| 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 |
| 27.1-a1 |
27.1-a |
$2$ |
$3$ |
4.4.11661.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{8} \) |
$14.56891$ |
$(-a), (a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 5 \) |
$1.561257313$ |
$7.917115044$ |
4.578608762 |
\( -\frac{43579931283769}{243} a^{3} + \frac{107777583890159}{243} a^{2} + \frac{123309264089110}{243} a - \frac{276240756518333}{243} \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - 2 a^{2} - 2 a + 3\) , \( a^{2} - a - 2\) , \( -7 a^{3} - 4 a^{2} + 17 a + 11\) , \( -18 a^{3} - 50 a^{2} - 38 a - 10\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-2a+3\right){x}^{2}+\left(-7a^{3}-4a^{2}+17a+11\right){x}-18a^{3}-50a^{2}-38a-10$ |
| 27.1-a2 |
27.1-a |
$2$ |
$3$ |
4.4.11661.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{24} \) |
$14.56891$ |
$(-a), (a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \cdot 5 \) |
$0.520419104$ |
$71.25403539$ |
4.578608762 |
\( -\frac{16988069332}{14348907} a^{3} + \frac{37528265996}{14348907} a^{2} + \frac{65539602622}{14348907} a - \frac{123694858235}{14348907} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( 0\) , \( a^{3} - a^{2} - 4 a + 1\) , \( -2 a^{3} + 6 a + 1\) , \( -21 a^{3} - 5 a^{2} + 65 a + 28\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{3}-a^{2}-4a+1\right){y}={x}^{3}+\left(-2a^{3}+6a+1\right){x}-21a^{3}-5a^{2}+65a+28$ |
| 27.1-b1 |
27.1-b |
$2$ |
$3$ |
4.4.11661.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{8} \) |
$14.56891$ |
$(-a), (a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$52.68757225$ |
0.975821282 |
\( -\frac{43579931283769}{243} a^{3} + \frac{107777583890159}{243} a^{2} + \frac{123309264089110}{243} a - \frac{276240756518333}{243} \) |
\( \bigl[a^{2} - 3\) , \( a^{3} - 2 a^{2} - 3 a + 5\) , \( 0\) , \( -3 a^{3} + 9 a^{2} + 7 a - 20\) , \( -27 a^{3} + 68 a^{2} + 75 a - 174\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}={x}^{3}+\left(a^{3}-2a^{2}-3a+5\right){x}^{2}+\left(-3a^{3}+9a^{2}+7a-20\right){x}-27a^{3}+68a^{2}+75a-174$ |
| 27.1-b2 |
27.1-b |
$2$ |
$3$ |
4.4.11661.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{24} \) |
$14.56891$ |
$(-a), (a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$52.68757225$ |
0.975821282 |
\( -\frac{16988069332}{14348907} a^{3} + \frac{37528265996}{14348907} a^{2} + \frac{65539602622}{14348907} a - \frac{123694858235}{14348907} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - 6 a - 2\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -3 a^{2} + 6 a\) , \( 4 a^{3} - 16 a^{2} + 10 a + 5\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-6a-2\right){x}^{2}+\left(-3a^{2}+6a\right){x}+4a^{3}-16a^{2}+10a+5$ |
| 27.1-c1 |
27.1-c |
$4$ |
$6$ |
4.4.11661.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{8} \) |
$14.56891$ |
$(-a), (a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$349.1848112$ |
3.233608568 |
\( \frac{5609257}{9} a^{3} - \frac{15326453}{9} a^{2} - \frac{13918606}{9} a + \frac{40495973}{9} \) |
\( \bigl[a^{3} - 5 a - 2\) , \( a^{3} - a^{2} - 4 a\) , \( a^{2} - 3\) , \( 5 a^{3} - 4 a^{2} - 22 a - 8\) , \( -5 a^{3} + 2 a^{2} + 25 a + 9\bigr] \) |
${y}^2+\left(a^{3}-5a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a\right){x}^{2}+\left(5a^{3}-4a^{2}-22a-8\right){x}-5a^{3}+2a^{2}+25a+9$ |
| 27.1-c2 |
27.1-c |
$4$ |
$6$ |
4.4.11661.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{12} \) |
$14.56891$ |
