| 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 |
| 1.1-a1 |
1.1-a |
$2$ |
$19$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$1.03054$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$7$ |
7Ns.2.1 |
$1$ |
\( 1 \) |
$1$ |
$17.56070946$ |
1.522706625 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 30 a - 188\) , \( -215 a + 1347\bigr] \) |
${y}^2+{y}={x}^{3}+\left(30a-188\right){x}-215a+1347$ |
| 1.1-a2 |
1.1-a |
$2$ |
$19$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$1.03054$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$7$ |
7Ns.2.1 |
$1$ |
\( 1 \) |
$1$ |
$17.56070946$ |
1.522706625 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -30 a - 158\) , \( 215 a + 1132\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-30a-158\right){x}+215a+1132$ |
| 3.1-a1 |
3.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{6} \) |
$1.35627$ |
$(-a+6)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 2 \) |
$1$ |
$10.42270844$ |
1.807526880 |
\( -\frac{5906432}{729} a - \frac{31166464}{729} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( 174 a - 1090\) , \( -13960 a + 87477\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(174a-1090\right){x}-13960a+87477$ |
| 3.1-b1 |
3.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{6} \) |
$1.35627$ |
$(-a+6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 2 \cdot 3 \) |
$0.090971937$ |
$19.05309288$ |
1.803550704 |
\( -\frac{5906432}{729} a - \frac{31166464}{729} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -a - 5\) , \( a + 5\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+\left(-a-5\right){x}+a+5$ |
| 3.2-a1 |
3.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( 3^{6} \) |
$1.35627$ |
$(-a-5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 2 \) |
$1$ |
$10.42270844$ |
1.807526880 |
\( \frac{5906432}{729} a - \frac{12357632}{243} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -174 a - 916\) , \( 13960 a + 73517\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(-174a-916\right){x}+13960a+73517$ |
| 3.2-b1 |
3.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( 3^{6} \) |
$1.35627$ |
$(-a-5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 2 \cdot 3 \) |
$0.090971937$ |
$19.05309288$ |
1.803550704 |
\( \frac{5906432}{729} a - \frac{12357632}{243} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( a - 6\) , \( -a + 6\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+\left(a-6\right){x}-a+6$ |
| 4.1-a1 |
4.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{6} \) |
$1.45740$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Nn |
$1$ |
\( 1 \) |
$1$ |
$19.84955375$ |
1.721174595 |
\( -\frac{27}{8} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 3 a + 20\) , \( -25 a - 130\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(3a+20\right){x}-25a-130$ |
| 4.1-b1 |
4.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{6} \) |
$1.45740$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Nn |
$1$ |
\( 1 \) |
$1$ |
$19.84955375$ |
1.721174595 |
\( -\frac{27}{8} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( -3 a + 23\) , \( 25 a - 155\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(-3a+23\right){x}+25a-155$ |
| 9.1-a1 |
9.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$1.78495$ |
$(-a+6), (-a-5)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.262450617$ |
$27.78537212$ |
2.529286276 |
\( -\frac{23496139271}{729} a + \frac{49078238762}{243} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -199 a - 1048\) , \( 549 a + 2891\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-199a-1048\right){x}+549a+2891$ |
| 9.1-a2 |
9.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$1.78495$ |
$(-a+6), (-a-5)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1.049802470$ |
$27.78537212$ |
2.529286276 |
\( -\frac{337183}{27} a + \frac{2653798}{27} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 124 a - 777\) , \( 1847 a - 11574\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(124a-777\right){x}+1847a-11574$ |
| 9.1-a3 |
9.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$1.78495$ |
$(-a+6), (-a-5)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1.049802470$ |
$27.78537212$ |
2.529286276 |
\( \frac{337183}{27} a + \frac{772205}{9} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -124 a - 653\) , \( -1847 a - 9727\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-124a-653\right){x}-1847a-9727$ |
| 9.1-a4 |
9.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$1.78495$ |
$(-a+6), (-a-5)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.262450617$ |
$27.78537212$ |
2.529286276 |
\( \frac{23496139271}{729} a + \frac{123738577015}{729} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 199 a - 1247\) , \( -549 a + 3440\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(199a-1247\right){x}-549a+3440$ |
| 9.1-b1 |
9.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$1.78495$ |
$(-a+6), (-a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$8.450537277$ |
2.198263536 |
