| 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 |
| 1372.2-a1 |
1372.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( 2^{2} \cdot 7^{14} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.990016086$ |
$3.774044288$ |
4.105833142 |
\( \frac{12088568367087}{686} a - \frac{175165680328615}{2401} \) |
\( \bigl[a + 1\) , \( -a\) , \( a\) , \( 97 a + 212\) , \( 12605 a + 39803\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(97a+212\right){x}+12605a+39803$ |
| 1372.2-b1 |
1372.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( 2^{2} \cdot 7^{14} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{3} \) |
$1$ |
$0.284974999$ |
0.313154613 |
\( -\frac{12088568367087}{686} a - \frac{265711382087621}{4802} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -1171 a - 3666\) , \( -45539 a - 143111\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-1171a-3666\right){x}-45539a-143111$ |
| 1372.2-c1 |
1372.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( 2^{2} \cdot 7^{12} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$3.457276630$ |
$1.163235840$ |
4.419304816 |
\( -\frac{6206419620}{117649} a - \frac{38967954713}{235298} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -10 a - 23\) , \( -81 a - 283\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-10a-23\right){x}-81a-283$ |
| 1372.2-d1 |
1372.2-d |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( - 2^{10} \cdot 7^{7} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 1 \) |
$3.135726275$ |
$6.306809770$ |
5.433002912 |
\( -\frac{520678477}{537824} a + \frac{2158802007}{537824} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 3 a + 8\) , \( -25 a - 78\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(3a+8\right){x}-25a-78$ |
| 1372.2-e1 |
1372.2-e |
$6$ |
$18$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( 2^{36} \cdot 7^{8} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$4.426489598$ |
$0.661753152$ |
7.242525569 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 852 a - 3756\) , \( -27108 a + 110707\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(852a-3756\right){x}-27108a+110707$ |
| 1372.2-e2 |
1372.2-e |
$6$ |
$18$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( 2^{4} \cdot 7^{8} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$4.426489598$ |
$5.955778371$ |
7.242525569 |
\( -\frac{15625}{28} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 2 a - 16\) , \( 10 a - 47\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(2a-16\right){x}+10a-47$ |
| 1372.2-e3 |
1372.2-e |
$6$ |
$18$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( 2^{12} \cdot 7^{12} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1.475496532$ |
$1.985259457$ |
7.242525569 |
\( \frac{9938375}{21952} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -23 a + 94\) , \( -207 a + 849\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-23a+94\right){x}-207a+849$ |
| 1372.2-e4 |
1372.2-e |
$6$ |
$18$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( 2^{6} \cdot 7^{18} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$0.737748266$ |
$1.985259457$ |
7.242525569 |
\( \frac{4956477625}{941192} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 177 a - 786\) , \( -2055 a + 8353\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(177a-786\right){x}-2055a+8353$ |
| 1372.2-e5 |
1372.2-e |
$6$ |
$18$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( 2^{2} \cdot 7^{10} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$2.213244799$ |
$5.955778371$ |
7.242525569 |
\( \frac{128787625}{98} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 52 a - 236\) , \( 444 a - 1839\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(52a-236\right){x}+444a-1839$ |
| 1372.2-e6 |
1372.2-e |
$6$ |
$18$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( 2^{18} \cdot 7^{10} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$2.213244799$ |
$0.661753152$ |
7.242525569 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 13652 a - 60076\) , \( -1751012 a + 7164019\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(13652a-60076\right){x}-1751012a+7164019$ |
| 1372.2-f1 |
1372.2-f |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( 2^{6} \cdot 7^{8} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$3.653198018$ |
2.007221360 |
\( -\frac{109125}{392} a + \frac{334375}{392} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -2 a - 2\) , \( -12 a - 40\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-2a-2\right){x}-12a-40$ |
| 1372.2-g1 |
1372.2-g |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( - 2^{36} \cdot 7^{11} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{4} \) |
$1$ |
$1.177458047$ |
2.587780823 |
\( \frac{23646177031723}{89915392} a + \frac{74258455843343}{89915392} \) |
\( \bigl[a + 1\) , \( -a\) , \( a\) , \( -803 a + 3094\) , \( -95375 a + 395943\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-803a+3094\right){x}-95375a+395943$ |
| 1372.2-g2 |
