| 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 |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{34} \) |
$0.92001$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$8.120880803$ |
1.115488766 |
\( -\frac{25153757}{131072} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -40 a - 124\) , \( 801 a + 2515\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-40a-124\right){x}+801a+2515$ |
| 4.1-b1 |
4.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{20} \) |
$0.92001$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B.1.2 |
$4$ |
\( 2 \) |
$1$ |
$0.853190410$ |
0.937557727 |
\( -\frac{9814089221}{1024} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 1\) , \( 311 a - 1287\) , \( 6283 a - 26012\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(311a-1287\right){x}+6283a-26012$ |
| 4.1-b2 |
4.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$0.92001$ |
$(2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B.1.1 |
$4$ |
\( 2 \) |
$1$ |
$21.32976027$ |
0.937557727 |
\( \frac{6859}{4} \) |
\( \bigl[a\) , \( 1\) , \( 1\) , \( 3 a + 15\) , \( 5 a + 17\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+{x}^{2}+\left(3a+15\right){x}+5a+17$ |
| 7.1-a1 |
7.1-a |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
7.1 |
\( 7 \) |
\( -7 \) |
$1.05816$ |
$(-a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.3 |
$1$ |
\( 1 \) |
$21.08909123$ |
$0.222906563$ |
1.291435685 |
\( -\frac{195338235078135808}{7} a - \frac{613353587397607424}{7} \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( 351 a - 3081\) , \( 11878 a - 72832\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(351a-3081\right){x}+11878a-72832$ |
| 7.1-a2 |
7.1-a |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
7.1 |
\( 7 \) |
\( - 7^{7} \) |
$1.05816$ |
$(-a-2)$ |
$1$ |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.1 |
$1$ |
\( 7 \) |
$3.012727319$ |
$10.92242160$ |
1.291435685 |
\( -\frac{1840001024}{823543} a + \frac{7615438848}{823543} \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( a - 1\) , \( -a - 4\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(a-1\right){x}-a-4$ |
| 7.2-a1 |
7.2-a |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
7.2 |
\( 7 \) |
\( - 7^{7} \) |
$1.05816$ |
$(-a+3)$ |
$1$ |
$\Z/7\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.1 |
$1$ |
\( 7 \) |
$3.012727319$ |
$10.92242160$ |
1.291435685 |
\( \frac{1840001024}{823543} a + \frac{5775437824}{823543} \) |
\( \bigl[0\) , \( a\) , \( a\) , \( -a\) , \( -4\bigr] \) |
${y}^2+a{y}={x}^{3}+a{x}^{2}-a{x}-4$ |
| 7.2-a2 |
7.2-a |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
7.2 |
\( 7 \) |
\( -7 \) |
$1.05816$ |
$(-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$7$ |
7B.1.3 |
$1$ |
\( 1 \) |
$21.08909123$ |
$0.222906563$ |
1.291435685 |
\( \frac{195338235078135808}{7} a - \frac{808691822475743232}{7} \) |
\( \bigl[0\) , \( a\) , \( a\) , \( -351 a - 2730\) , \( -11879 a - 60953\bigr] \) |
${y}^2+a{y}={x}^{3}+a{x}^{2}+\left(-351a-2730\right){x}-11879a-60953$ |
| 17.1-a1 |
17.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( - 17^{2} \) |
$1.32096$ |
$(a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$24.46621578$ |
1.680346598 |
\( -\frac{9844754985}{289} a + \frac{40758297444}{289} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 4 a + 12\) , \( -3 a - 9\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(4a+12\right){x}-3a-9$ |
| 17.1-a2 |
17.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17 \) |
$1.32096$ |
$(a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$48.93243156$ |
1.680346598 |
\( -\frac{34263}{17} a + \frac{172989}{17} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -a - 3\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-a-3\right){x}$ |
| 17.2-a1 |
17.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
17.2 |
\( 17 \) |
\( 17 \) |
$1.32096$ |
$(a-6)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$48.93243156$ |
1.680346598 |
\( \frac{34263}{17} a + \frac{138726}{17} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( a - 4\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(a-4\right){x}$ |
| 17.2-a2 |
17.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
17.2 |
\( 17 \) |
\( - 17^{2} \) |
$1.32096$ |
$(a-6)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$24.46621578$ |
1.680346598 |
\( \frac{9844754985}{289} a + \frac{30913542459}{289} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -4 a + 16\) , \( 3 a - 12\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-4a+16\right){x}+3a-12$ |
| 25.1-a1 |
25.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$1.45466$ |
$(5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.147790867$ |
$25.86029770$ |
2.099922056 |
\( -\frac{373457}{25} a - \frac{1134461}{25} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( -2 a - 5\) , \( a + 2\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-2a-5\right){x}+a+2$ |
| 25.1-b1 |
25.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$1.45466$ |
$(5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.147790867$ |
$25.86029770$ |
2.099922056 |
