| 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 |
| 4.1-a1 |
4.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{10} \) |
$1.84435$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5Ns |
$1$ |
\( 5 \) |
$0.321242509$ |
$12.08960761$ |
2.661064575 |
\( \frac{3131359847}{32} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a\) , \( 149 a - 1144\) , \( 3584 a - 27889\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(149a-1144\right){x}+3584a-27889$ |
| 4.1-b1 |
4.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{2} \) |
$1.84435$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$7.878457783$ |
$3.320117318$ |
3.584551597 |
\( -\frac{165189978245875}{2} a + 644014585066625 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 814 a - 6158\) , \( 33472 a - 260349\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(814a-6158\right){x}+33472a-260349$ |
| 4.1-b2 |
4.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{6} \) |
$1.84435$ |
$(2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$2.626152594$ |
$29.88105586$ |
3.584551597 |
\( \frac{857375}{8} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 24 a + 2\) , \( 92 a - 75\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(24a+2\right){x}+92a-75$ |
| 4.1-b3 |
4.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{2} \) |
$1.84435$ |
$(2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$7.878457783$ |
$29.88105586$ |
3.584551597 |
\( \frac{165189978245875}{2} a + \frac{1122839191887375}{2} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 84 a - 468\) , \( -778 a + 6691\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(84a-468\right){x}-778a+6691$ |
| 4.1-c1 |
4.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{2} \) |
$1.84435$ |
$(2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$7.878457783$ |
$29.88105586$ |
3.584551597 |
\( -\frac{165189978245875}{2} a + 644014585066625 \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( -58 a - 382\) , \( 310 a + 2111\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-58a-382\right){x}+310a+2111$ |
| 4.1-c2 |
4.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{6} \) |
$1.84435$ |
$(2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$2.626152594$ |
$29.88105586$ |
3.584551597 |
\( \frac{857375}{8} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( 2 a + 28\) , \( -90 a - 605\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2a+28\right){x}-90a-605$ |
| 4.1-c3 |
4.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{2} \) |
$1.84435$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$7.878457783$ |
$3.320117318$ |
3.584551597 |
\( \frac{165189978245875}{2} a + \frac{1122839191887375}{2} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( -788 a - 5342\) , \( -39630 a - 269369\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-788a-5342\right){x}-39630a-269369$ |
| 4.1-d1 |
4.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{10} \) |
$1.84435$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5Ns |
$1$ |
\( 5 \) |
$0.321242509$ |
$12.08960761$ |
2.661064575 |
\( \frac{3131359847}{32} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( -151 a - 994\) , \( -3585 a - 24305\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-151a-994\right){x}-3585a-24305$ |
| 12.1-a1 |
12.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{14} \cdot 3^{6} \) |
$2.42730$ |
$(a+7), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 7 \) |
$0.263415359$ |
$7.087637130$ |
3.581878078 |
\( \frac{4826809}{3456} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( 19 a + 162\) , \( 153 a + 1105\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(19a+162\right){x}+153a+1105$ |
| 12.1-b1 |
12.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{14} \cdot 3^{6} \) |
$2.42730$ |
$(a+7), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 7 \) |
$0.263415359$ |
$7.087637130$ |
3.581878078 |
\( \frac{4826809}{3456} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a\) , \( -21 a + 182\) , \( -154 a + 1259\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-21a+182\right){x}-154a+1259$ |
| 25.1-a1 |
25.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{2} \) |
$2.91617$ |
$(5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.966248860$ |
$21.17104369$ |
2.803312137 |
\( \frac{4096}{5} \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( -2 a + 31\) , \( 3 a - 12\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2a+31\right){x}+3a-12$ |
| 25.1-b1 |
25.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{2} \) |
$2.91617$ |
$(5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.966248860$ |
$21.17104369$ |
2.803312137 |
\( \frac{4096}{5} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 2 a + 29\) , \( -3 a - 9\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(2a+29\right){x}-3a-9$ |
| 36.1-a1 |
36.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{38} \cdot 3^{13} \) |
$3.19451$ |
$(a+7), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 19 \) |
$1$ |
$0.766541008$ |
1.995855929 |
\( -\frac{37325374883}{42467328} a - \frac{37227730897}{5308416} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( 448 a - 3495\) , \( -21691 a + 169114\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(448a-3495\right){x}-21691a+169114$ |
| 36.1-b1 |
