| 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 |
| 55.1-a1 |
55.1-a |
$4$ |
$4$ |
3.3.1825.1 |
$3$ |
$[3, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( - 5^{3} \cdot 11^{3} \) |
$7.44441$ |
$(a-2), (a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 3 \) |
$1$ |
$183.6867657$ |
3.224836468 |
\( -\frac{155241403769}{6655} a^{2} + \frac{4279876738}{1331} a + \frac{252074062995}{1331} \) |
\( \bigl[a^{2} - 6\) , \( -a - 1\) , \( a + 1\) , \( -109 a^{2} - 206 a + 272\) , \( 497 a^{2} + 955 a - 1189\bigr] \) |
${y}^2+\left(a^{2}-6\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-109a^{2}-206a+272\right){x}+497a^{2}+955a-1189$ |
| 55.1-a2 |
55.1-a |
$4$ |
$4$ |
3.3.1825.1 |
$3$ |
$[3, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{12} \cdot 11^{3} \) |
$7.44441$ |
$(a-2), (a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$4$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$11.48042286$ |
3.224836468 |
\( \frac{12753842239372040877}{831875} a^{2} - \frac{48225896996235685519}{831875} a + \frac{32099167947172480586}{831875} \) |
\( \bigl[a^{2} - 6\) , \( -a - 1\) , \( a + 1\) , \( -329 a^{2} - 486 a + 677\) , \( -95993 a^{2} - 185350 a + 231171\bigr] \) |
${y}^2+\left(a^{2}-6\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-329a^{2}-486a+677\right){x}-95993a^{2}-185350a+231171$ |
| 55.1-a3 |
55.1-a |
$4$ |
$4$ |
3.3.1825.1 |
$3$ |
$[3, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{6} \cdot 11^{6} \) |
$7.44441$ |
$(a-2), (a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3Ns |
$4$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$45.92169144$ |
3.224836468 |
\( \frac{38401441560864}{44289025} a^{2} - \frac{143054085407499}{44289025} a + \frac{95412829327063}{44289025} \) |
\( \bigl[a^{2} - 6\) , \( -a - 1\) , \( a + 1\) , \( -994 a^{2} - 1896 a + 2387\) , \( -40006 a^{2} - 76786 a + 95948\bigr] \) |
${y}^2+\left(a^{2}-6\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-994a^{2}-1896a+2387\right){x}-40006a^{2}-76786a+95948$ |
| 55.1-a4 |
55.1-a |
$4$ |
$4$ |
3.3.1825.1 |
$3$ |
$[3, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{3} \cdot 11^{12} \) |
$7.44441$ |
$(a-2), (a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$16$ |
\( 2 \cdot 3 \) |
$1$ |
$5.740211430$ |
3.224836468 |
\( \frac{17152936432430680778629}{15692141883605} a^{2} + \frac{6583602216565772008917}{3138428376721} a - \frac{8226561740559112701822}{3138428376721} \) |
\( \bigl[a^{2} - 6\) , \( -a - 1\) , \( a + 1\) , \( -15819 a^{2} - 30346 a + 37937\) , \( -2609091 a^{2} - 5007086 a + 6256633\bigr] \) |
${y}^2+\left(a^{2}-6\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-15819a^{2}-30346a+37937\right){x}-2609091a^{2}-5007086a+6256633$ |
| 55.1-b1 |
55.1-b |
$2$ |
$2$ |
3.3.1825.1 |
$3$ |
$[3, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( - 5 \cdot 11^{3} \) |
$7.44441$ |
$(a-2), (a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$146.3705383$ |
3.426275145 |
\( -\frac{86056}{6655} a^{2} + \frac{1570644}{6655} a + \frac{10210811}{6655} \) |
\( \bigl[a^{2} + a - 5\) , \( -a^{2} + 5\) , \( a + 1\) , \( -3 a^{2} - a + 23\) , \( -3 a^{2} - 2 a + 16\bigr] \) |
${y}^2+\left(a^{2}+a-5\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+5\right){x}^{2}+\left(-3a^{2}-a+23\right){x}-3a^{2}-2a+16$ |
| 55.1-b2 |
55.1-b |
$2$ |
$2$ |
3.3.1825.1 |
$3$ |
$[3, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{2} \cdot 11^{6} \) |
$7.44441$ |
$(a-2), (a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$146.3705383$ |
3.426275145 |
\( -\frac{296729465242}{8857805} a^{2} + \frac{144963398044}{8857805} a + \frac{545956197079}{1771561} \) |
\( \bigl[a^{2} + a - 5\) , \( -a^{2} + 5\) , \( a + 1\) , \( -8 a^{2} - 11 a + 33\) , \( 5 a^{2} + 12 a - 6\bigr] \) |
${y}^2+\left(a^{2}+a-5\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+5\right){x}^{2}+\left(-8a^{2}-11a+33\right){x}+5a^{2}+12a-6$ |
| 55.1-c1 |
55.1-c |
$2$ |
$2$ |
3.3.1825.1 |
