| 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 |
| 625.1-a1 |
625.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{10} \) |
$2.04747$ |
$(-a), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.691211587$ |
$9.930593711$ |
2.995756926 |
\( \frac{56224}{5} a + 21877 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -15 a - 28\) , \( -62 a - 112\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-15a-28\right){x}-62a-112$ |
| 625.1-a2 |
625.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( - 5^{11} \) |
$2.04747$ |
$(-a), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.382423174$ |
$2.482648427$ |
2.995756926 |
\( \frac{15046148446}{25} a + \frac{5390405181}{5} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -280 a + 780\) , \( -350 a + 975\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(-280a+780\right){x}-350a+975$ |
| 625.1-b1 |
625.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{16} \) |
$2.04747$ |
$(-a), (-a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.1.3 |
$1$ |
\( 2^{2} \) |
$4.684105865$ |
$0.470844196$ |
3.850208633 |
\( \frac{162783232}{5} a - \frac{454397952}{5} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 342 a - 965\) , \( 5472 a - 15284\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(342a-965\right){x}+5472a-15284$ |
| 625.1-b2 |
625.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{20} \) |
$2.04747$ |
$(-a), (-a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.1.4 |
$1$ |
\( 2^{2} \) |
$0.936821173$ |
$2.354220982$ |
3.850208633 |
\( -\frac{729088}{3125} a + \frac{2019328}{3125} \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( -117 a + 327\) , \( 928 a - 2587\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-117a+327\right){x}+928a-2587$ |
| 625.1-c1 |
625.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{4} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5, 7$ |
5Ns.2.1, 7Ns.3.1 |
$1$ |
\( 1 \) |
$1$ |
$16.42661250$ |
3.584580724 |
\( 0 \) |
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( a + 2\) , \( -a - 2\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+2\right){x}-a-2$ |
| 625.1-c2 |
625.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{4} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5, 7$ |
5Ns.2.1, 7Ns.3.1 |
$1$ |
\( 1 \) |
$1$ |
$16.42661250$ |
3.584580724 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( a + 2\) , \( a - 1\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+2\right){x}+a-1$ |
| 625.1-d1 |
625.1-d |
$2$ |
$5$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{16} \) |
$2.04747$ |
$(-a), (-a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2^{3} \) |
$0.106273201$ |
$7.704201446$ |
2.858654902 |
\( -\frac{162783232}{5} a - 58322944 \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( 50 a - 208\) , \( -475 a + 1568\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(50a-208\right){x}-475a+1568$ |
| 625.1-d2 |
625.1-d |
$2$ |
$5$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{20} \) |
$2.04747$ |
$(-a), (-a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.1 |
$1$ |
\( 2^{3} \) |
$0.531366008$ |
$1.540840289$ |
2.858654902 |
\( \frac{729088}{3125} a + \frac{258048}{625} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( 8 a + 2\) , \( 97 a - 251\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(8a+2\right){x}+97a-251$ |
| 625.1-e1 |
625.1-e |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{20} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$9$ |
\( 2 \) |
$1$ |
$0.933393352$ |
3.666296305 |
\( -\frac{4659111497728}{244140625} a - \frac{8015912824832}{244140625} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -45 a - 8\) , \( -144 a - 158\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}+\left(-45a-8\right){x}-144a-158$ |
| 625.1-e2 |
625.1-e |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{12} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$8.400540170$ |
3.666296305 |
\( \frac{6279168}{625} a - \frac{16887808}{625} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( 5 a - 8\) , \( -9 a + 17\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}+\left(5a-8\right){x}-9a+17$ |
| 625.1-f1 |
625.1-f |
$2$ |
$5$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{16} \) |
$2.04747$ |
$(-a), (-a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.1.3 |
$1$ |
\( 2^{2} \) |
$4.684105865$ |
$0.470844196$ |
3.850208633 |
\( -\frac{162783232}{5} a - 58322944 \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( -342 a - 623\) , \( -5472 a - 9812\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-342a-623\right){x}-5472a-9812$ |
| 625.1-f2 |
625.1-f |
$2$ |
$5$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{20} \) |
$2.04747$ |
$(-a), (-a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.1.4 |
$1$ |
\( 2^{2} \) |
$0.936821173$ |
$2.354220982$ |
3.850208633 |
