| 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 |
| 16.1-a1 |
16.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$2.75133$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2, 79$ |
2B, 79Ns.2.1 |
$9$ |
\( 1 \) |
$1$ |
$17.69503190$ |
2.586185635 |
\( 0 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 20\) , \( 6 a - 78\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+20\right){x}+6a-78$ |
| 16.1-a2 |
16.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{8} \) |
$2.75133$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$2, 79$ |
2B, 79Ns.2.1 |
$9$ |
\( 1 \) |
$1$ |
$17.69503190$ |
2.586185635 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 20\) , \( -6 a - 52\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a+20\right){x}-6a-52$ |
| 16.1-a3 |
16.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{16} \) |
$2.75133$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$79$ |
79Ns.2.1 |
$9$ |
\( 1 \) |
$1$ |
$17.69503190$ |
2.586185635 |
\( 54000 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 26 a - 185\) , \( 235 a - 1955\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(26a-185\right){x}+235a-1955$ |
| 16.1-a4 |
16.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{16} \) |
$2.75133$ |
$(2)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-12$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$79$ |
79Ns.2.1 |
$9$ |
\( 1 \) |
$1$ |
$17.69503190$ |
2.586185635 |
\( 54000 \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -24 a - 160\) , \( -260 a - 1880\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-24a-160\right){x}-260a-1880$ |
| 19.1-a1 |
19.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( 19^{6} \) |
$2.87211$ |
$(2a-17)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.583453087$ |
$17.75857733$ |
2.435442809 |
\( -\frac{820169978705}{47045881} a + \frac{9033569250077}{47045881} \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( 28 a - 100\) , \( -70 a + 869\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(28a-100\right){x}-70a+869$ |
| 19.1-a2 |
19.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$2.87211$ |
$(2a-17)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$0.527817695$ |
$17.75857733$ |
2.435442809 |
\( \frac{1106097}{361} a + \frac{12390028}{361} \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( 8 a + 65\) , \( 24 a + 105\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(8a+65\right){x}+24a+105$ |
| 19.1-b1 |
19.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( 19^{6} \) |
$2.87211$ |
$(2a-17)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1.003556666$ |
$5.131639142$ |
4.014256158 |
\( -\frac{820169978705}{47045881} a + \frac{9033569250077}{47045881} \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( -445 a - 3210\) , \( -18448 a - 132776\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-445a-3210\right){x}-18448a-132776$ |
| 19.1-b2 |
19.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$2.87211$ |
$(2a-17)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$3.010669998$ |
$46.18475228$ |
4.014256158 |
\( \frac{1106097}{361} a + \frac{12390028}{361} \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( -40 a - 295\) , \( 153 a + 1103\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-40a-295\right){x}+153a+1103$ |
| 19.2-a1 |
19.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
19.2 |
\( 19 \) |
\( 19^{2} \) |
$2.87211$ |
$(-2a-15)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$0.527817695$ |
$17.75857733$ |
2.435442809 |
\( -\frac{1106097}{361} a + \frac{13496125}{361} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( -9 a + 74\) , \( -24 a + 129\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-9a+74\right){x}-24a+129$ |
| 19.2-a2 |
19.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
19.2 |
\( 19 \) |
\( 19^{6} \) |
$2.87211$ |
$(-2a-15)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.583453087$ |
$17.75857733$ |
2.435442809 |
\( \frac{820169978705}{47045881} a + \frac{8213399271372}{47045881} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( -29 a - 71\) , \( 70 a + 799\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-29a-71\right){x}+70a+799$ |
| 19.2-b1 |
19.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
19.2 |
\( 19 \) |
\( 19^{2} \) |
$2.87211$ |
$(-2a-15)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$3.010669998$ |
$46.18475228$ |
4.014256158 |