\( -\frac{23496139271}{729} a + \frac{49078238762}{243} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 84 a - 472\) , \( 1065 a - 6589\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(84a-472\right){x}+1065a-6589$ |
| 9.1-b2 |
9.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$1.78495$ |
$(-a+6), (-a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$1$ |
$33.80214910$ |
2.198263536 |
\( -\frac{337183}{27} a + \frac{2653798}{27} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -10 a + 8\) , \( -29 a - 68\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-10a+8\right){x}-29a-68$ |
| 9.1-b3 |
9.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$1.78495$ |
$(-a+6), (-a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$1$ |
$33.80214910$ |
2.198263536 |
\( \frac{337183}{27} a + \frac{772205}{9} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 9 a - 2\) , \( 29 a - 97\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(9a-2\right){x}+29a-97$ |
| 9.1-b4 |
9.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$1.78495$ |
$(-a+6), (-a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$8.450537277$ |
2.198263536 |
\( \frac{23496139271}{729} a + \frac{123738577015}{729} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -85 a - 387\) , \( -1065 a - 5524\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-85a-387\right){x}-1065a-5524$ |
| 9.2-a1 |
9.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{12} \) |
$1.78495$ |
$(-a-5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Ns |
$1$ |
\( 2 \) |
$3.481040408$ |
$3.017191929$ |
3.642890967 |
\( \frac{5906432}{729} a - \frac{12357632}{243} \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( -a + 7\) , \( -4 a - 29\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-a+7\right){x}-4a-29$ |
| 9.2-b1 |
9.2-b |
$2$ |
$19$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.78495$ |
$(-a-5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.3.1 |
$1$ |
\( 2 \) |
$0.104154583$ |
$44.19348301$ |
1.596506862 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -3808 a - 20054\) , \( 307454 a + 1619139\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-3808a-20054\right){x}+307454a+1619139$ |
| 9.2-b2 |
9.2-b |
$2$ |
$19$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.78495$ |
$(-a-5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.3.1 |
$1$ |
\( 2 \) |
$1.978937092$ |
$2.325972790$ |
1.596506862 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 2 a - 14\) , \( 4 a - 26\bigr] \) |
${y}^2+{y}={x}^{3}+\left(2a-14\right){x}+4a-26$ |
| 9.2-c1 |
9.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{12} \) |
$1.78495$ |
$(-a-5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Ns |
$1$ |
\( 2 \) |
$0.263815516$ |
$21.93925530$ |
2.007503858 |
\( \frac{5906432}{729} a - \frac{12357632}{243} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( 2134 a - 13363\) , \( -125614 a + 787127\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2134a-13363\right){x}-125614a+787127$ |
| 9.3-a1 |
9.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{12} \) |
$1.78495$ |
$(-a+6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Ns |
$1$ |
\( 2 \) |
$3.481040408$ |
$3.017191929$ |
3.642890967 |
\( -\frac{5906432}{729} a - \frac{31166464}{729} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( a + 6\) , \( 4 a - 33\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(a+6\right){x}+4a-33$ |
| 9.3-b1 |
9.3-b |
$2$ |
$19$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.78495$ |
$(-a+6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.3.1 |
$1$ |
\( 2 \) |
$0.104154583$ |
$44.19348301$ |
1.596506862 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 3808 a - 23862\) , \( -307454 a + 1926593\bigr] \) |
${y}^2+{y}={x}^{3}+\left(3808a-23862\right){x}-307454a+1926593$ |
| 9.3-b2 |
9.3-b |
$2$ |
$19$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{6} \) |
$1.78495$ |
$(-a+6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.3.1 |
$1$ |
\( 2 \) |
$1.978937092$ |
$2.325972790$ |
1.596506862 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -2 a - 12\) , \( -4 a - 22\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-2a-12\right){x}-4a-22$ |
| 9.3-c1 |
9.3-c |
$1$ |
$1$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{12} \) |
$1.78495$ |
$(-a+6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Ns |
$1$ |
\( 2 \) |
$0.263815516$ |
$21.93925530$ |
2.007503858 |
\( -\frac{5906432}{729} a - \frac{31166464}{729} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( -2134 a - 11229\) , \( 125614 a + 661513\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(-2134a-11229\right){x}+125614a+661513$ |
| 12.1-a1 |
12.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{10} \cdot 3 \) |
$1.91805$ |
$(-a+6), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$19.23248614$ |
1.667668047 |
\( -\frac{1503260928590689}{96} a + \frac{9419855832943555}{96} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -4998 a - 26307\) , \( 474428 a + 2498481\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4998a-26307\right){x}+474428a+2498481$ |
| 12.1-a2 |
12.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{2} \cdot 3^{5} \) |
$1.91805$ |