36.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{3} \) |
$3.19451$ |
$(a+7), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$7.042042709$ |
0.965025631 |
\( -\frac{46370553}{2} a - \frac{315207591}{2} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 17 a + 116\) , \( 45 a + 306\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(17a+116\right){x}+45a+306$ |
| 36.1-c1 |
36.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{6} \) |
$3.19451$ |
$(a+7), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 1 \) |
$1$ |
$36.17494452$ |
2.478666356 |
\( -473040 a + \frac{7386147}{2} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -480 a - 3263\) , \( 12384 a + 84177\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-480a-3263\right){x}+12384a+84177$ |
| 36.1-d1 |
36.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{10} \) |
$3.19451$ |
$(a+7), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$7.573262403$ |
2.075645558 |
\( \frac{2119}{18} a + \frac{4801}{6} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 4 a - 14\) , \( 3300 a - 25731\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(4a-14\right){x}+3300a-25731$ |
| 36.1-e1 |
36.1-e |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{6} \) |
$3.19451$ |
$(a+7), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 1 \) |
$1$ |
$36.17494452$ |
2.478666356 |
\( 473040 a + \frac{6440067}{2} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 480 a - 3743\) , \( -12384 a + 96561\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(480a-3743\right){x}-12384a+96561$ |
| 36.1-f1 |
36.1-f |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{10} \) |
$3.19451$ |
$(a+7), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$12.97391297$ |
3.555831477 |
\( \frac{2119}{18} a + \frac{4801}{6} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 17 a + 101\) , \( 61 a + 407\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(17a+101\right){x}+61a+407$ |
| 36.1-g1 |
36.1-g |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{3} \) |
$3.19451$ |
$(a+7), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$15.35329757$ |
2.103981231 |
\( \frac{46370553}{2} a - 180789072 \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -663 a - 4485\) , \( -26387 a - 179340\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-663a-4485\right){x}-26387a-179340$ |
| 36.1-h1 |
36.1-h |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{10} \) |
$3.19451$ |
$(a+7), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$7.573262403$ |
2.075645558 |
\( -\frac{2119}{18} a + \frac{8261}{9} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -5 a - 9\) , \( -3301 a - 22430\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-5a-9\right){x}-3301a-22430$ |
| 36.1-i1 |
36.1-i |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{6} \) |
$3.19451$ |
$(a+7), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$4$ |
\( 1 \) |
$1$ |
$10.98053125$ |
3.009494417 |
\( -473040 a + \frac{7386147}{2} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -6 a + 69\) , \( -9 a + 88\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-6a+69\right){x}-9a+88$ |
| 36.1-j1 |
36.1-j |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{38} \cdot 3^{13} \) |
$3.19451$ |
$(a+7), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 19 \) |
$1$ |
$0.990291659$ |
2.578439324 |
\( \frac{37325374883}{42467328} a - \frac{335147222059}{42467328} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 4461 a - 34748\) , \( -448698 a + 3498699\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(4461a-34748\right){x}-448698a+3498699$ |
| 36.1-k1 |
36.1-k |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{38} \cdot 3^{13} \) |
$3.19451$ |
$(a+7), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 19 \) |
$1$ |
$0.990291659$ |
2.578439324 |
\( -\frac{37325374883}{42467328} a - \frac{37227730897}{5308416} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -4463 a - 30287\) , \( 448697 a + 3050001\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-4463a-30287\right){x}+448697a+3050001$ |
| 36.1-l1 |
36.1-l |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{3} \) |
$3.19451$ |
$(a+7), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$15.35329757$ |
2.103981231 |
\( -\frac{46370553}{2} a - \frac{315207591}{2} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 663 a - 5148\) , \( 26387 a - 205727\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(663a-5148\right){x}+26387a-205727$ |
| 36.1-m1 |
36.1-m |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{3} \) |
$3.19451$ |
$(a+7), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$7.042042709$ |
0.965025631 |
\( \frac{46370553}{2} a - 180789072 \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 12 a + 77\) , \( 33 a + 180\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(12a+77\right){x}+33a+180$ |
| 36.1-n1 |
36.1-n |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{10} \) |
$3.19451$ |
$(a+7), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$12.97391297$ |
3.555831477 |
\( -\frac{2119}{18} a + \frac{8261}{9} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( 10 a + 64\) , \( 29 a + 205\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(10a+64\right){x}+29a+205$ |
| 36.1-o1 |
36.1-o |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{6} \) |