\( -\frac{1106097}{361} a + \frac{13496125}{361} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 70 a - 306\) , \( -489 a + 4852\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(70a-306\right){x}-489a+4852$ |
| 19.2-b2 |
19.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
19.2 |
\( 19 \) |
\( 19^{6} \) |
$2.87211$ |
$(-2a-15)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1.003556666$ |
$5.131639142$ |
4.014256158 |
\( \frac{820169978705}{47045881} a + \frac{8213399271372}{47045881} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 475 a - 3626\) , \( 14792 a - 120413\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(475a-3626\right){x}+14792a-120413$ |
| 39.1-a1 |
39.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3^{3} \cdot 13 \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$1$ |
$17.71929219$ |
3.452975138 |
\( \frac{5042176}{117} a + \frac{36290560}{117} \) |
\( \bigl[0\) , \( -a\) , \( a + 1\) , \( -128 a - 904\) , \( -1845 a - 13275\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-128a-904\right){x}-1845a-13275$ |
| 39.1-b1 |
39.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3^{9} \cdot 13 \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
|
\( 1 \) |
$1$ |
$16.36502275$ |
3.589617723 |
\( -\frac{483530009234047}{3159} a + \frac{3963689866496444}{3159} \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( 2084 a - 16953\) , \( -137012 a + 1123443\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(2084a-16953\right){x}-137012a+1123443$ |
| 39.1-b2 |
39.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{36} \cdot 13 \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
|
\( 2 \) |
$1$ |
$2.045627844$ |
3.589617723 |
\( \frac{7571581861819}{5036466357} a + \frac{65356091301080}{5036466357} \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( 2489 a - 20278\) , \( -76128 a + 624323\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(2489a-20278\right){x}-76128a+624323$ |
| 39.1-b3 |
39.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{18} \cdot 13^{2} \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
$0 \le r \le 1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
|
\( 2^{2} \) |
$1$ |
$8.182511377$ |
3.589617723 |
\( \frac{129876906937691}{3326427} a + \frac{104431672740049}{369603} \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( 2084 a - 16958\) , \( -136966 a + 1123037\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(2084a-16958\right){x}-136966a+1123037$ |
| 39.1-b4 |
39.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3^{9} \cdot 13^{4} \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
$0 \le r \le 1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
|
\( 2^{2} \) |
$1$ |
$4.091255688$ |
3.589617723 |
\( \frac{163710734479379606571667}{6940323} a + \frac{1178291993821242306674440}{6940323} \) |
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( 1679 a - 13718\) , \( -194800 a + 1596887\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(1679a-13718\right){x}-194800a+1596887$ |
| 39.1-c1 |
39.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3 \cdot 13 \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$22.75821193$ |
1.478304722 |
\( -\frac{5617}{39} a + \frac{45869}{39} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 2 a + 44\) , \( 6 a + 109\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(2a+44\right){x}+6a+109$ |
| 39.1-c2 |
39.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{4} \cdot 13 \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$22.75821193$ |
1.478304722 |
\( -\frac{2447848046349}{13} a + \frac{180593953971325}{117} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 7 a - 6\) , \( 7 a + 54\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(7a-6\right){x}+7a+54$ |
| 39.1-c3 |
39.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{2} \cdot 13^{2} \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$22.75821193$ |
1.478304722 |
\( -\frac{17629353}{169} a + \frac{482928916}{507} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 2 a + 39\) , \( 5 a + 85\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(2a+39\right){x}+5a+85$ |
| 39.1-c4 |
39.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3 \cdot 13^{4} \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$5.689552982$ |
1.478304722 |
\( \frac{51999945252787}{85683} a + \frac{374264458305925}{85683} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -3 a + 4\) , \( -41 a - 240\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-3a+4\right){x}-41a-240$ |