\( \frac{729088}{3125} a + \frac{258048}{625} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 117 a + 210\) , \( -928 a - 1659\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(117a+210\right){x}-928a-1659$ |
| 625.1-g1 |
625.1-g |
$2$ |
$5$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{16} \) |
$2.04747$ |
$(-a), (-a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2^{3} \) |
$0.106273201$ |
$7.704201446$ |
2.858654902 |
\( \frac{162783232}{5} a - \frac{454397952}{5} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -50 a - 158\) , \( 475 a + 1093\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(-50a-158\right){x}+475a+1093$ |
| 625.1-g2 |
625.1-g |
$2$ |
$5$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{20} \) |
$2.04747$ |
$(-a), (-a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.1 |
$1$ |
\( 2^{3} \) |
$0.531366008$ |
$1.540840289$ |
2.858654902 |
\( -\frac{729088}{3125} a + \frac{2019328}{3125} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( -8 a + 10\) , \( -97 a - 154\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(-8a+10\right){x}-97a-154$ |
| 625.1-h1 |
625.1-h |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{10} \) |
$2.04747$ |
$(-a), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.073335063$ |
$16.90647484$ |
1.082218806 |
\( \frac{56224}{5} a + 21877 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 3 a - 7\) , \( -2 a + 6\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(3a-7\right){x}-2a+6$ |
| 625.1-h2 |
625.1-h |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( - 5^{11} \) |
$2.04747$ |
$(-a), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.146670126$ |
$8.453237420$ |
1.082218806 |
\( \frac{15046148446}{25} a + \frac{5390405181}{5} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -22 a + 18\) , \( 23 a - 19\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-22a+18\right){x}+23a-19$ |
| 625.1-i1 |
625.1-i |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{20} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$2.967381797$ |
1.295071590 |
\( -\frac{4659111497728}{244140625} a - \frac{8015912824832}{244140625} \) |
\( \bigl[0\) , \( -a\) , \( a\) , \( -363 a + 1010\) , \( 2783 a - 7760\bigr] \) |
${y}^2+a{y}={x}^{3}-a{x}^{2}+\left(-363a+1010\right){x}+2783a-7760$ |
| 625.1-i2 |
625.1-i |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{12} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$2.967381797$ |
1.295071590 |
\( \frac{6279168}{625} a - \frac{16887808}{625} \) |
\( \bigl[0\) , \( a - 1\) , \( a\) , \( 28 a + 52\) , \( 119 a + 212\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(28a+52\right){x}+119a+212$ |
| 625.1-j1 |
625.1-j |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{16} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.3.1 |
$1$ |
\( 2 \) |
$1$ |
$3.285322500$ |
1.433832289 |
\( 0 \) |
\( \bigl[0\) , \( -a - 1\) , \( a + 1\) , \( a + 2\) , \( -213 a + 588\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+2\right){x}-213a+588$ |
| 625.1-j2 |
625.1-j |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{16} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.3.1 |
$1$ |
\( 2 \) |
$1$ |
$3.285322500$ |
1.433832289 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( a + 1\) , \( a + 2\) , \( 24 a + 34\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+2\right){x}+24a+34$ |
| 625.1-k1 |
625.1-k |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{28} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{4} \) |
$1$ |
$0.389954337$ |
1.361520203 |
\( -\frac{359104782699}{244140625} a - \frac{52148361654}{48828125} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -525 a + 1330\) , \( -3211 a + 8322\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-525a+1330\right){x}-3211a+8322$ |
| 625.1-k2 |
625.1-k |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{28} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{4} \) |
$1$ |
$0.389954337$ |
1.361520203 |
\( \frac{359104782699}{244140625} a - \frac{619846590969}{244140625} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 523 a + 807\) , \( 3210 a + 5112\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(523a+807\right){x}+3210a+5112$ |
| 625.1-k3 |
625.1-k |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{20} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$4$ |
\( 2^{4} \) |
$1$ |
$1.559817348$ |
1.361520203 |
\( -\frac{118077162021}{15625} a + \frac{329617083726}{15625} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -102 a - 318\) , \( 460 a + 237\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-102a-318\right){x}+460a+237$ |
| 625.1-k4 |
625.1-k |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{16} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$6.239269393$ |
1.361520203 |