$3.19451$ |
$(a+7), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$4$ |
\( 1 \) |
$1$ |
$10.98053125$ |
3.009494417 |
\( 473040 a + \frac{6440067}{2} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 6 a + 63\) , \( 9 a + 79\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(6a+63\right){x}+9a+79$ |
| 36.1-p1 |
36.1-p |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{38} \cdot 3^{13} \) |
$3.19451$ |
$(a+7), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 19 \) |
$1$ |
$0.766541008$ |
1.995855929 |
\( \frac{37325374883}{42467328} a - \frac{335147222059}{42467328} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -449 a - 3047\) , \( 21690 a + 147423\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-449a-3047\right){x}+21690a+147423$ |
| 49.1-a1 |
49.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{2} \) |
$3.45046$ |
$(7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$39.75665665$ |
0.681020306 |
\( \frac{12167}{7} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 10 a + 79\) , \( 15 a + 109\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(10a+79\right){x}+15a+109$ |
| 49.1-a2 |
49.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{4} \) |
$3.45046$ |
$(7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$19.87832832$ |
0.681020306 |
\( \frac{18191447}{49} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -15 a - 91\) , \( -395 a - 2678\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-15a-91\right){x}-395a-2678$ |
| 49.1-b1 |
49.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{2} \) |
$3.45046$ |
$(7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$39.75665665$ |
0.681020306 |
\( \frac{12167}{7} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 17 a + 61\) , \( 46 a + 288\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(17a+61\right){x}+46a+288$ |
| 49.1-b2 |
49.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{4} \) |
$3.45046$ |
$(7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$19.87832832$ |
0.681020306 |
\( \frac{18191447}{49} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 42 a - 134\) , \( 261 a - 1389\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(42a-134\right){x}+261a-1389$ |
| 68.1-a1 |
68.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{2} \cdot 17^{2} \) |
$3.74503$ |
$(2a-15), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$2.243393028$ |
$18.50007136$ |
11.37493593 |
\( \frac{26750817}{578} a + \frac{91817145}{289} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -126 a - 871\) , \( -3777 a - 25681\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-126a-871\right){x}-3777a-25681$ |
| 68.1-b1 |
68.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{2} \cdot 17^{2} \) |
$3.74503$ |
$(2a-15), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$8.281422611$ |
$2.627376862$ |
5.963448987 |
\( \frac{5144098397625}{578} a - \frac{20054935668375}{289} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 5 a - 20\) , \( 7 a - 47\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(5a-20\right){x}+7a-47$ |
| 68.1-c1 |
68.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{2} \cdot 17^{2} \) |
$3.74503$ |
$(2a-15), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$14.82108197$ |
2.031047607 |
\( \frac{5144098397625}{578} a - \frac{20054935668375}{289} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 673 a + 4560\) , \( -68446 a - 465253\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(673a+4560\right){x}-68446a-465253$ |
| 68.1-d1 |
68.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{2} \cdot 17^{2} \) |
$3.74503$ |
$(2a-15), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$32.11976393$ |
4.401619922 |
\( \frac{26750817}{578} a + \frac{91817145}{289} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( a + 9\) , \( -1\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(a+9\right){x}-1$ |
| 68.2-a1 |
68.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
68.2 |
\( 2^{2} \cdot 17 \) |
\( 2^{2} \cdot 17^{2} \) |
$3.74503$ |
$(2a+13), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$2.243393028$ |
$18.50007136$ |
11.37493593 |
\( -\frac{26750817}{578} a + \frac{210385107}{578} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( 153 a - 1051\) , \( 2752 a - 21027\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(153a-1051\right){x}+2752a-21027$ |
| 68.2-b1 |
68.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
68.2 |
\( 2^{2} \cdot 17 \) |
\( 2^{2} \cdot 17^{2} \) |
$3.74503$ |
$(2a+13), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$8.281422611$ |
$2.627376862$ |
5.963448987 |
\( -\frac{5144098397625}{578} a - \frac{34965772939125}{578} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -6 a - 14\) , \( -8 a - 39\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-6a-14\right){x}-8a-39$ |
| 68.2-c1 |
68.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
68.2 |
\( 2^{2} \cdot 17 \) |
\( 2^{2} \cdot 17^{2} \) |
$3.74503$ |
$(2a+13), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$14.82108197$ |
2.031047607 |
\( -\frac{5144098397625}{578} a - \frac{34965772939125}{578} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -646 a + 5179\) , \( 73651 a - 573845\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-646a+5179\right){x}+73651a-573845$ |
| 68.2-d1 |
68.2-d |