| 39.1-d1 |
39.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3^{3} \cdot 13^{3} \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 3^{2} \) |
$1$ |
$5.716729918$ |
3.342073611 |
\( -\frac{8290352125}{19773} a + \frac{67959714500}{19773} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 277 a - 2079\) , \( 7107 a - 57625\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(277a-2079\right){x}+7107a-57625$ |
| 39.1-d2 |
39.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{6} \cdot 13^{6} \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$5.716729918$ |
3.342073611 |
\( \frac{162826068625}{130323843} a + \frac{1610509497875}{130323843} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 302 a - 2284\) , \( 5863 a - 47428\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(302a-2284\right){x}+5863a-47428$ |
| 39.1-e1 |
39.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3 \cdot 13 \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$3.421644004$ |
$36.41573265$ |
4.046874215 |
\( -\frac{5617}{39} a + \frac{45869}{39} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( -31 a - 219\) , \( 4974 a + 35787\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-31a-219\right){x}+4974a+35787$ |
| 39.1-e2 |
39.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{4} \cdot 13 \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$13.68657601$ |
$4.551966581$ |
4.046874215 |
\( -\frac{2447848046349}{13} a + \frac{180593953971325}{117} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( -3501 a - 25194\) , \( -105197 a - 757158\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-3501a-25194\right){x}-105197a-757158$ |
| 39.1-e3 |
39.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{2} \cdot 13^{2} \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$6.843288008$ |
$18.20786632$ |
4.046874215 |
\( -\frac{17629353}{169} a + \frac{482928916}{507} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( -1956 a - 14074\) , \( 128615 a + 925681\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-1956a-14074\right){x}+128615a+925681$ |
| 39.1-e4 |
39.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3 \cdot 13^{4} \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$3.421644004$ |
$9.103933163$ |
4.046874215 |
\( \frac{51999945252787}{85683} a + \frac{374264458305925}{85683} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 1716 a - 13822\) , \( 198441 a - 1625786\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(1716a-13822\right){x}+198441a-1625786$ |
| 39.1-f1 |
39.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3^{3} \cdot 13^{3} \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$1.417588570$ |
$22.68731870$ |
3.133649804 |
\( -\frac{8290352125}{19773} a + \frac{67959714500}{19773} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( a + 25\) , \( 6 a + 57\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+\left(a+25\right){x}+6a+57$ |
| 39.1-f2 |
39.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{6} \cdot 13^{6} \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.708794285$ |
$11.34365935$ |
3.133649804 |
\( \frac{162826068625}{130323843} a + \frac{1610509497875}{130323843} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( -24 a - 155\) , \( 235 a + 1705\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+\left(-24a-155\right){x}+235a+1705$ |
| 39.1-g1 |
39.1-g |
$1$ |
$1$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3^{3} \cdot 13 \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1.742570068$ |
$21.53318794$ |
4.874773075 |
\( \frac{5042176}{117} a + \frac{36290560}{117} \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( 0\) , \( -21\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}-21$ |
| 39.1-h1 |
39.1-h |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3^{9} \cdot 13 \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 3^{2} \) |
$1$ |
$1.121760308$ |
0.655795459 |
\( -\frac{483530009234047}{3159} a + \frac{3963689866496444}{3159} \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( -114 a - 833\) , \( -3297 a - 23745\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-114a-833\right){x}-3297a-23745$ |
| 39.1-h2 |
39.1-h |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{36} \cdot 13 \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$1.121760308$ |
0.655795459 |
