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 |
55.2-a1 |
55.2-a |
$2$ |
$2$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5^{2} \cdot 11^{2} \) |
$10.55676$ |
$(-a^3+a^2+4a+1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$134.2078364$ |
1.874696378 |
\( \frac{287491768525988172527}{605} a^{3} + \frac{203073306864646217091}{605} a^{2} - \frac{235072139333994482191}{121} a - \frac{106228121417968802248}{55} \) |
\( \bigl[a^{3} - 4 a - 4\) , \( -a - 1\) , \( a\) , \( 64 a^{3} - a^{2} - 383 a - 342\) , \( 655 a^{3} + 104 a^{2} - 3702 a - 3363\bigr] \) |
${y}^2+\left(a^{3}-4a-4\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(64a^{3}-a^{2}-383a-342\right){x}+655a^{3}+104a^{2}-3702a-3363$ |
55.2-a2 |
55.2-a |
$2$ |
$2$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5^{4} \cdot 11^{4} \) |
$10.55676$ |
$(-a^3+a^2+4a+1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$134.2078364$ |
1.874696378 |
\( -\frac{16065231177563}{73205} a^{3} - \frac{11346843517992}{73205} a^{2} + \frac{65679504182957}{73205} a + \frac{5935761172007}{6655} \) |
\( \bigl[a^{3} - 4 a - 4\) , \( -a - 1\) , \( a\) , \( -6 a^{3} - a^{2} + 32 a + 28\) , \( 45 a^{3} + 6 a^{2} - 259 a - 235\bigr] \) |
${y}^2+\left(a^{3}-4a-4\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6a^{3}-a^{2}+32a+28\right){x}+45a^{3}+6a^{2}-259a-235$ |
55.2-b1 |
55.2-b |
$2$ |
$2$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5^{4} \cdot 11^{4} \) |
$10.55676$ |
$(-a^3+a^2+4a+1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.768685734$ |
$30.87409782$ |
2.652077121 |
\( -\frac{5932926689}{73205} a^{3} - \frac{838885349}{14641} a^{2} + \frac{4852982231}{14641} a + \frac{2192718217}{6655} \) |
\( \bigl[a^{2} - a - 4\) , \( -a^{3} + 4 a + 3\) , \( a^{2} - a - 3\) , \( a^{3} - a^{2} - a + 1\) , \( a^{3} - 6 a^{2} + 2 a + 11\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{3}+4a+3\right){x}^{2}+\left(a^{3}-a^{2}-a+1\right){x}+a^{3}-6a^{2}+2a+11$ |
55.2-b2 |
55.2-b |
$2$ |
$2$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5^{8} \cdot 11^{2} \) |
$10.55676$ |
$(-a^3+a^2+4a+1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.537371468$ |
$30.87409782$ |
2.652077121 |
\( \frac{196392867540579}{3025} a^{3} + \frac{27744964314013}{605} a^{2} - \frac{802919148292512}{3025} a - \frac{72567111214406}{275} \) |
\( \bigl[a^{2} - a - 4\) , \( -a^{3} + 4 a + 3\) , \( a^{2} - a - 3\) , \( 16 a^{3} - 41 a^{2} - 21 a + 46\) , \( 66 a^{3} - 198 a^{2} - 52 a + 280\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{3}+4a+3\right){x}^{2}+\left(16a^{3}-41a^{2}-21a+46\right){x}+66a^{3}-198a^{2}-52a+280$ |
55.2-c1 |
55.2-c |
$4$ |
$6$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5 \cdot 11 \) |
$10.55676$ |
$(-a^3+a^2+4a+1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 1 \) |
$1$ |
$117.3818148$ |
1.639660313 |
\( \frac{19204557701238}{55} a^{3} - \frac{59961076285437}{55} a^{2} - \frac{47938019656374}{55} a + \frac{17112248159944}{5} \) |
\( \bigl[a^{2} - 3\) , \( -a^{3} + a^{2} + 4 a - 1\) , \( 1\) , \( -32 a^{3} + 63 a^{2} + 97 a - 182\) , \( -141 a^{3} + 286 a^{2} + 415 a - 821\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+a^{2}+4a-1\right){x}^{2}+\left(-32a^{3}+63a^{2}+97a-182\right){x}-141a^{3}+286a^{2}+415a-821$ |
55.2-c2 |
55.2-c |
$4$ |
$6$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5^{2} \cdot 11^{2} \) |
$10.55676$ |
$(-a^3+a^2+4a+1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$117.3818148$ |
1.639660313 |
\( -\frac{131300318}{605} a^{3} + \frac{408882018}{605} a^{2} + \frac{331836833}{605} a - \frac{23469337}{11} \) |
