| 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 |
| 14.1-a1 |
14.1-a |
$6$ |
$8$ |
4.4.10889.1 |
$4$ |
$[4, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{4} \cdot 7^{8} \) |
$12.96876$ |
$(-a^3+a^2+4a), (-a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1.109601231$ |
$45.02824921$ |
3.830433959 |
\( -\frac{132052758893986755}{92236816} a^{3} + \frac{21874107228363709}{46118408} a^{2} + \frac{689337767575288381}{92236816} a + \frac{197273590459273121}{92236816} \) |
\( \bigl[a^{3} - 6 a\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -20 a^{3} + 16 a^{2} + 127 a - 92\) , \( -43 a^{3} - 20 a^{2} + 457 a - 309\bigr] \) |
${y}^2+\left(a^{3}-6a\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(-20a^{3}+16a^{2}+127a-92\right){x}-43a^{3}-20a^{2}+457a-309$ |
| 14.1-a2 |
14.1-a |
$6$ |
$8$ |
4.4.10889.1 |
$4$ |
$[4, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{8} \cdot 7^{4} \) |
$12.96876$ |
$(-a^3+a^2+4a), (-a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.554800615$ |
$720.4519873$ |
3.830433959 |
\( -\frac{10156308195}{614656} a^{3} + \frac{1932332501}{307328} a^{2} + \frac{53871409229}{614656} a + \frac{16864249233}{614656} \) |
\( \bigl[a^{3} - 6 a\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -5 a^{3} + 6 a^{2} + 22 a - 12\) , \( 6 a^{3} - 10 a^{2} - 22 a + 19\bigr] \) |
${y}^2+\left(a^{3}-6a\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(-5a^{3}+6a^{2}+22a-12\right){x}+6a^{3}-10a^{2}-22a+19$ |
| 14.1-a3 |
14.1-a |
$6$ |
$8$ |
4.4.10889.1 |
$4$ |
$[4, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{2} \cdot 7^{16} \) |
$12.96876$ |
$(-a^3+a^2+4a), (-a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$2.219202462$ |
$2.814265575$ |
3.830433959 |
\( -\frac{214923712255265581976283013}{132931722278404} a^{3} + \frac{317224068276547277595953029}{66465861139202} a^{2} - \frac{163804519569114127964110717}{132931722278404} a - \frac{110106115348997558168173573}{132931722278404} \) |
\( \bigl[a^{3} - 6 a\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{3} - a^{2} - 4 a + 2\) , \( 120 a^{3} - 349 a^{2} + 77 a + 63\) , \( 2081 a^{3} - 6717 a^{2} + 3463 a + 42\bigr] \) |
${y}^2+\left(a^{3}-6a\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(120a^{3}-349a^{2}+77a+63\right){x}+2081a^{3}-6717a^{2}+3463a+42$ |
| 14.1-a4 |
14.1-a |
$6$ |
$8$ |
4.4.10889.1 |
$4$ |
$[4, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{4} \cdot 7^{2} \) |
$12.96876$ |
$(-a^3+a^2+4a), (-a+2)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.109601231$ |
$720.4519873$ |
3.830433959 |
\( -\frac{847965}{784} a^{3} + \frac{1431539}{392} a^{2} - \frac{1769197}{784} a + \frac{154367}{784} \) |
\( \bigl[a^{3} - 6 a\) , \( a^{3} - a^{2} - 5 a + 1\) , \( a^{3} - a^{2} - 5 a + 3\) , \( a^{3} - 6 a - 2\) , \( 27 a^{3} - 8 a^{2} - 142 a - 44\bigr] \) |
${y}^2+\left(a^{3}-6a\right){x}{y}+\left(a^{3}-a^{2}-5a+3\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a+1\right){x}^{2}+\left(a^{3}-6a-2\right){x}+27a^{3}-8a^{2}-142a-44$ |
| 14.1-a5 |
14.1-a |
$6$ |
$8$ |
4.4.10889.1 |
$4$ |
$[4, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{16} \cdot 7^{2} \) |
$12.96876$ |
$(-a^3+a^2+4a), (-a+2)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.277400307$ |
$720.4519873$ |
3.830433959 |
\( \frac{17198704073165}{3211264} a^{3} + \frac{11232184290309}{1605632} a^{2} - \frac{21948649503491}{3211264} a - \frac{520692768767}{3211264} \) |
\( \bigl[a\) , \( -a + 1\) , \( a^{3} - 5 a\) , \( 3 a^{3} - 21 a - 6\) , \( -2 a^{3} + a^{2} + 9 a + 2\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(3a^{3}-21a-6\right){x}-2a^{3}+a^{2}+9a+2$ |
| 14.1-a6 |
14.1-a |
$6$ |
$8$ |
4.4.10889.1 |
$4$ |
$[4, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{2} \cdot 7^{4} \) |
$12.96876$ |
$(-a^3+a^2+4a), (-a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$2.219202462$ |
$2.814265575$ |
3.830433959 |
\( -\frac{95179358417704702844187}{9604} a^{3} + \frac{15741481610760652692763}{4802} a^{2} + \frac{496965974535884420520925}{9604} a + \frac{142223280344254765184677}{9604} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( a^{2} - 3\) , \( a^{2} - a - 2\) , \( 98 a^{3} - 287 a^{2} + 54 a + 32\) , \( 2311 a^{3} - 6868 a^{2} + 1761 a + 1157\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(98a^{3}-287a^{2}+54a+32\right){x}+2311a^{3}-6868a^{2}+1761a+1157$ |