\( \frac{7571581861819}{5036466357} a + \frac{65356091301080}{5036466357} \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( -2059 a - 14833\) , \( -155406 a - 1118535\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-2059a-14833\right){x}-155406a-1118535$ |
| 39.1-h3 |
39.1-h |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{18} \cdot 13^{2} \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$1.121760308$ |
0.655795459 |
\( \frac{129876906937691}{3326427} a + \frac{104431672740049}{369603} \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( -2039 a - 14688\) , \( -158118 a - 1138054\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-2039a-14688\right){x}-158118a-1138054$ |
| 39.1-h4 |
39.1-h |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3^{9} \cdot 13^{4} \) |
$3.43779$ |
$(-a-7), (-a+9)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$0.280440077$ |
0.655795459 |
\( \frac{163710734479379606571667}{6940323} a + \frac{1178291993821242306674440}{6940323} \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( -32819 a - 236223\) , \( -9268674 a - 66710389\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-32819a-236223\right){x}-9268674a-66710389$ |
| 39.2-a1 |
39.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.2 |
\( 3 \cdot 13 \) |
\( - 3^{3} \cdot 13 \) |
$3.43779$ |
$(-a-7), (a+8)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$1$ |
$17.71929219$ |
3.452975138 |
\( -\frac{5042176}{117} a + \frac{41332736}{117} \) |
\( \bigl[0\) , \( a - 1\) , \( a\) , \( 128 a - 1032\) , \( 1844 a - 15119\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(128a-1032\right){x}+1844a-15119$ |
| 39.2-b1 |
39.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.2 |
\( 3 \cdot 13 \) |
\( - 3^{9} \cdot 13^{4} \) |
$3.43779$ |
$(-a-7), (a+8)$ |
$0 \le r \le 1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
|
\( 2^{2} \) |
$1$ |
$4.091255688$ |
3.589617723 |
\( -\frac{163710734479379606571667}{6940323} a + \frac{1342002728300621913246107}{6940323} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( -1680 a - 12038\) , \( 194800 a + 1402087\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-1680a-12038\right){x}+194800a+1402087$ |
| 39.2-b2 |
39.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.2 |
\( 3 \cdot 13 \) |
\( 3^{36} \cdot 13 \) |
$3.43779$ |
$(-a-7), (a+8)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
|
\( 2 \) |
$1$ |
$2.045627844$ |
3.589617723 |
\( -\frac{7571581861819}{5036466357} a + \frac{24309224387633}{1678822119} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( -2490 a - 17788\) , \( 76128 a + 548195\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-2490a-17788\right){x}+76128a+548195$ |
| 39.2-b3 |
39.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.2 |
\( 3 \cdot 13 \) |
\( 3^{18} \cdot 13^{2} \) |
$3.43779$ |
$(-a-7), (a+8)$ |
$0 \le r \le 1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
|
\( 2^{2} \) |
$1$ |
$8.182511377$ |
3.589617723 |
\( -\frac{129876906937691}{3326427} a + \frac{1069761961598132}{3326427} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( -2085 a - 14873\) , \( 136966 a + 986071\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-2085a-14873\right){x}+136966a+986071$ |
| 39.2-b4 |
39.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.2 |
\( 3 \cdot 13 \) |
\( - 3^{9} \cdot 13 \) |
$3.43779$ |
$(-a-7), (a+8)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
|
\( 1 \) |
$1$ |
$16.36502275$ |
3.589617723 |
\( \frac{483530009234047}{3159} a + \frac{3480159857262397}{3159} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( -2085 a - 14868\) , \( 137012 a + 986431\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-2085a-14868\right){x}+137012a+986431$ |
| 39.2-c1 |
39.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.2 |
\( 3 \cdot 13 \) |
\( - 3 \cdot 13^{4} \) |
$3.43779$ |
$(-a-7), (a+8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$5.689552982$ |
1.478304722 |
\( -\frac{51999945252787}{85683} a + \frac{426264403558712}{85683} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( a + 1\) , \( 40 a - 281\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(a+1\right){x}+40a-281$ |
| 39.2-c2 |
39.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.2 |
\( 3 \cdot 13 \) |
\( - 3 \cdot 13 \) |
$3.43779$ |
$(-a-7), (a+8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$22.75821193$ |
1.478304722 |
