| 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 |
| 196.1-a1 |
196.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( - 2^{6} \cdot 7^{6} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$0.271490846$ |
$9.921996894$ |
2.657080224 |
\( -\frac{73237625}{134456} a + \frac{311928625}{134456} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 17 a - 60\) , \( -24 a + 85\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(17a-60\right){x}-24a+85$ |
| 196.1-b1 |
196.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{2} \cdot 7^{6} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.850552065$ |
$13.88766351$ |
3.883821150 |
\( -\frac{23815258332914609}{4802} a + \frac{42169455120363273}{2401} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 253 a - 914\) , \( -3966 a + 14055\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(253a-914\right){x}-3966a+14055$ |
| 196.1-b2 |
196.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( - 2^{4} \cdot 7^{3} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.425276032$ |
$27.77532702$ |
3.883821150 |
\( \frac{426385990349}{196} a + \frac{1082532991005}{196} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 13 a - 64\) , \( -64 a + 237\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(13a-64\right){x}-64a+237$ |
| 196.1-c1 |
196.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( - 2^{2} \cdot 7^{2} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$34.99668541$ |
5.753419641 |
\( \frac{6497}{7} a + \frac{34137}{14} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -8 a - 22\) , \( 3 a + 7\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-8a-22\right){x}+3a+7$ |
| 196.1-d1 |
196.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( - 2^{2} \cdot 7^{12} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{3} \) |
$1.275895211$ |
$0.262301958$ |
2.971046427 |
\( -\frac{195023039462640289}{40353607} a - \frac{963487859994700925}{80707214} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 907 a - 3788\) , \( 29235 a - 109976\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(907a-3788\right){x}+29235a-109976$ |
| 196.1-d2 |
196.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( - 2^{6} \cdot 7^{12} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{4} \) |
$0.425298403$ |
$2.360717626$ |
2.971046427 |
\( -\frac{110498869585}{161414428} a + \frac{10434469447}{322828856} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 17 a - 58\) , \( 11 a - 54\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(17a-58\right){x}+11a-54$ |
| 196.1-d3 |
196.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( - 2^{2} \cdot 7^{4} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1.275895211$ |
$21.24645863$ |
2.971046427 |
\( -\frac{7988898965}{686} a + \frac{28292550037}{686} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 12 a - 33\) , \( -29 a + 109\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(12a-33\right){x}-29a+109$ |
| 196.1-e1 |
196.1-e |
$1$ |
$1$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( - 2^{2} \cdot 7^{2} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$34.99668541$ |
5.753419641 |
\( -\frac{6497}{7} a + \frac{6733}{2} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 13 a - 25\) , \( -21 a + 103\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(13a-25\right){x}-21a+103$ |
| 196.1-f1 |
196.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( - 2^{4} \cdot 7^{3} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.425276032$ |
$27.77532702$ |
3.883821150 |
\( -\frac{426385990349}{196} a + \frac{754459490677}{98} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -6 a - 47\) , \( a + 85\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-6a-47\right){x}+a+85$ |
| 196.1-f2 |
196.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{2} \cdot 7^{6} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.850552065$ |
$13.88766351$ |
3.883821150 |
\( \frac{23815258332914609}{4802} a + \frac{60523651907811937}{4802} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -246 a - 657\) , \( 3053 a + 7841\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-246a-657\right){x}+3053a+7841$ |
| 196.1-g1 |
196.1-g |
$3$ |
$9$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( - 2^{6} \cdot 7^{12} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{4} \) |
$0.425298403$ |
$2.360717626$ |
2.971046427 |
\( \frac{110498869585}{161414428} a - \frac{210563269723}{322828856} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -17 a - 41\) , \( -11 a - 43\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-17a-41\right){x}-11a-43$ |
| 196.1-g2 |
196.1-g |
$3$ |
$9$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( - 2^{2} \cdot 7^{4} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1.275895211$ |
$21.24645863$ |
2.971046427 |
\( \frac{7988898965}{686} a + \frac{10151825536}{343} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -12 a - 21\) , \( 29 a + 80\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-12a-21\right){x}+29a+80$ |
| 196.1-g3 |
196.1-g |
$3$ |
$9$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( - 2^{2} \cdot 7^{12} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3^{3} \) |
$1.275895211$ |
$0.262301958$ |
2.971046427 |
\( \frac{195023039462640289}{40353607} a - \frac{1353533938919981503}{80707214} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -907 a - 2881\) , \( -29235 a - 80741\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-907a-2881\right){x}-29235a-80741$ |
| 196.1-h1 |
196.1-h |
$6$ |
$18$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{36} \cdot 7^{2} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$2.817069585$ |
$0.436190660$ |
1.818090865 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -171\) , \( -874\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-171{x}-874$ |
| 196.1-h2 |
196.1-h |
$6$ |
$18$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{4} \cdot 7^{2} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$2.817069585$ |
$35.33144352$ |
1.818090865 |
\( -\frac{15625}{28} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}$ |
| 196.1-h3 |
196.1-h |
$6$ |
$18$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{12} \cdot 7^{6} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{3} \) |
$0.939023195$ |
$3.925715946$ |
1.818090865 |
\( \frac{9938375}{21952} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 4\) , \( -6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+4{x}-6$ |
| 196.1-h4 |
196.1-h |
$6$ |
$18$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{6} \cdot 7^{12} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{2} \cdot 3^{3} \) |
$0.469511597$ |
$3.925715946$ |
1.818090865 |
\( \frac{4956477625}{941192} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -36\) , \( -70\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-36{x}-70$ |
| 196.1-h5 |
196.1-h |
$6$ |
$18$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{2} \cdot 7^{4} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1.408534792$ |
$35.33144352$ |
1.818090865 |
\( \frac{128787625}{98} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -11\) , \( 12\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-11{x}+12$ |
| 196.1-h6 |
196.1-h |
$6$ |
$18$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{18} \cdot 7^{4} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1.408534792$ |
$0.436190660$ |
1.818090865 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -2731\) , \( -55146\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-2731{x}-55146$ |
| 196.1-i1 |
196.1-i |
$1$ |
$1$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( - 2^{14} \cdot 7^{4} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1.393975167$ |
$3.860221112$ |
1.769279120 |
\( \frac{794295801}{43904} a + \frac{1988948007}{43904} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 129 a - 459\) , \( 2606 a - 9234\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(129a-459\right){x}+2606a-9234$ |
| 196.1-j1 |
196.1-j |
$1$ |
$1$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( - 2^{6} \cdot 7^{6} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$0.271490846$ |
$9.921996894$ |
2.657080224 |
\( \frac{73237625}{134456} a + \frac{29836375}{16807} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -17 a - 43\) , \( 24 a + 61\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-17a-43\right){x}+24a+61$ |
| 196.1-k1 |
196.1-k |
$1$ |
$1$ |
\(\Q(\sqrt{37}) \) |
$2$ |
$[2, 0]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( - 2^{14} \cdot 7^{4} \) |
$2.03378$ |
$(-a-1), (a-2), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1.393975167$ |
$3.860221112$ |
1.769279120 |
\( -\frac{794295801}{43904} a + \frac{86976369}{1372} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -130 a - 329\) , \( -2607 a - 6627\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-130a-329\right){x}-2607a-6627$ |
*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.