\( \bigl[a^{2} - 3\) , \( -a^{3} + a^{2} + 4 a - 1\) , \( 1\) , \( -2 a^{3} + 3 a^{2} + 7 a - 7\) , \( -2 a^{3} + 2 a^{2} + 7 a - 4\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+a^{2}+4a-1\right){x}^{2}+\left(-2a^{3}+3a^{2}+7a-7\right){x}-2a^{3}+2a^{2}+7a-4$ |
55.2-c3 |
55.2-c |
$4$ |
$6$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5^{3} \cdot 11^{3} \) |
$10.55676$ |
$(-a^3+a^2+4a+1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 1 \) |
$1$ |
$117.3818148$ |
1.639660313 |
\( -\frac{133990069789633150033955762628}{6655} a^{3} + \frac{496615724716951900546923293583}{6655} a^{2} - \frac{8693172568676612045861200086}{1331} a - \frac{78523816804016589414106934741}{605} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - 4 a - 5\) , \( a\) , \( -155 a^{3} + 482 a^{2} + 395 a - 1557\) , \( -2048 a^{3} + 6363 a^{2} + 5087 a - 19778\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-4a-5\right){x}^{2}+\left(-155a^{3}+482a^{2}+395a-1557\right){x}-2048a^{3}+6363a^{2}+5087a-19778$ |
55.2-c4 |
55.2-c |
$4$ |
$6$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5^{6} \cdot 11^{6} \) |
$10.55676$ |
$(-a^3+a^2+4a+1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$117.3818148$ |
1.639660313 |
\( \frac{63264852088520783254}{44289025} a^{3} - \frac{234482453934311554272}{44289025} a^{2} + \frac{20522874463215378149}{44289025} a + \frac{37075864375131467618}{4026275} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - 4 a - 5\) , \( a\) , \( 10 a^{3} - 28 a^{2} - 25 a + 88\) , \( -134 a^{3} + 427 a^{2} + 330 a - 1341\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-4a-5\right){x}^{2}+\left(10a^{3}-28a^{2}-25a+88\right){x}-134a^{3}+427a^{2}+330a-1341$ |
55.2-d1 |
55.2-d |
$4$ |
$6$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5^{2} \cdot 11^{2} \) |
$10.55676$ |
$(-a^3+a^2+4a+1), (a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$669.0430099$ |
1.038399794 |
\( -\frac{131300318}{605} a^{3} + \frac{408882018}{605} a^{2} + \frac{331836833}{605} a - \frac{23469337}{11} \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( -a^{3} + 2 a^{2} + 4 a - 5\) , \( a^{3} - a^{2} - 3 a + 1\) , \( a^{3} - 6 a^{2} + a + 19\) , \( -a^{3} + a^{2} + 5 a - 3\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a^{3}-a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-5\right){x}^{2}+\left(a^{3}-6a^{2}+a+19\right){x}-a^{3}+a^{2}+5a-3$ |
55.2-d2 |
55.2-d |
$4$ |
$6$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5 \cdot 11 \) |
$10.55676$ |
$(-a^3+a^2+4a+1), (a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 1 \) |
$1$ |
$669.0430099$ |
1.038399794 |
\( \frac{19204557701238}{55} a^{3} - \frac{59961076285437}{55} a^{2} - \frac{47938019656374}{55} a + \frac{17112248159944}{5} \) |
\( \bigl[a^{3} - a^{2} - 3 a\) , \( -a^{3} + 2 a^{2} + 4 a - 5\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 6 a^{3} - 21 a^{2} - 4 a + 34\) , \( 14 a^{3} - 58 a^{2} + 10 a + 113\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a\right){x}{y}+\left(a^{3}-a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-5\right){x}^{2}+\left(6a^{3}-21a^{2}-4a+34\right){x}+14a^{3}-58a^{2}+10a+113$ |
55.2-d3 |
55.2-d |
$4$ |
$6$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5^{3} \cdot 11^{3} \) |
$10.55676$ |
$(-a^3+a^2+4a+1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$36$ |
\( 1 \) |
$1$ |
$8.259790245$ |
1.038399794 |
\( -\frac{133990069789633150033955762628}{6655} a^{3} + \frac{496615724716951900546923293583}{6655} a^{2} - \frac{8693172568676612045861200086}{1331} a - \frac{78523816804016589414106934741}{605} \) |