\( -\frac{22825881}{125} a + \frac{12909294}{25} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 100 a - 295\) , \( -836 a + 2322\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(100a-295\right){x}-836a+2322$ |
| 625.1-k5 |
625.1-k |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{16} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{4} \) |
$1$ |
$0.389954337$ |
1.361520203 |
\( -\frac{32714515537919631}{125} a + \frac{91315629670496661}{125} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -727 a - 3443\) , \( -25790 a - 84138\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-727a-3443\right){x}-25790a-84138$ |
| 625.1-k6 |
625.1-k |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{16} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$6.239269393$ |
1.361520203 |
\( \frac{22825881}{125} a + \frac{41720589}{125} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -102 a - 193\) , \( 835 a + 1487\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-102a-193\right){x}+835a+1487$ |
| 625.1-k7 |
625.1-k |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{20} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$4$ |
\( 2^{4} \) |
$1$ |
$1.559817348$ |
1.361520203 |
\( \frac{118077162021}{15625} a + \frac{42307984341}{3125} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 100 a - 420\) , \( -461 a + 697\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(100a-420\right){x}-461a+697$ |
| 625.1-k8 |
625.1-k |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{16} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{4} \) |
$1$ |
$0.389954337$ |
1.361520203 |
\( \frac{32714515537919631}{125} a + \frac{11720222826515406}{25} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 725 a - 4170\) , \( 25789 a - 109928\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(725a-4170\right){x}+25789a-109928$ |
| 625.1-l1 |
625.1-l |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{28} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.992306378$ |
0.866156017 |
\( -\frac{359104782699}{244140625} a - \frac{52148361654}{48828125} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -665 a - 1093\) , \( 15322 a + 27318\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-665a-1093\right){x}+15322a+27318$ |
| 625.1-l2 |
625.1-l |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{28} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.992306378$ |
0.866156017 |
\( \frac{359104782699}{244140625} a - \frac{619846590969}{244140625} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 663 a - 1757\) , \( -15323 a + 42641\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(663a-1757\right){x}-15323a+42641$ |
| 625.1-l3 |
625.1-l |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{20} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$3.969225515$ |
0.866156017 |
\( -\frac{118077162021}{15625} a + \frac{329617083726}{15625} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 663 a - 1882\) , \( -14073 a + 39266\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(663a-1882\right){x}-14073a+39266$ |
| 625.1-l4 |
625.1-l |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{16} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.969225515$ |
0.866156017 |
\( -\frac{22825881}{125} a + \frac{12909294}{25} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -40 a - 93\) , \( 197 a + 318\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-40a-93\right){x}+197a+318$ |
| 625.1-l5 |
625.1-l |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{16} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.969225515$ |
0.866156017 |
\( -\frac{32714515537919631}{125} a + \frac{91315629670496661}{125} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 10663 a - 30007\) , \( -910323 a + 2542391\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(10663a-30007\right){x}-910323a+2542391$ |
| 625.1-l6 |
625.1-l |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{16} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.969225515$ |
0.866156017 |
\( \frac{22825881}{125} a + \frac{41720589}{125} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 38 a - 132\) , \( -198 a + 516\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(38a-132\right){x}-198a+516$ |
| 625.1-l7 |
625.1-l |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{20} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$3.969225515$ |
0.866156017 |
\( \frac{118077162021}{15625} a + \frac{42307984341}{3125} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -665 a - 1218\) , \( 14072 a + 25193\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-665a-1218\right){x}+14072a+25193$ |
| 625.1-l8 |
625.1-l |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{16} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.969225515$ |
0.866156017 |
\( \frac{32714515537919631}{125} a + \frac{11720222826515406}{25} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -10665 a - 19343\) , \( 910322 a + 1632068\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-10665a-19343\right){x}+910322a+1632068$ |