$3$ |
$[3, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{2} \cdot 11^{6} \) |
$7.44441$ |
$(a-2), (a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.504313358$ |
$58.02562122$ |
6.164975369 |
\( -\frac{296729465242}{8857805} a^{2} + \frac{144963398044}{8857805} a + \frac{545956197079}{1771561} \) |
\( \bigl[a + 1\) , \( a^{2} - a - 6\) , \( 1\) , \( 16 a^{2} - 7 a - 133\) , \( 51 a^{2} - 6 a - 406\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a^{2}-a-6\right){x}^{2}+\left(16a^{2}-7a-133\right){x}+51a^{2}-6a-406$ |
| 55.1-c2 |
55.1-c |
$2$ |
$2$ |
3.3.1825.1 |
$3$ |
$[3, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( - 5 \cdot 11^{3} \) |
$7.44441$ |
$(a-2), (a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$1.008626716$ |
$116.0512424$ |
6.164975369 |
\( -\frac{86056}{6655} a^{2} + \frac{1570644}{6655} a + \frac{10210811}{6655} \) |
\( \bigl[a + 1\) , \( a^{2} - a - 6\) , \( 1\) , \( a^{2} - 2 a - 3\) , \( -2 a^{2} - a + 18\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a^{2}-a-6\right){x}^{2}+\left(a^{2}-2a-3\right){x}-2a^{2}-a+18$ |
| 55.1-d1 |
55.1-d |
$4$ |
$4$ |
3.3.1825.1 |
$3$ |
$[3, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( - 5^{3} \cdot 11^{3} \) |
$7.44441$ |
$(a-2), (a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 3^{2} \) |
$1.482586566$ |
$22.88309867$ |
5.360523849 |
\( -\frac{155241403769}{6655} a^{2} + \frac{4279876738}{1331} a + \frac{252074062995}{1331} \) |
\( \bigl[a^{2} - 5\) , \( -a^{2} - a + 7\) , \( a^{2} + a - 5\) , \( -33 a^{2} - 58 a + 99\) , \( 67 a^{2} + 136 a - 138\bigr] \) |
${y}^2+\left(a^{2}-5\right){x}{y}+\left(a^{2}+a-5\right){y}={x}^{3}+\left(-a^{2}-a+7\right){x}^{2}+\left(-33a^{2}-58a+99\right){x}+67a^{2}+136a-138$ |
| 55.1-d2 |
55.1-d |
$4$ |
$4$ |
3.3.1825.1 |
$3$ |
$[3, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{12} \cdot 11^{3} \) |
$7.44441$ |
$(a-2), (a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$4$ |
\( 2^{2} \cdot 3^{2} \) |
$1.482586566$ |
$1.430193667$ |
5.360523849 |
\( \frac{12753842239372040877}{831875} a^{2} - \frac{48225896996235685519}{831875} a + \frac{32099167947172480586}{831875} \) |
\( \bigl[a^{2} - 5\) , \( -a^{2} - a + 7\) , \( a^{2} + a - 5\) , \( -88 a^{2} - 153 a + 224\) , \( -13618 a^{2} - 26089 a + 32652\bigr] \) |
${y}^2+\left(a^{2}-5\right){x}{y}+\left(a^{2}+a-5\right){y}={x}^{3}+\left(-a^{2}-a+7\right){x}^{2}+\left(-88a^{2}-153a+224\right){x}-13618a^{2}-26089a+32652$ |
| 55.1-d3 |
55.1-d |
$4$ |
$4$ |
3.3.1825.1 |
$3$ |
$[3, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{6} \cdot 11^{6} \) |
$7.44441$ |
$(a-2), (a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3Ns |
$4$ |
\( 2^{2} \cdot 3^{2} \) |
$0.741293283$ |
$11.44154933$ |
5.360523849 |
\( \frac{38401441560864}{44289025} a^{2} - \frac{143054085407499}{44289025} a + \frac{95412829327063}{44289025} \) |
\( \bigl[a^{2} - 5\) , \( -a^{2} - a + 7\) , \( a^{2} + a - 5\) , \( -273 a^{2} - 518 a + 674\) , \( -5654 a^{2} - 10842 a + 13580\bigr] \) |
${y}^2+\left(a^{2}-5\right){x}{y}+\left(a^{2}+a-5\right){y}={x}^{3}+\left(-a^{2}-a+7\right){x}^{2}+\left(-273a^{2}-518a+674\right){x}-5654a^{2}-10842a+13580$ |
| 55.1-d4 |
55.1-d |
$4$ |
$4$ |
3.3.1825.1 |
$3$ |
$[3, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5^{3} \cdot 11^{12} \) |
$7.44441$ |
$(a-2), (a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1.482586566$ |
$5.720774668$ |
5.360523849 |
\( \frac{17152936432430680778629}{15692141883605} a^{2} + \frac{6583602216565772008917}{3138428376721} a - \frac{8226561740559112701822}{3138428376721} \) |
\( \bigl[a^{2} - 5\) , \( -a^{2} - a + 7\) , \( a^{2} + a - 5\) , \( -4298 a^{2} - 8243 a + 10324\) , \( -368974 a^{2} - 708087 a + 884820\bigr] \) |
${y}^2+\left(a^{2}-5\right){x}{y}+\left(a^{2}+a-5\right){y}={x}^{3}+\left(-a^{2}-a+7\right){x}^{2}+\left(-4298a^{2}-8243a+10324\right){x}-368974a^{2}-708087a+884820$ |
*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.
