a - 814\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(148a+155\right){x}-446a-814$ |
3969.1-d3 |
3969.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( - 3^{18} \cdot 7^{7} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$1.350443682$ |
1.432361828 |
\( \frac{211597056}{117649} a - \frac{25299648}{117649} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( -149 a - 223\) , \( -689 a - 1003\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-149a-223\right){x}-689a-1003$ |
3969.1-d4 |
3969.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3969.1 |
\( 3^{4} \cdot 7^{2} \) |
\( - 3^{6} \cdot 7^{5} \) |
$2.00611$ |
$(-2a+1), (2a+1), (3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$12.15399314$ |
1.432361828 |
\( \frac{1441034496}{343} a + \frac{2038364352}{343} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( -17 a + 17\) , \( -28 a + 41\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-17a+17\right){x}-28a+41$ |
*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.