\( \bigl[a\) , \( -a + 1\) , \( a^{3} - a^{2} - 3 a\) , \( 189 a^{3} - 180 a^{2} - 756 a - 392\) , \( 2259 a^{3} - 2017 a^{2} - 9352 a - 4703\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-a^{2}-3a\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(189a^{3}-180a^{2}-756a-392\right){x}+2259a^{3}-2017a^{2}-9352a-4703$ |
55.2-d4 |
55.2-d |
$4$ |
$6$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5^{6} \cdot 11^{6} \) |
$10.55676$ |
$(-a^3+a^2+4a+1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$8.259790245$ |
1.038399794 |
\( \frac{63264852088520783254}{44289025} a^{3} - \frac{234482453934311554272}{44289025} a^{2} + \frac{20522874463215378149}{44289025} a + \frac{37075864375131467618}{4026275} \) |
\( \bigl[a\) , \( -a + 1\) , \( a^{3} - a^{2} - 3 a\) , \( 4 a^{3} - 15 a^{2} - a + 28\) , \( 85 a^{3} - 41 a^{2} - 410 a - 280\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-a^{2}-3a\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(4a^{3}-15a^{2}-a+28\right){x}+85a^{3}-41a^{2}-410a-280$ |
55.2-e1 |
55.2-e |
$2$ |
$2$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5^{8} \cdot 11^{2} \) |
$10.55676$ |
$(-a^3+a^2+4a+1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.047398503$ |
$856.4096892$ |
2.268084662 |
\( \frac{196392867540579}{3025} a^{3} + \frac{27744964314013}{605} a^{2} - \frac{802919148292512}{3025} a - \frac{72567111214406}{275} \) |
\( \bigl[a^{2} - 4\) , \( a^{3} - 5 a - 4\) , \( a^{2} - 3\) , \( -22 a^{3} + 47 a^{2} + 69 a - 141\) , \( 67 a^{3} - 189 a^{2} - 181 a + 592\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-5a-4\right){x}^{2}+\left(-22a^{3}+47a^{2}+69a-141\right){x}+67a^{3}-189a^{2}-181a+592$ |
55.2-e2 |
55.2-e |
$2$ |
$2$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5^{4} \cdot 11^{4} \) |
$10.55676$ |
$(-a^3+a^2+4a+1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.023699251$ |
$856.4096892$ |
2.268084662 |
\( -\frac{5932926689}{73205} a^{3} - \frac{838885349}{14641} a^{2} + \frac{4852982231}{14641} a + \frac{2192718217}{6655} \) |
\( \bigl[a^{2} - 4\) , \( a^{3} - 5 a - 4\) , \( a^{2} - 3\) , \( -2 a^{3} + 2 a^{2} + 9 a - 1\) , \( 2 a^{3} - 5 a^{2} - 7 a + 13\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-5a-4\right){x}^{2}+\left(-2a^{3}+2a^{2}+9a-1\right){x}+2a^{3}-5a^{2}-7a+13$ |
55.2-f1 |
55.2-f |
$2$ |
$2$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5^{2} \cdot 11^{2} \) |
$10.55676$ |
$(-a^3+a^2+4a+1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$70.55402936$ |
0.985541431 |
\( \frac{287491768525988172527}{605} a^{3} + \frac{203073306864646217091}{605} a^{2} - \frac{235072139333994482191}{121} a - \frac{106228121417968802248}{55} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 1\) , \( -a^{3} + a^{2} + 5 a - 1\) , \( a + 1\) , \( 219 a^{3} - 805 a^{2} + 81 a + 1343\) , \( 3458 a^{3} - 12790 a^{2} + 996 a + 22447\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-1\right){x}^{2}+\left(219a^{3}-805a^{2}+81a+1343\right){x}+3458a^{3}-12790a^{2}+996a+22447$ |
55.2-f2 |
55.2-f |
$2$ |
$2$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5^{4} \cdot 11^{4} \) |
$10.55676$ |
$(-a^3+a^2+4a+1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$70.55402936$ |
0.985541431 |
\( -\frac{16065231177563}{73205} a^{3} - \frac{11346843517992}{73205} a^{2} + \frac{65679504182957}{73205} a + \frac{5935761172007}{6655} \) |
\( \bigl[1\) , \( -a^{3} + 4 a + 4\) , \( a^{3} - 5 a - 4\) , \( -8 a^{3} - a^{2} + 32 a + 20\) , \( -10 a^{3} - 9 a^{2} + 41 a + 47\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-5a-4\right){y}={x}^{3}+\left(-a^{3}+4a+4\right){x}^{2}+\left(-8a^{3}-a^{2}+32a+20\right){x}-10a^{3}-9a^{2}+41a+47$ |
*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.