$1$ |
$1$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
68.2 |
\( 2^{2} \cdot 17 \) |
\( 2^{2} \cdot 17^{2} \) |
$3.74503$ |
$(2a+13), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$32.11976393$ |
4.401619922 |
\( -\frac{26750817}{578} a + \frac{210385107}{578} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -2 a + 11\) , \( -a\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2a+11\right){x}-a$ |
| 69.1-a1 |
69.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
69.1 |
\( 3 \cdot 23 \) |
\( 3^{4} \cdot 23^{2} \) |
$3.75873$ |
$(a+7), (a+5)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$5.381879077$ |
$19.64697124$ |
7.245022582 |
\( \frac{1350349}{4761} a - \frac{5517055}{4761} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 5 a - 19\) , \( -12 a + 51\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(5a-19\right){x}-12a+51$ |
| 69.1-a2 |
69.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
69.1 |
\( 3 \cdot 23 \) |
\( 3^{2} \cdot 23^{4} \) |
$3.75873$ |
$(a+7), (a+5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$10.76375815$ |
$19.64697124$ |
7.245022582 |
\( -\frac{821839657067}{839523} a + \frac{2136799665574}{279841} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 80 a - 604\) , \( -888 a + 6882\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(80a-604\right){x}-888a+6882$ |
| 69.1-a3 |
69.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
69.1 |
\( 3 \cdot 23 \) |
\( 3 \cdot 23^{8} \) |
$3.75873$ |
$(a+7), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$21.52751631$ |
$4.911742811$ |
7.245022582 |
\( \frac{504423560186689}{234932955843} a + \frac{4504150076599399}{234932955843} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 85 a - 644\) , \( -737 a + 5707\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(85a-644\right){x}-737a+5707$ |
| 69.1-a4 |
69.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
69.1 |
\( 3 \cdot 23 \) |
\( 3 \cdot 23^{2} \) |
$3.75873$ |
$(a+7), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$5.381879077$ |
$19.64697124$ |
7.245022582 |
\( -\frac{22202258344574737}{1587} a + \frac{173116781217095513}{1587} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 1275 a - 9924\) , \( -64363 a + 501821\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1275a-9924\right){x}-64363a+501821$ |
| 69.1-b1 |
69.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
69.1 |
\( 3 \cdot 23 \) |
\( 3^{4} \cdot 23^{2} \) |
$3.75873$ |
$(a+7), (a+5)$ |
$0 \le r \le 2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{3} \) |
$1$ |
$6.241577796$ |
3.421326909 |
\( \frac{1350349}{4761} a - \frac{5517055}{4761} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 18 a + 135\) , \( 51 a + 369\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(18a+135\right){x}+51a+369$ |
| 69.1-b2 |
69.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
69.1 |
\( 3 \cdot 23 \) |
\( 3^{2} \cdot 23^{4} \) |
$3.75873$ |
$(a+7), (a+5)$ |
$0 \le r \le 2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$16$ |
\( 2^{3} \) |
$1$ |
$6.241577796$ |
3.421326909 |
\( -\frac{821839657067}{839523} a + \frac{2136799665574}{279841} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -57 a - 375\) , \( -855 a - 5790\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-57a-375\right){x}-855a-5790$ |
| 69.1-b3 |
69.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
69.1 |
\( 3 \cdot 23 \) |
\( 3 \cdot 23^{8} \) |
$3.75873$ |
$(a+7), (a+5)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$6.241577796$ |
3.421326909 |
\( \frac{504423560186689}{234932955843} a + \frac{4504150076599399}{234932955843} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -427 a - 2890\) , \( 10006 a + 68035\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-427a-2890\right){x}+10006a+68035$ |
| 69.1-b4 |
69.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
69.1 |
\( 3 \cdot 23 \) |
\( 3 \cdot 23^{2} \) |
$3.75873$ |
$(a+7), (a+5)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$64$ |
\( 2 \) |
$1$ |
$1.560394449$ |
3.421326909 |
\( -\frac{22202258344574737}{1587} a + \frac{173116781217095513}{1587} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -887 a - 6020\) , \( -45280 a - 307771\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-887a-6020\right){x}-45280a-307771$ |
| 69.2-a1 |
69.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
69.2 |
\( 3 \cdot 23 \) |
\( 3^{4} \cdot 23^{2} \) |
$3.75873$ |
$(a+7), (a-6)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$5.381879077$ |
$19.64697124$ |
7.245022582 |
\( -\frac{1350349}{4761} a - \frac{1388902}{1587} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -4 a - 14\) , \( 6 a + 26\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-4a-14\right){x}+6a+26$ |
| 69.2-a2 |
69.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{213}) \) |
$2$ |
$[2, 0]$ |
69.2 |
\( 3 \cdot 23 \) |
\( 3 \cdot 23^{8} \) |
$3.75873$ |
$(a+7), (a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$21.52751631$ |
$4.911742811$ |
7.245022582 |
\( -\frac{504423560186689}{234932955843} a + \frac{5008573636786088}{234932955843} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -84 a - 559\) , \( 651 a + 4412\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-84a-559\right){x}+651a+4412$ |
*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.