$(-a+6), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$19.23248614$ |
1.667668047 |
\( -\frac{60451}{486} a + \frac{272513}{243} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 52 a + 288\) , \( -1090 a - 5730\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(52a+288\right){x}-1090a-5730$ |
| 12.1-b1 |
12.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{10} \cdot 3 \) |
$1.91805$ |
$(-a+6), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 1 \) |
$16.49862578$ |
$1.010440818$ |
2.891098107 |
\( -\frac{1503260928590689}{96} a + \frac{9419855832943555}{96} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 736 a - 4514\) , \( 25456 a - 159268\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(736a-4514\right){x}+25456a-159268$ |
| 12.1-b2 |
12.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{2} \cdot 3^{5} \) |
$1.91805$ |
$(-a+6), (2)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5 \) |
$3.299725157$ |
$25.26102046$ |
2.891098107 |
\( -\frac{60451}{486} a + \frac{272513}{243} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 11 a + 31\) , \( 23 a + 116\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(11a+31\right){x}+23a+116$ |
| 12.2-a1 |
12.2-a |
$2$ |
$5$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( - 2^{2} \cdot 3^{5} \) |
$1.91805$ |
$(-a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$19.23248614$ |
1.667668047 |
\( \frac{60451}{486} a + \frac{161525}{162} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -52 a + 340\) , \( 1090 a - 6820\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-52a+340\right){x}+1090a-6820$ |
| 12.2-a2 |
12.2-a |
$2$ |
$5$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( - 2^{10} \cdot 3 \) |
$1.91805$ |
$(-a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$19.23248614$ |
1.667668047 |
\( \frac{1503260928590689}{96} a + \frac{1319432484058811}{16} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 4998 a - 31305\) , \( -474428 a + 2972909\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(4998a-31305\right){x}-474428a+2972909$ |
| 12.2-b1 |
12.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( - 2^{2} \cdot 3^{5} \) |
$1.91805$ |
$(-a-5), (2)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5 \) |
$3.299725157$ |
$25.26102046$ |
2.891098107 |
\( \frac{60451}{486} a + \frac{161525}{162} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( 4 a + 10\) , \( 2 a + 6\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(4a+10\right){x}+2a+6$ |
| 12.2-b2 |
12.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( - 2^{10} \cdot 3 \) |
$1.91805$ |
$(-a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 1 \) |
$16.49862578$ |
$1.010440818$ |
2.891098107 |
\( \frac{1503260928590689}{96} a + \frac{1319432484058811}{16} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( -721 a - 3810\) , \( -29251 a - 154050\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-721a-3810\right){x}-29251a-154050$ |
| 19.1-a1 |
19.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$2.15156$ |
$(-5a-26)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$17.03289160$ |
2.953878024 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( 1996420 a - 12510129\) , \( -3687799230 a + 23108787343\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(1996420a-12510129\right){x}-3687799230a+23108787343$ |
| 19.1-a2 |
19.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( 19^{6} \) |
$2.15156$ |
$(-5a-26)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$1$ |
$17.03289160$ |
2.953878024 |
\( -\frac{89915392}{6859} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -24220 a - 127549\) , \( 5249630 a + 27646028\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(-24220a-127549\right){x}+5249630a+27646028$ |
| 19.1-a3 |
19.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$2.15156$ |
$(-5a-26)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$17.03289160$ |
2.953878024 |
\( \frac{32768}{19} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( 1730 a + 9111\) , \( 3580 a + 18853\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(1730a+9111\right){x}+3580a+18853$ |
| 19.1-b1 |
19.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$2.15156$ |
$(-5a-26)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$10.04336916$ |
$0.205438503$ |
0.715641372 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -769\) , \( -8470\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}-769{x}-8470$ |
| 19.1-b2 |
19.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( 19^{6} \) |
$2.15156$ |
$(-5a-26)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$3.347789720$ |
$1.848946532$ |
0.715641372 |
\( -\frac{89915392}{6859} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -9\) , \( -15\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}-9{x}-15$ |
| 19.1-b3 |
19.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$2.15156$ |
$(-5a-26)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1.115929906$ |
$16.64051879$ |
0.715641372 |
\( \frac{32768}{19} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+{x}$ |
| 21.1-a1 |
21.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{3} \cdot 7^{3} \) |
$2.20607$ |