1372.2-g |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( 2^{18} \cdot 7^{16} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{4} \) |
$1$ |
$1.177458047$ |
2.587780823 |
\( \frac{30307733850309423723}{60236288} a + \frac{47583975156364490471}{30118144} \) |
\( \bigl[a + 1\) , \( -a\) , \( a\) , \( 11997 a - 53226\) , \( -1319567 a + 5541543\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(11997a-53226\right){x}-1319567a+5541543$ |
| 1372.2-h1 |
1372.2-h |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( 2^{8} \cdot 7^{17} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
|
\( 2^{3} \) |
$1$ |
$0.492883394$ |
5.019084647 |
\( -\frac{15665187166391045301}{4519603984} a + \frac{16213685929436816735}{1129900996} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -1546 a - 5370\) , \( -16333 a - 55964\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-1546a-5370\right){x}-16333a-55964$ |
| 1372.2-h2 |
1372.2-h |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( - 2^{16} \cdot 7^{13} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
|
\( 2^{3} \) |
$1$ |
$0.985766789$ |
5.019084647 |
\( \frac{182436157159}{4302592} a - \frac{107410375309}{4302592} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -986 a - 3130\) , \( 30035 a + 94228\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-986a-3130\right){x}+30035a+94228$ |
| 1372.2-i1 |
1372.2-i |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( - 2^{10} \cdot 7^{13} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 3 \) |
$1$ |
$1.712735810$ |
0.705787070 |
\( -\frac{520678477}{537824} a + \frac{2158802007}{537824} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( 9 a - 21\) , \( -31 a - 162\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(9a-21\right){x}-31a-162$ |
| 1372.2-j1 |
1372.2-j |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( - 2^{8} \cdot 7^{9} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.976445967$ |
$10.94615819$ |
5.872621262 |
\( -\frac{3528949}{686} a - \frac{20337669}{5488} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( 5 a - 35\) , \( -16 a + 78\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(5a-35\right){x}-16a+78$ |
| 1372.2-j2 |
1372.2-j |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( - 2^{4} \cdot 7^{12} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.952891934$ |
$5.473079098$ |
5.872621262 |
\( \frac{21413443562485}{235298} a + \frac{134478824713501}{470596} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( -35 a - 55\) , \( 60 a + 802\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-35a-55\right){x}+60a+802$ |
| 1372.2-k1 |
1372.2-k |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( - 2^{36} \cdot 7^{11} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
|
\( 2^{3} \) |
$1$ |
$0.341813771$ |
4.421778299 |
\( -\frac{23646177031723}{89915392} a + \frac{48952316437533}{44957696} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( 910 a - 3576\) , \( 26751 a - 115139\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(910a-3576\right){x}+26751a-115139$ |
| 1372.2-k2 |
1372.2-k |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( 2^{18} \cdot 7^{16} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
|
\( 2^{3} \) |
$1$ |
$0.170906885$ |
4.421778299 |
\( -\frac{30307733850309423723}{60236288} a + \frac{125475684163038404665}{60236288} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( 13710 a - 59896\) , \( 1738879 a - 7262147\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(13710a-59896\right){x}+1738879a-7262147$ |
| 1372.2-l1 |
1372.2-l |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( 2^{4} \cdot 7^{10} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{3} \) |
$1$ |
$2.859615246$ |
3.142386903 |
\( \frac{12774075}{9604} a - \frac{50451713}{9604} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 10 a - 3\) , \( 19 a - 6\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(10a-3\right){x}+19a-6$ |
| 1372.2-m1 |
1372.2-m |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( - 2^{16} \cdot 7^{13} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 5 \) |
$1$ |
$1.981199053$ |
2.721386192 |
\( -\frac{182436157159}{4302592} a + \frac{37512890925}{2151296} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -383 a - 1285\) , \( 7356 a + 22790\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-383a-1285\right){x}+7356a+22790$ |
| 1372.2-m2 |
1372.2-m |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
1372.2 |
\( 2^{2} \cdot 7^{3} \) |
\( 2^{8} \cdot 7^{17} \) |
$3.95927$ |
$(-a-2), (-a+3), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 5 \) |
$1$ |
$1.981199053$ |
2.721386192 |
\( \frac{15665187166391045301}{4519603984} a + \frac{49189556551356221639}{4519603984} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -6303 a - 19925\) , \( 508044 a + 1595350\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-6303a-19925\right){x}+508044a+1595350$ |
*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.