$(-a), (a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$116.3949370$ |
3.233608568 |
\( -\frac{64986344}{729} a^{3} - \frac{40985144}{729} a^{2} + \frac{153470132}{729} a + \frac{77142629}{729} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( a^{3} - 6 a - 3\) , \( 0\) , \( a^{3} + 3 a^{2} - 3 a - 2\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}={x}^{3}+\left(a^{3}-6a-3\right){x}^{2}+\left(a^{3}+3a^{2}-3a-2\right){x}$ |
| 27.1-c3 |
27.1-c |
$4$ |
$6$ |
4.4.11661.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{18} \) |
$14.56891$ |
$(-a), (a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$58.19746854$ |
3.233608568 |
\( \frac{63791259979876622}{531441} a^{3} + \frac{40426016608642742}{531441} a^{2} - \frac{148694095429555436}{531441} a - \frac{72662822899947101}{531441} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( a^{3} - 6 a - 3\) , \( 0\) , \( -4 a^{3} - 12 a^{2} + 12 a + 8\) , \( -87 a^{3} - 64 a^{2} + 247 a + 122\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}={x}^{3}+\left(a^{3}-6a-3\right){x}^{2}+\left(-4a^{3}-12a^{2}+12a+8\right){x}-87a^{3}-64a^{2}+247a+122$ |
| 27.1-c4 |
27.1-c |
$4$ |
$6$ |
4.4.11661.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{10} \) |
$14.56891$ |
$(-a), (a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$174.5924056$ |
3.233608568 |
\( -\frac{3383788332109}{81} a^{3} + \frac{2705242125866}{81} a^{2} + \frac{14167293434752}{81} a + \frac{5740179127237}{81} \) |
\( \bigl[a\) , \( -a^{3} + 2 a^{2} + 2 a - 5\) , \( a\) , \( -23 a^{3} + 68 a^{2} + 47 a - 205\) , \( 285 a^{3} - 764 a^{2} - 755 a + 2048\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a^{3}+2a^{2}+2a-5\right){x}^{2}+\left(-23a^{3}+68a^{2}+47a-205\right){x}+285a^{3}-764a^{2}-755a+2048$ |
| 27.1-d1 |
27.1-d |
$2$ |
$3$ |
4.4.11661.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{18} \) |
$14.56891$ |
$(-a), (a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$76.48416653$ |
1.416555636 |
\( -\frac{16988069332}{14348907} a^{3} + \frac{37528265996}{14348907} a^{2} + \frac{65539602622}{14348907} a - \frac{123694858235}{14348907} \) |
\( \bigl[1\) , \( a^{3} - 5 a - 2\) , \( a^{3} - 5 a - 2\) , \( -2 a^{3} + 2 a^{2} + 9 a + 1\) , \( 16 a^{3} - 9 a^{2} - 76 a - 31\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-5a-2\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(-2a^{3}+2a^{2}+9a+1\right){x}+16a^{3}-9a^{2}-76a-31$ |
| 27.1-d2 |
27.1-d |
$2$ |
$3$ |
4.4.11661.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{14} \) |
$14.56891$ |
$(-a), (a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$76.48416653$ |
1.416555636 |
\( -\frac{43579931283769}{243} a^{3} + \frac{107777583890159}{243} a^{2} + \frac{123309264089110}{243} a - \frac{276240756518333}{243} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( -a^{3} + 2 a^{2} + 4 a - 5\) , \( a^{2} - 2\) , \( -391 a^{3} + 969 a^{2} + 1105 a - 2478\) , \( 8426 a^{3} - 20840 a^{2} - 23837 a + 53420\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-5\right){x}^{2}+\left(-391a^{3}+969a^{2}+1105a-2478\right){x}+8426a^{3}-20840a^{2}-23837a+53420$ |
| 27.1-e1 |
27.1-e |
$2$ |
$3$ |
4.4.11661.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{14} \) |
$14.56891$ |
$(-a), (a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \cdot 5 \) |
$0.021370245$ |
$408.4651630$ |
3.233379162 |