\( \frac{373457}{25} a - \frac{1507918}{25} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( a - 6\) , \( -2 a + 4\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-6\right){x}-2a+4$ |
| 28.1-a1 |
28.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{2} \cdot 7^{6} \) |
$1.49646$ |
$(-a-2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$4.778422636$ |
1.312733656 |
\( -\frac{6206419620}{117649} a - \frac{38967954713}{235298} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( -10 a - 29\) , \( -43 a - 135\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-10a-29\right){x}-43a-135$ |
| 28.1-b1 |
28.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{4} \) |
$1.49646$ |
$(-a-2), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.095218482$ |
$22.21799898$ |
2.324760677 |
\( \frac{12774075}{9604} a - \frac{50451713}{9604} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( -2 a\) , \( -a + 1\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}-2a{x}-a+1$ |
| 28.1-c1 |
28.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( - 2^{8} \cdot 7^{3} \) |
$1.49646$ |
$(-a-2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$19.42800747$ |
1.334321031 |
\( -\frac{3528949}{686} a - \frac{20337669}{5488} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( 3 a + 6\) , \( 3 a + 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(3a+6\right){x}+3a+7$ |
| 28.1-c2 |
28.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( - 2^{4} \cdot 7^{6} \) |
$1.49646$ |
$(-a-2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$9.714003739$ |
1.334321031 |
\( \frac{21413443562485}{235298} a + \frac{134478824713501}{470596} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( -17 a - 54\) , \( 31 a + 91\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-17a-54\right){x}+31a+91$ |
| 28.1-d1 |
28.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{6} \cdot 7^{2} \) |
$1.49646$ |
$(-a-2), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.397928512$ |
$9.608278544$ |
2.100741912 |
\( -\frac{109125}{392} a + \frac{334375}{392} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 4 a + 14\) , \( -5 a - 16\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4a+14\right){x}-5a-16$ |
| 28.2-a1 |
28.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{2} \cdot 7^{6} \) |
$1.49646$ |
$(-a+3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$4.778422636$ |
1.312733656 |
\( \frac{6206419620}{117649} a - \frac{51380793953}{235298} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( 9 a - 39\) , \( 43 a - 178\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(9a-39\right){x}+43a-178$ |
| 28.2-b1 |
28.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{4} \) |
$1.49646$ |
$(-a+3), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.095218482$ |
$22.21799898$ |
2.324760677 |
\( -\frac{12774075}{9604} a - \frac{18838819}{4802} \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( -1\) , \( 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}-{x}+1$ |
| 28.2-c1 |
28.2-c |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( - 2^{4} \cdot 7^{6} \) |
$1.49646$ |
$(-a+3), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$9.714003739$ |
1.334321031 |
\( -\frac{21413443562485}{235298} a + \frac{177305711838471}{470596} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( 23 a - 84\) , \( -109 a + 463\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(23a-84\right){x}-109a+463$ |
| 28.2-c2 |
28.2-c |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( - 2^{8} \cdot 7^{3} \) |
$1.49646$ |
$(-a+3), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$19.42800747$ |
1.334321031 |
\( \frac{3528949}{686} a - \frac{48569261}{5488} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( 3 a - 4\) , \( -a + 11\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(3a-4\right){x}-a+11$ |
| 28.2-d1 |
28.2-d |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{6} \cdot 7^{2} \) |
$1.49646$ |
$(-a+3), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.397928512$ |
$9.608278544$ |
2.100741912 |
\( \frac{109125}{392} a + \frac{112625}{196} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 5 a + 3\) , \( 9 a - 15\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(5a+3\right){x}+9a-15$ |
| 36.1-a1 |
36.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{4} \) |
$1.59350$ |
$(2), (3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$7.106415608$ |
1.952282511 |
\( \frac{20833}{18} a - \frac{43171}{9} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 3 a + 6\) , \( 4 a + 11\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(3a+6\right){x}+4a+11$ |
| 36.1-b1 |
36.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{18} \) |
$1.59350$ |
$(2), (3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$13.36566751$ |
1.835915627 |
\( -\frac{6362477477}{39366} \) |
\( \bigl[a\) , \( 1\) , \( 1\) , \( -270 a - 843\) , \( 4165 a + 13079\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-270a-843\right){x}+4165a+13079$ |
| 36.1-c1 |
36.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{22} \cdot 3^{2} \) |
$1.59350$ |