\( \frac{5617}{39} a + \frac{40252}{39} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -4 a + 46\) , \( -7 a + 115\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-4a+46\right){x}-7a+115$ |
| 39.2-c3 |
39.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.2 |
\( 3 \cdot 13 \) |
\( 3^{2} \cdot 13^{2} \) |
$3.43779$ |
$(-a-7), (a+8)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$22.75821193$ |
1.478304722 |
\( \frac{17629353}{169} a + \frac{430040857}{507} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -4 a + 41\) , \( -6 a + 90\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-4a+41\right){x}-6a+90$ |
| 39.2-c4 |
39.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.2 |
\( 3 \cdot 13 \) |
\( 3^{4} \cdot 13 \) |
$3.43779$ |
$(-a-7), (a+8)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$22.75821193$ |
1.478304722 |
\( \frac{2447848046349}{13} a + \frac{158563321554184}{117} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -9 a + 1\) , \( -8 a + 61\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-9a+1\right){x}-8a+61$ |
| 39.2-d1 |
39.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.2 |
\( 3 \cdot 13 \) |
\( 3^{6} \cdot 13^{6} \) |
$3.43779$ |
$(-a-7), (a+8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$5.716729918$ |
3.342073611 |
\( -\frac{162826068625}{130323843} a + \frac{591111855500}{43441281} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -272 a - 1954\) , \( -7846 a - 56471\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-272a-1954\right){x}-7846a-56471$ |
| 39.2-d2 |
39.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.2 |
\( 3 \cdot 13 \) |
\( - 3^{3} \cdot 13^{3} \) |
$3.43779$ |
$(-a-7), (a+8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 3^{2} \) |
$1$ |
$5.716729918$ |
3.342073611 |
\( \frac{8290352125}{19773} a + \frac{59669362375}{19773} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -247 a - 1774\) , \( -8910 a - 64129\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-247a-1774\right){x}-8910a-64129$ |
| 39.2-e1 |
39.2-e |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.2 |
\( 3 \cdot 13 \) |
\( - 3 \cdot 13^{4} \) |
$3.43779$ |
$(-a-7), (a+8)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$3.421644004$ |
$9.103933163$ |
4.046874215 |
\( -\frac{51999945252787}{85683} a + \frac{426264403558712}{85683} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( -1688 a - 12164\) , \( -210577 a - 1515613\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-1688a-12164\right){x}-210577a-1515613$ |
| 39.2-e2 |
39.2-e |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.2 |
\( 3 \cdot 13 \) |
\( - 3 \cdot 13 \) |
$3.43779$ |
$(-a-7), (a+8)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$3.421644004$ |
$36.41573265$ |
4.046874215 |
\( \frac{5617}{39} a + \frac{40252}{39} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( 30 a - 250\) , \( -4975 a + 40761\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(30a-250\right){x}-4975a+40761$ |
| 39.2-e3 |
39.2-e |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.2 |
\( 3 \cdot 13 \) |
\( 3^{2} \cdot 13^{2} \) |
$3.43779$ |
$(-a-7), (a+8)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$6.843288008$ |
$18.20786632$ |
4.046874215 |
\( \frac{17629353}{169} a + \frac{430040857}{507} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( 1955 a - 16030\) , \( -128616 a + 1054296\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(1955a-16030\right){x}-128616a+1054296$ |
| 39.2-e4 |
39.2-e |
$4$ |
$4$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.2 |
\( 3 \cdot 13 \) |
\( 3^{4} \cdot 13 \) |
$3.43779$ |
$(-a-7), (a+8)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$13.68657601$ |
$4.551966581$ |
4.046874215 |
\( \frac{2447848046349}{13} a + \frac{158563321554184}{117} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( 3500 a - 28695\) , \( 105196 a - 862355\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(3500a-28695\right){x}+105196a-862355$ |
| 39.2-f1 |
39.2-f |
$2$ |
$2$ |
\(\Q(\sqrt{237}) \) |
$2$ |
$[2, 0]$ |
39.2 |
\( 3 \cdot 13 \) |
\( 3^{6} \cdot 13^{6} \) |
$3.43779$ |
$(-a-7), (a+8)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.708794285$ |
$11.34365935$ |
3.133649804 |
\( -\frac{162826068625}{130323843} a + \frac{591111855500}{43441281} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 24 a - 179\) , \( -235 a + 1940\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(24a-179\right){x}-235a+1940$ |
*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.