| 14.1-b1 |
14.1-b |
$6$ |
$8$ |
4.4.10889.1 |
$4$ |
$[4, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{4} \cdot 7^{2} \) |
$12.96876$ |
$(-a^3+a^2+4a), (-a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.061612995$ |
$877.2347552$ |
2.071830130 |
\( -\frac{847965}{784} a^{3} + \frac{1431539}{392} a^{2} - \frac{1769197}{784} a + \frac{154367}{784} \) |
\( \bigl[1\) , \( a^{2} - 2 a - 3\) , \( a^{3} - 5 a - 1\) , \( -a^{3} + a^{2} + 3 a + 2\) , \( -8 a^{3} + 3 a^{2} + 40 a + 11\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(-a^{3}+a^{2}+3a+2\right){x}-8a^{3}+3a^{2}+40a+11$ |
| 14.1-b2 |
14.1-b |
$6$ |
$8$ |
4.4.10889.1 |
$4$ |
$[4, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{2} \cdot 7^{4} \) |
$12.96876$ |
$(-a^3+a^2+4a), (-a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.492903962$ |
$54.82717220$ |
2.071830130 |
\( -\frac{95179358417704702844187}{9604} a^{3} + \frac{15741481610760652692763}{4802} a^{2} + \frac{496965974535884420520925}{9604} a + \frac{142223280344254765184677}{9604} \) |
\( \bigl[1\) , \( -a^{2} + 4\) , \( a^{2} - 3\) , \( 2 a^{3} - 32 a^{2} + 117 a - 133\) , \( 29 a^{3} - 583 a^{2} + 1566 a - 642\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(2a^{3}-32a^{2}+117a-133\right){x}+29a^{3}-583a^{2}+1566a-642$ |
| 14.1-b3 |
14.1-b |
$6$ |
$8$ |
4.4.10889.1 |
$4$ |
$[4, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{8} \cdot 7^{4} \) |
$12.96876$ |
$(-a^3+a^2+4a), (-a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.123225990$ |
$877.2347552$ |
2.071830130 |
\( -\frac{10156308195}{614656} a^{3} + \frac{1932332501}{307328} a^{2} + \frac{53871409229}{614656} a + \frac{16864249233}{614656} \) |
\( \bigl[1\) , \( -a^{2} + 4\) , \( a^{2} - 3\) , \( 2 a^{3} - 7 a^{2} + 2 a + 7\) , \( 3 a^{3} - 10 a^{2} + 4 a + 1\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(2a^{3}-7a^{2}+2a+7\right){x}+3a^{3}-10a^{2}+4a+1$ |
| 14.1-b4 |
14.1-b |
$6$ |
$8$ |
4.4.10889.1 |
$4$ |
$[4, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{4} \cdot 7^{8} \) |
$12.96876$ |
$(-a^3+a^2+4a), (-a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.246451981$ |
$219.3086888$ |
2.071830130 |
\( -\frac{132052758893986755}{92236816} a^{3} + \frac{21874107228363709}{46118408} a^{2} + \frac{689337767575288381}{92236816} a + \frac{197273590459273121}{92236816} \) |
\( \bigl[1\) , \( -a^{2} + 4\) , \( a^{2} - 3\) , \( 27 a^{3} - 82 a^{2} + 27 a + 12\) , \( 253 a^{3} - 757 a^{2} + 225 a + 111\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(27a^{3}-82a^{2}+27a+12\right){x}+253a^{3}-757a^{2}+225a+111$ |
| 14.1-b5 |
14.1-b |
$6$ |
$8$ |
4.4.10889.1 |
$4$ |
$[4, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{2} \cdot 7^{16} \) |
$12.96876$ |
$(-a^3+a^2+4a), (-a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.492903962$ |
$13.70679305$ |
2.071830130 |
\( -\frac{214923712255265581976283013}{132931722278404} a^{3} + \frac{317224068276547277595953029}{66465861139202} a^{2} - \frac{163804519569114127964110717}{132931722278404} a - \frac{110106115348997558168173573}{132931722278404} \) |
\( \bigl[a^{3} - a^{2} - 5 a + 2\) , \( -a^{3} + 6 a + 2\) , \( a^{3} - 6 a - 1\) , \( 52 a^{3} - 116 a^{2} - 75 a + 73\) , \( -13214 a^{3} + 16685 a^{2} + 62449 a - 45306\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a+2\right){x}{y}+\left(a^{3}-6a-1\right){y}={x}^{3}+\left(-a^{3}+6a+2\right){x}^{2}+\left(52a^{3}-116a^{2}-75a+73\right){x}-13214a^{3}+16685a^{2}+62449a-45306$ |
| 14.1-b6 |
14.1-b |
$6$ |
$8$ |
4.4.10889.1 |
$4$ |
$[4, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{16} \cdot 7^{2} \) |
$12.96876$ |
$(-a^3+a^2+4a), (-a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.246451981$ |
$219.3086888$ |
2.071830130 |
\( \frac{17198704073165}{3211264} a^{3} + \frac{11232184290309}{1605632} a^{2} - \frac{21948649503491}{3211264} a - \frac{520692768767}{3211264} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( -a\) , \( a^{3} - a^{2} - 5 a + 3\) , \( 7 a^{3} - 39 a - 21\) , \( 6 a^{3} - a^{2} - 32 a - 14\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{3}-a^{2}-5a+3\right){y}={x}^{3}-a{x}^{2}+\left(7a^{3}-39a-21\right){x}+6a^{3}-a^{2}-32a-14$ |
*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.