\( -\frac{43579931283769}{243} a^{3} + \frac{107777583890159}{243} a^{2} + \frac{123309264089110}{243} a - \frac{276240756518333}{243} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - a^{2} - 5 a\) , \( a^{2} - 3\) , \( -22 a^{3} + 46 a^{2} + 68 a - 105\) , \( -106 a^{3} - 64 a^{2} + 626 a + 732\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-22a^{3}+46a^{2}+68a-105\right){x}-106a^{3}-64a^{2}+626a+732$ |
| 27.1-e2 |
27.1-e |
$2$ |
$3$ |
4.4.11661.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{18} \) |
$14.56891$ |
$(-a), (a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \cdot 5 \) |
$0.007123415$ |
$408.4651630$ |
3.233379162 |
\( -\frac{16988069332}{14348907} a^{3} + \frac{37528265996}{14348907} a^{2} + \frac{65539602622}{14348907} a - \frac{123694858235}{14348907} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 2\) , \( a^{3} - 6 a - 3\) , \( a^{2} - a - 2\) , \( -9 a^{3} + 18 a^{2} + 30 a - 42\) , \( 19 a^{3} - 44 a^{2} - 57 a + 107\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{3}-6a-3\right){x}^{2}+\left(-9a^{3}+18a^{2}+30a-42\right){x}+19a^{3}-44a^{2}-57a+107$ |
| 27.1-f1 |
27.1-f |
$4$ |
$6$ |
4.4.11661.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{12} \) |
$14.56891$ |
$(-a), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.072858208$ |
$687.1793671$ |
1.854559163 |
\( -\frac{64986344}{729} a^{3} - \frac{40985144}{729} a^{2} + \frac{153470132}{729} a + \frac{77142629}{729} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - 5 a - 1\) , \( -8 a^{3} + 29 a^{2} - 17 a - 12\) , \( -59 a^{3} + 217 a^{2} - 121 a - 112\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(-8a^{3}+29a^{2}-17a-12\right){x}-59a^{3}+217a^{2}-121a-112$ |
| 27.1-f2 |
27.1-f |
$4$ |
$6$ |
4.4.11661.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{18} \) |
$14.56891$ |
$(-a), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.145716416$ |
$343.5896835$ |
1.854559163 |
\( \frac{63791259979876622}{531441} a^{3} + \frac{40426016608642742}{531441} a^{2} - \frac{148694095429555436}{531441} a - \frac{72662822899947101}{531441} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - 5 a - 1\) , \( 62 a^{3} - 226 a^{2} + 118 a + 118\) , \( -538 a^{3} + 1957 a^{2} - 1048 a - 990\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(62a^{3}-226a^{2}+118a+118\right){x}-538a^{3}+1957a^{2}-1048a-990$ |
| 27.1-f3 |
27.1-f |
$4$ |
$6$ |
4.4.11661.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{10} \) |
$14.56891$ |
$(-a), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.048572138$ |
$1030.769050$ |
1.854559163 |
\( -\frac{3383788332109}{81} a^{3} + \frac{2705242125866}{81} a^{2} + \frac{14167293434752}{81} a + \frac{5740179127237}{81} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a^{2} - 2\) , \( -687 a^{3} + 1699 a^{2} + 1945 a - 4356\) , \( 26168 a^{3} - 64720 a^{2} - 74043 a + 165881\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(-687a^{3}+1699a^{2}+1945a-4356\right){x}+26168a^{3}-64720a^{2}-74043a+165881$ |
| 27.1-f4 |
27.1-f |
$4$ |
$6$ |
4.4.11661.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{8} \) |
$14.56891$ |
$(-a), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.024286069$ |
$2061.538101$ |
1.854559163 |
\( \frac{5609257}{9} a^{3} - \frac{15326453}{9} a^{2} - \frac{13918606}{9} a + \frac{40495973}{9} \) |
\( \bigl[1\) , \( a^{3} - 5 a - 2\) , \( a^{3} - 4 a - 1\) , \( 8 a^{3} - 35 a^{2} + 30 a + 18\) , \( -43 a^{3} + 157 a^{2} - 89 a - 79\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(8a^{3}-35a^{2}+30a+18\right){x}-43a^{3}+157a^{2}-89a-79$ |
*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.
