| 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 |
| 343.1-a1 |
343.1-a |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
343.1 |
\( 7^{3} \) |
\( - 7^{13} \) |
$2.79963$ |
$(-a-2), (-a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B.6.1 |
$1$ |
\( 7 \) |
$1$ |
$2.688240606$ |
2.584807720 |
\( \frac{1840001024}{823543} a + \frac{5775437824}{823543} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -51 a - 160\) , \( -341 a - 1074\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}+\left(-51a-160\right){x}-341a-1074$ |
| 343.1-a2 |
343.1-a |
$2$ |
$7$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
343.1 |
\( 7^{3} \) |
\( - 7^{7} \) |
$2.79963$ |
$(-a-2), (-a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B.6.3 |
$49$ |
\( 1 \) |
$1$ |
$0.384034372$ |
2.584807720 |
\( \frac{195338235078135808}{7} a - \frac{808691822475743232}{7} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -21401 a - 69320\) , \( 3330266 a + 10420351\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}+\left(-21401a-69320\right){x}+3330266a+10420351$ |
| 343.1-b1 |
343.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
343.1 |
\( 7^{3} \) |
\( 7^{14} \) |
$2.79963$ |
$(-a-2), (-a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$5.855614784$ |
3.217322196 |
\( -\frac{48788531}{117649} a - \frac{152983814}{117649} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -5 a - 16\) , \( 32\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-5a-16\right){x}+32$ |
| 343.1-c1 |
343.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
343.1 |
\( 7^{3} \) |
\( 7^{14} \) |
$2.79963$ |
$(-a-2), (-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$3.207607389$ |
$1.344318664$ |
4.738441093 |
\( \frac{48788531}{117649} a - \frac{201772345}{117649} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 11 a + 33\) , \( -679 a - 2135\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(11a+33\right){x}-679a-2135$ |
| 343.1-d1 |
343.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
343.1 |
\( 7^{3} \) |
\( - 7^{15} \) |
$2.79963$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$4.254424669$ |
1.753170515 |
\( -\frac{8183558401}{117649} a - \frac{23271540090}{117649} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 1\) , \( 827 a - 3425\) , \( 25168 a - 104197\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(827a-3425\right){x}+25168a-104197$ |
| 343.1-d2 |
343.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
343.1 |
\( 7^{3} \) |
\( - 7^{9} \) |
$2.79963$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$12.76327400$ |
1.753170515 |
\( -\frac{1063343}{49} a + \frac{4416725}{49} \) |
\( \bigl[a\) , \( 1\) , \( 1\) , \( 20 a + 68\) , \( 236 a + 742\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+{x}^{2}+\left(20a+68\right){x}+236a+742$ |
| 343.1-d3 |
343.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
343.1 |
\( 7^{3} \) |
\( 7^{9} \) |
$2.79963$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$12.76327400$ |
1.753170515 |
\( -\frac{167034552579}{49} a + \frac{691532566484}{49} \) |
\( \bigl[a\) , \( 1\) , \( 1\) , \( -190 a - 597\) , \( 2042 a + 6412\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-190a-597\right){x}+2042a+6412$ |
| 343.1-d4 |
343.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
343.1 |
\( 7^{3} \) |
\( 7^{15} \) |
$2.79963$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$4.254424669$ |
1.753170515 |
\( \frac{3825287113585893}{117649} a + \frac{12011612154623447}{117649} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 1\) , \( 2297 a - 9550\) , \( -80084 a + 331609\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2297a-9550\right){x}-80084a+331609$ |
| 343.1-e1 |
343.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
343.1 |
\( 7^{3} \) |
\( 7^{15} \) |
$2.79963$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$0.803813119$ |
0.662473340 |
\( -\frac{3825287113585893}{117649} a + \frac{15836899268209340}{117649} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 489 a - 2202\) , \( 11618 a - 49152\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(489a-2202\right){x}+11618a-49152$ |
| 343.1-e2 |
343.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
343.1 |
\( 7^{3} \) |
\( - 7^{15} \) |
$2.79963$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$1.607626239$ |
0.662473340 |
\( \frac{8183558401}{117649} a - \frac{31455098491}{117649} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 24 a - 157\) , \( 181 a - 661\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(24a-157\right){x}+181a-661$ |
| 343.1-e3 |
343.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
343.1 |
\( 7^{3} \) |
\( - 7^{9} \) |
$2.79963$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.822878717$ |
0.662473340 |
\( \frac{1063343}{49} a + \frac{3353382}{49} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( -a + 3\) , \( -2 a - 19\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-a+3\right){x}-2a-19$ |
| 343.1-e4 |
343.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
343.1 |
\( 7^{3} \) |
\( 7^{9} \) |
$2.79963$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$2.411439358$ |
0.662473340 |
\( \frac{167034552579}{49} a + \frac{524498013905}{49} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( -16 a - 97\) , \( -151 a - 555\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-16a-97\right){x}-151a-555$ |
*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.