$(2), (3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$13.90397201$ |
1.909857437 |
\( -\frac{1953125}{6144} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -15 a - 47\) , \( 157 a + 493\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-15a-47\right){x}+157a+493$ |
| 36.1-d1 |
36.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{4} \) |
$1.59350$ |
$(2), (3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$7.106415608$ |
1.952282511 |
\( -\frac{20833}{18} a - \frac{65509}{18} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( 3 a + 11\) , \( 2 a + 6\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a+11\right){x}+2a+6$ |
| 36.1-e1 |
36.1-e |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{2} \cdot 3^{4} \) |
$1.59350$ |
$(2), (3)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$48.14860131$ |
0.529097522 |
\( -\frac{3307949}{18} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -21 a - 62\) , \( 68 a + 215\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-21a-62\right){x}+68a+215$ |
| 36.1-e2 |
36.1-e |
$2$ |
$5$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{20} \) |
$1.59350$ |
$(2), (3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$1.925944052$ |
0.529097522 |
\( \frac{1160935651}{1889568} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 154 a + 488\) , \( -2384 a - 7484\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(154a+488\right){x}-2384a-7484$ |
| 37.1-a1 |
37.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
37.1 |
\( 37 \) |
\( 37 \) |
$1.60445$ |
$(-2a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$14.86342269$ |
2.041648123 |
\( \frac{385513}{37} a - \frac{1668167}{37} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( a + 4\) , \( a + 3\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(a+4\right){x}+a+3$ |
| 37.2-a1 |
37.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
37.2 |
\( 37 \) |
\( 37 \) |
$1.60445$ |
$(2a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$14.86342269$ |
2.041648123 |
\( -\frac{385513}{37} a - \frac{1282654}{37} \) |
\( \bigl[a + 1\) , \( a\) , \( 1\) , \( 6 a + 11\) , \( 3 a + 33\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(6a+11\right){x}+3a+33$ |
| 49.1-a1 |
49.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{9} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$2.700428314$ |
1.669195602 |
\( -\frac{3825287113585893}{117649} a + \frac{15836899268209340}{117649} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 392 a - 1636\) , \( 9025 a - 37367\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(392a-1636\right){x}+9025a-37367$ |
| 49.1-a2 |
49.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{9} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$2.700428314$ |
1.669195602 |
\( \frac{8183558401}{117649} a - \frac{31455098491}{117649} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 22 a - 101\) , \( 180 a - 751\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(22a-101\right){x}+180a-751$ |
| 49.1-a3 |
49.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{3} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$9$ |
\( 2 \) |
$1$ |
$24.30385482$ |
1.669195602 |
\( \frac{1063343}{49} a + \frac{3353382}{49} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -3 a + 4\) , \( -2 a + 4\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-3a+4\right){x}-2a+4$ |
| 49.1-a4 |
49.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{3} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$9$ |
\( 2 \) |
$1$ |
$24.30385482$ |
1.669195602 |
\( \frac{167034552579}{49} a + \frac{524498013905}{49} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 2 a - 21\) , \( 20 a - 85\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2a-21\right){x}+20a-85$ |
| 49.1-b1 |
49.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{9} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$2.700428314$ |
1.669195602 |
\( -\frac{8183558401}{117649} a - \frac{23271540090}{117649} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -24 a - 79\) , \( -181 a - 571\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-24a-79\right){x}-181a-571$ |
| 49.1-b2 |
49.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( - 7^{3} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$9$ |
\( 2 \) |
$1$ |
$24.30385482$ |
1.669195602 |
\( -\frac{1063343}{49} a + \frac{4416725}{49} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( a + 1\) , \( a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}+a+2$ |
| 49.1-b3 |
49.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{3} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$9$ |
\( 2 \) |
$1$ |
$24.30385482$ |
1.669195602 |
\( -\frac{167034552579}{49} a + \frac{691532566484}{49} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -4 a - 19\) , \( -21 a - 65\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-4a-19\right){x}-21a-65$ |
| 49.1-b4 |
49.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{9} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$2.700428314$ |
1.669195602 |