$(-a+6), (-3a+19)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 3 \) |
$0.814975320$ |
$13.31588418$ |
5.645987291 |
\( \frac{1299826}{1323} a + \frac{8750387}{1323} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -1389 a + 8688\) , \( 23987 a - 150319\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-1389a+8688\right){x}+23987a-150319$ |
| 21.1-b1 |
21.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{3} \cdot 7^{3} \) |
$2.20607$ |
$(-a+6), (-3a+19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 1 \) |
$1$ |
$22.26667828$ |
1.930765873 |
\( \frac{1299826}{1323} a + \frac{8750387}{1323} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 4 a + 34\) , \( 7 a + 44\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(4a+34\right){x}+7a+44$ |
| 21.2-a1 |
21.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( - 3^{3} \cdot 7^{3} \) |
$2.20607$ |
$(-a-5), (-3a+19)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 3 \) |
$0.814975320$ |
$13.31588418$ |
5.645987291 |
\( -\frac{1299826}{1323} a + \frac{3350071}{441} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( 1387 a + 7300\) , \( -23988 a - 126331\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(1387a+7300\right){x}-23988a-126331$ |
| 21.2-b1 |
21.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( - 3^{3} \cdot 7^{3} \) |
$2.20607$ |
$(-a-5), (-3a+19)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 1 \) |
$1$ |
$22.26667828$ |
1.930765873 |
\( -\frac{1299826}{1323} a + \frac{3350071}{441} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -4 a + 5\) , \( -4 a + 13\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-4a+5\right){x}-4a+13$ |
| 27.1-a1 |
27.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{14} \) |
$2.34912$ |
$(-a+6), (-a-5)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$15.32322821$ |
0.885794930 |
\( -\frac{23496139271}{729} a + \frac{49078238762}{243} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -4 a - 24\) , \( -79 a - 411\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-4a-24\right){x}-79a-411$ |
| 27.1-a2 |
27.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{10} \) |
$2.34912$ |
$(-a+6), (-a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$10.21548547$ |
0.885794930 |
\( -\frac{337183}{27} a + \frac{2653798}{27} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( -10751 a - 56613\) , \( -1428083 a - 7520679\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-10751a-56613\right){x}-1428083a-7520679$ |
| 27.1-a3 |
27.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{10} \) |
$2.34912$ |
$(-a+6), (-a-5)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$30.64645642$ |
0.885794930 |
\( \frac{337183}{27} a + \frac{772205}{9} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( a + 6\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(a+6\right){x}$ |
| 27.1-a4 |
27.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{14} \) |
$2.34912$ |
$(-a+6), (-a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$1$ |
$5.107742738$ |
0.885794930 |
\( \frac{23496139271}{729} a + \frac{123738577015}{729} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 25269 a - 158229\) , \( -918081 a + 5753187\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(25269a-158229\right){x}-918081a+5753187$ |
| 27.1-b1 |
27.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{12} \) |
$2.34912$ |
$(-a+6), (-a-5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Ns |
$1$ |
\( 2^{2} \) |
$0.220002425$ |
$21.93925530$ |
3.348216379 |
\( \frac{5906432}{729} a - \frac{12357632}{243} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 128 a - 789\) , \( -1680 a + 10518\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(128a-789\right){x}-1680a+10518$ |
| 27.1-c1 |
27.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{12} \) |
$2.34912$ |
$(-a+6), (-a-5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Ns |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$3.017191929$ |
3.139484642 |
\( \frac{5906432}{729} a - \frac{12357632}{243} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( -12 a - 54\) , \( -287 a - 1517\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(-12a-54\right){x}-287a-1517$ |
| 27.1-d1 |
27.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{14} \) |
$2.34912$ |
$(-a+6), (-a-5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1.371342968$ |
$5.107742738$ |
2.429457297 |
\( -\frac{23496139271}{729} a + \frac{49078238762}{243} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 10170 a - 63645\) , \( 1344874 a - 8427205\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(10170a-63645\right){x}+1344874a-8427205$ |
| 27.1-d2 |
27.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{133}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{10} \) |
$2.34912$ |
$(-a+6), (-a-5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.914228645$ |
$30.64645642$ |
2.429457297 |
\( -\frac{337183}{27} a + \frac{2653798}{27} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( a + 6\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+6\right){x}$ |
*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.