| 625.1-m1 |
625.1-m |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{12} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$8.400540170$ |
3.666296305 |
\( -\frac{6279168}{625} a - \frac{2121728}{125} \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( -5 a - 3\) , \( 8 a + 8\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-5a-3\right){x}+8a+8$ |
| 625.1-m2 |
625.1-m |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{20} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$9$ |
\( 2 \) |
$1$ |
$0.933393352$ |
3.666296305 |
\( \frac{4659111497728}{244140625} a - \frac{2535004864512}{48828125} \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( 45 a - 53\) , \( 143 a - 302\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(45a-53\right){x}+143a-302$ |
| 625.1-n1 |
625.1-n |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( - 5^{11} \) |
$2.04747$ |
$(-a), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.382423174$ |
$2.482648427$ |
2.995756926 |
\( -\frac{15046148446}{25} a + \frac{41998174351}{25} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 280 a + 500\) , \( 350 a + 625\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(280a+500\right){x}+350a+625$ |
| 625.1-n2 |
625.1-n |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{10} \) |
$2.04747$ |
$(-a), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.691211587$ |
$9.930593711$ |
2.995756926 |
\( -\frac{56224}{5} a + \frac{165609}{5} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 13 a - 43\) , \( 61 a - 174\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(13a-43\right){x}+61a-174$ |
| 625.1-o1 |
625.1-o |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{12} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$2.967381797$ |
1.295071590 |
\( -\frac{6279168}{625} a - \frac{2121728}{125} \) |
\( \bigl[0\) , \( -a\) , \( a + 1\) , \( -28 a + 80\) , \( -120 a + 331\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-28a+80\right){x}-120a+331$ |
| 625.1-o2 |
625.1-o |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{20} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$2.967381797$ |
1.295071590 |
\( \frac{4659111497728}{244140625} a - \frac{2535004864512}{48828125} \) |
\( \bigl[0\) , \( a - 1\) , \( a + 1\) , \( 363 a + 647\) , \( -2784 a - 4977\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(363a+647\right){x}-2784a-4977$ |
| 625.1-p1 |
625.1-p |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{16} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5, 7$ |
5Ns.2.1, 7Ns.2.1 |
$1$ |
\( 3 \) |
$1$ |
$3.285322500$ |
2.150748434 |
\( 0 \) |
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( a + 2\) , \( 69 a - 192\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+2\right){x}+69a-192$ |
| 625.1-p2 |
625.1-p |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{16} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$5, 7$ |
5Ns.2.1, 7Ns.2.1 |
$1$ |
\( 3 \) |
$1$ |
$3.285322500$ |
2.150748434 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( a + 2\) , \( -69 a - 121\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+2\right){x}-69a-121$ |
| 625.1-q1 |
625.1-q |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{16} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.3.1 |
$1$ |
\( 2 \) |
$1$ |
$3.285322500$ |
1.433832289 |
\( 0 \) |
\( \bigl[0\) , \( -a - 1\) , \( a\) , \( a + 2\) , \( -25 a + 57\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+2\right){x}-25a+57$ |
| 625.1-q2 |
625.1-q |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{16} \) |
$2.04747$ |
$(-a), (-a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.3.1 |
$1$ |
\( 2 \) |
$1$ |
$3.285322500$ |
1.433832289 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( a\) , \( a + 2\) , \( 212 a + 378\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+2\right){x}+212a+378$ |
| 625.1-r1 |
625.1-r |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( - 5^{11} \) |
$2.04747$ |
$(-a), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.146670126$ |
$8.453237420$ |
1.082218806 |
\( -\frac{15046148446}{25} a + \frac{41998174351}{25} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 25 a - 3\) , \( -5 a + 118\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(25a-3\right){x}-5a+118$ |
| 625.1-r2 |
625.1-r |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
625.1 |
\( 5^{4} \) |
\( 5^{10} \) |
$2.04747$ |
$(-a), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.073335063$ |
$16.90647484$ |
1.082218806 |
\( -\frac{56224}{5} a + \frac{165609}{5} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -3\) , \( -5 a - 7\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}-3{x}-5a-7$ |
*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.