\( \frac{3825287113585893}{117649} a + \frac{12011612154623447}{117649} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -394 a - 1244\) , \( -9026 a - 28342\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-394a-1244\right){x}-9026a-28342$ |
| 49.1-c1 |
49.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{8} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.442032891$ |
$5.921998271$ |
2.876569790 |
\( \frac{48788531}{117649} a - \frac{201772345}{117649} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( a + 5\) , \( 4 a + 10\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+5\right){x}+4a+10$ |
| 49.1-d1 |
49.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.1 |
\( 7^{2} \) |
\( 7^{8} \) |
$1.72118$ |
$(-a-2), (-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.442032891$ |
$5.921998271$ |
2.876569790 |
\( -\frac{48788531}{117649} a - \frac{152983814}{117649} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 5\) , \( -4 a + 9\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+5{x}-4a+9$ |
| 49.2-a1 |
49.2-a |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.2 |
\( 7^{2} \) |
\( - 7^{7} \) |
$1.72118$ |
$(-a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B.6.3 |
$1$ |
\( 2 \) |
$4.195650442$ |
$1.068764585$ |
2.463788418 |
\( -\frac{195338235078135808}{7} a - \frac{613353587397607424}{7} \) |
\( \bigl[0\) , \( -a - 1\) , \( a + 1\) , \( -7698 a - 29615\) , \( 842473 a + 2486905\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-7698a-29615\right){x}+842473a+2486905$ |
| 49.2-a2 |
49.2-a |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.2 |
\( 7^{2} \) |
\( - 7^{13} \) |
$1.72118$ |
$(-a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B.6.1 |
$1$ |
\( 2 \) |
$0.599378634$ |
$7.481352100$ |
2.463788418 |
\( -\frac{1840001024}{823543} a + \frac{7615438848}{823543} \) |
\( \bigl[0\) , \( -a - 1\) , \( a + 1\) , \( 2 a - 5\) , \( 16 a + 57\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(2a-5\right){x}+16a+57$ |
| 49.2-b1 |
49.2-b |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.2 |
\( 7^{2} \) |
\( - 7^{9} \) |
$1.72118$ |
$(-a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$7$ |
7B |
$1$ |
\( 2 \) |
$1.501562525$ |
$2.678376117$ |
2.209718956 |
\( -68910804992 a + 285294501888 \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( -232 a - 746\) , \( 3993 a + 12514\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-232a-746\right){x}+3993a+12514$ |
| 49.2-b2 |
49.2-b |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.2 |
\( 7^{2} \) |
\( - 7^{3} \) |
$1.72118$ |
$(-a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$7$ |
7B |
$1$ |
\( 2 \) |
$0.214508932$ |
$18.74863282$ |
2.209718956 |
\( -4096 a + 16384 \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( -2 a - 6\) , \( -6 a - 20\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-2a-6\right){x}-6a-20$ |
| 49.2-c1 |
49.2-c |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.2 |
\( 7^{2} \) |
\( - 7^{3} \) |
$1.72118$ |
$(-a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$7$ |
7B.2.3 |
$1$ |
\( 2 \) |
$1.635744836$ |
$2.274807683$ |
2.044477338 |
\( -68910804992 a + 285294501888 \) |
\( \bigl[0\) , \( a\) , \( a + 1\) , \( -4 a - 23\) , \( -25 a - 86\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-4a-23\right){x}-25a-86$ |
| 49.2-c2 |
49.2-c |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.2 |
\( 7^{2} \) |
\( - 7^{9} \) |
$1.72118$ |
$(-a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$7$ |
7B.2.1 |
$1$ |
\( 2 \) |
$0.233677833$ |
$15.92365378$ |
2.044477338 |
\( -4096 a + 16384 \) |
\( \bigl[0\) , \( -a\) , \( a + 1\) , \( 25 a - 98\) , \( -99 a + 401\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(25a-98\right){x}-99a+401$ |
| 49.3-a1 |
49.3-a |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.3 |
\( 7^{2} \) |
\( - 7^{13} \) |
$1.72118$ |
$(-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B.6.1 |
$1$ |
\( 2 \) |
$0.599378634$ |
$7.481352100$ |
2.463788418 |
\( \frac{1840001024}{823543} a + \frac{5775437824}{823543} \) |
\( \bigl[0\) , \( a + 1\) , \( a\) , \( -4\) , \( -18 a + 70\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a+1\right){x}^{2}-4{x}-18a+70$ |
| 49.3-a2 |
49.3-a |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.3 |
\( 7^{2} \) |
\( - 7^{7} \) |
$1.72118$ |
$(-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B.6.3 |
$1$ |
\( 2 \) |
$4.195650442$ |
$1.068764585$ |
2.463788418 |
\( \frac{195338235078135808}{7} a - \frac{808691822475743232}{7} \) |
\( \bigl[0\) , \( a + 1\) , \( a\) , \( 7700 a - 37314\) , \( -834775 a + 3292065\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(7700a-37314\right){x}-834775a+3292065$ |
| 49.3-b1 |
49.3-b |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
49.3 |
\( 7^{2} \) |
\( - 7^{3} \) |
$1.72118$ |
$(-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$7$ |
7B |
$1$ |
\( 2 \) |
$0.214508932$ |
$18.74863282$ |
2.209718956 |
\( 4096 a + 12288 \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( 2 a - 8\) , \( 5 a - 25\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}+\left(2a-8\right){x}+5a-25$ |
*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.
