| 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 |
| 847.5-a1 |
847.5-a |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
847.5 |
\( 7 \cdot 11^{2} \) |
\( - 7 \cdot 11^{8} \) |
$3.50952$ |
$(-a+3), (a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 3 \) |
$0.893638274$ |
$6.018092713$ |
4.432349019 |
\( \frac{190386176}{7} a - \frac{786636800}{7} \) |
\( \bigl[0\) , \( a + 1\) , \( a + 1\) , \( -259 a - 821\) , \( -4819 a - 15116\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-259a-821\right){x}-4819a-15116$ |
| 847.5-b1 |
847.5-b |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
847.5 |
\( 7 \cdot 11^{2} \) |
\( - 7 \cdot 11^{8} \) |
$3.50952$ |
$(-a+3), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$7.014524249$ |
1.927038013 |
\( \frac{570322944}{847} a + \frac{1791700992}{847} \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( -8 a + 5\) , \( 15 a - 9\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-8a+5\right){x}+15a-9$ |
| 847.5-c1 |
847.5-c |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
847.5 |
\( 7 \cdot 11^{2} \) |
\( - 7 \cdot 11^{2} \) |
$3.50952$ |
$(-a+3), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 1 \) |
$1$ |
$39.13337478$ |
5.375382430 |
\( \frac{190386176}{7} a - \frac{786636800}{7} \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( -57 a - 179\) , \( 464 a + 1456\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-57a-179\right){x}+464a+1456$ |
| 847.5-d1 |
847.5-d |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
847.5 |
\( 7 \cdot 11^{2} \) |
\( 7^{2} \cdot 11^{11} \) |
$3.50952$ |
$(-a+3), (a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{3} \) |
$0.284290461$ |
$3.192034936$ |
1.994398654 |
\( \frac{5601444922}{7891499} a - \frac{15922836637}{7891499} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( 12 a - 21\) , \( -34 a + 54\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(12a-21\right){x}-34a+54$ |
| 847.5-e1 |
847.5-e |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
847.5 |
\( 7 \cdot 11^{2} \) |
\( - 7^{7} \cdot 11^{6} \) |
$3.50952$ |
$(-a+3), (a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B.6.1 |
$1$ |
\( 2 \) |
$5.561627057$ |
$2.144474144$ |
6.553068899 |
\( \frac{1840001024}{823543} a + \frac{5775437824}{823543} \) |
\( \bigl[0\) , \( a - 1\) , \( a\) , \( -6 a - 17\) , \( -12 a - 111\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-6a-17\right){x}-12a-111$ |
| 847.5-e2 |
847.5-e |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
847.5 |
\( 7 \cdot 11^{2} \) |
\( - 7 \cdot 11^{6} \) |
$3.50952$ |
$(-a+3), (a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B.6.3 |
$1$ |
\( 2 \) |
$38.93138940$ |
$0.306353449$ |
6.553068899 |
\( \frac{195338235078135808}{7} a - \frac{808691822475743232}{7} \) |
\( \bigl[0\) , \( a - 1\) , \( a\) , \( 3284 a - 32777\) , \( 602541 a - 1491888\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(3284a-32777\right){x}+602541a-1491888$ |
| 847.5-f1 |
847.5-f |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
847.5 |
\( 7 \cdot 11^{2} \) |
\( - 7^{11} \cdot 11^{8} \) |
$3.50952$ |
$(-a+3), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 3 \cdot 11 \) |
$1$ |
$0.383542456$ |
1.738559066 |
\( \frac{103693635190784}{1977326743} a + \frac{247945746214912}{1977326743} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 5939 a - 24600\) , \( 479701 a - 1986040\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(5939a-24600\right){x}+479701a-1986040$ |
| 847.5-g1 |
847.5-g |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
847.5 |
\( 7 \cdot 11^{2} \) |
\( 7^{4} \cdot 11^{3} \) |
$3.50952$ |
$(-a+3), (a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{3} \) |
$0.341143242$ |
$4.091590931$ |
3.067686878 |
\( -\frac{739250}{2401} a - \frac{1941375}{2401} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -4 a + 7\) , \( 3 a - 18\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-4a+7\right){x}+3a-18$ |
| 847.5-h1 |
847.5-h |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
847.5 |
\( 7 \cdot 11^{2} \) |
\( - 7^{11} \cdot 11^{2} \) |
$3.50952$ |
$(-a+3), (a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 1 \) |
$5.790871932$ |
$2.579662359$ |
4.103920017 |
\( \frac{103693635190784}{1977326743} a + \frac{247945746214912}{1977326743} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( 224 a - 928\) , \( 3889 a - 16095\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(224a-928\right){x}+3889a-16095$ |
| 847.5-i1 |
847.5-i |
$2$ |
$3$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
847.5 |
\( 7 \cdot 11^{2} \) |
\( 7^{6} \cdot 11^{9} \) |
$3.50952$ |
$(-a+3), (a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{3} \) |
$1.213039273$ |
$2.305068603$ |
6.145267118 |
\( -\frac{219351059533}{156590819} a - \frac{677739978147}{156590819} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( -118 a + 485\) , \( 6169 a - 25540\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-118a+485\right){x}+6169a-25540$ |
| 847.5-i2 |
847.5-i |
$2$ |
$3$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
847.5 |
\( 7 \cdot 11^{2} \) |
\( 7^{2} \cdot 11^{7} \) |
$3.50952$ |
$(-a+3), (a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{3} \) |
$0.404346424$ |
$6.915205810$ |
6.145267118 |
\( \frac{132943}{539} a + \frac{378320}{539} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( 12 a - 50\) , \( -225 a + 932\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(12a-50\right){x}-225a+932$ |
| 847.5-j1 |
847.5-j |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
847.5 |
\( 7 \cdot 11^{2} \) |
\( 7^{2} \cdot 11^{7} \) |
$3.50952$ |
$(-a+3), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{3} \) |
$1$ |
$1.759786003$ |
1.933801583 |
\( -\frac{22727773069}{539} a - \frac{156569497107}{539} \) |
\( \bigl[a + 1\) , \( -a\) , \( a\) , \( -1401 a - 4414\) , \( 53364 a + 167513\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-1401a-4414\right){x}+53364a+167513$ |
| 847.5-k1 |
847.5-k |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
847.5 |
\( 7 \cdot 11^{2} \) |
\( 7^{4} \cdot 11^{9} \) |
$3.50952$ |
$(-a+3), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$3.111061334$ |
1.709348557 |
\( -\frac{739250}{2401} a - \frac{1941375}{2401} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -43 a + 198\) , \( 1017 a - 4192\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-43a+198\right){x}+1017a-4192$ |
*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.
