| 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 |
| 27.2-a1 |
27.2-a |
$2$ |
$3$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{3} \) |
$18.99423$ |
$(-1/3a^3-1/3a^2+3a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$9$ |
\( 1 \) |
$0.205990629$ |
$89.20040611$ |
4.698443578 |
\( -\frac{2659993414}{3} a^{3} + \frac{3059637059}{3} a^{2} + 7384788197 a - 8211399121 \) |
\( \bigl[a^{2} - 4\) , \( -\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 1\) , \( 0\) , \( -\frac{14}{3} a^{3} - \frac{11}{3} a^{2} + 45 a + 19\) , \( -\frac{28}{3} a^{3} + \frac{20}{3} a^{2} + 54 a + 17\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}={x}^{3}+\left(-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+3a+1\right){x}^{2}+\left(-\frac{14}{3}a^{3}-\frac{11}{3}a^{2}+45a+19\right){x}-\frac{28}{3}a^{3}+\frac{20}{3}a^{2}+54a+17$ |
| 27.2-a2 |
27.2-a |
$2$ |
$3$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{9} \) |
$18.99423$ |
$(-1/3a^3-1/3a^2+3a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 3 \) |
$0.068663543$ |
$802.8036549$ |
4.698443578 |
\( \frac{2272}{3} a^{3} + \frac{1171}{3} a^{2} - 5297 a - 3398 \) |
\( \bigl[a + 1\) , \( \frac{2}{3} a^{3} - \frac{1}{3} a^{2} - 6 a + 1\) , \( a^{2} - 3\) , \( -\frac{5}{3} a^{3} + \frac{1}{3} a^{2} + 10 a - 2\) , \( -\frac{7}{3} a^{3} + \frac{8}{3} a^{2} + 17 a - 22\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(\frac{2}{3}a^{3}-\frac{1}{3}a^{2}-6a+1\right){x}^{2}+\left(-\frac{5}{3}a^{3}+\frac{1}{3}a^{2}+10a-2\right){x}-\frac{7}{3}a^{3}+\frac{8}{3}a^{2}+17a-22$ |
| 27.2-b1 |
27.2-b |
$1$ |
$1$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{11} \) |
$18.99423$ |
$(-1/3a^3-1/3a^2+3a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1.883293612$ |
$90.82575439$ |
4.859865586 |
\( \frac{6364868}{3} a^{3} + \frac{8738927}{3} a^{2} - 8598468 a - 3255814 \) |
\( \bigl[\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 1\) , \( -\frac{2}{3} a^{3} + \frac{1}{3} a^{2} + 4 a - 3\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\) , \( \frac{10}{3} a^{3} - \frac{2}{3} a^{2} - 28 a - 4\) , \( \frac{28}{3} a^{3} + \frac{7}{3} a^{2} - 76 a - 33\bigr] \) |
${y}^2+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a-1\right){x}{y}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-3\right){y}={x}^{3}+\left(-\frac{2}{3}a^{3}+\frac{1}{3}a^{2}+4a-3\right){x}^{2}+\left(\frac{10}{3}a^{3}-\frac{2}{3}a^{2}-28a-4\right){x}+\frac{28}{3}a^{3}+\frac{7}{3}a^{2}-76a-33$ |
| 27.2-c1 |
27.2-c |
$2$ |
$3$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{3} \) |
$18.99423$ |
$(-1/3a^3-1/3a^2+3a+2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.1 |
$9$ |
\( 1 \) |
$1$ |
$241.7504940$ |
1.717135589 |
\( \frac{18512}{3} a^{3} + \frac{1769}{3} a^{2} - 48583 a - 15892 \) |
\( \bigl[\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 1\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 2\) , \( a\) , \( -\frac{1}{3} a^{3} + \frac{5}{3} a^{2} - a + 2\) , \( a^{3} - 3 a^{2} + 2 a\bigr] \) |
${y}^2+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a-1\right){x}{y}+a{y}={x}^{3}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+2a-2\right){x}^{2}+\left(-\frac{1}{3}a^{3}+\frac{5}{3}a^{2}-a+2\right){x}+a^{3}-3a^{2}+2a$ |
| 27.2-c2 |
27.2-c |
$2$ |
$3$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{9} \) |
$18.99423$ |
$(-1/3a^3-1/3a^2+3a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$81$ |
\( 1 \) |
$1$ |
$2.984574000$ |
1.717135589 |
\( \frac{69390538551526}{3} a^{3} + \frac{7310207971123}{3} a^{2} - 182347992864521 a - 62777053162061 \) |
\( \bigl[\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 1\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 2\) , \( a\) , \( 18 a^{3} - 65 a^{2} + 34 a + 12\) , \( \frac{701}{3} a^{3} - \frac{2560}{3} a^{2} + 416 a + 253\bigr] \) |
${y}^2+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a-1\right){x}{y}+a{y}={x}^{3}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+2a-2\right){x}^{2}+\left(18a^{3}-65a^{2}+34a+12\right){x}+\frac{701}{3}a^{3}-\frac{2560}{3}a^{2}+416a+253$ |
| 27.2-d1 |
27.2-d |
$2$ |
$3$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{3} \) |
$18.99423$ |
$(-1/3a^3-1/3a^2+3a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$9$ |
\( 1 \) |
$1$ |
$21.79844721$ |
1.393494589 |
\( \frac{69390538551526}{3} a^{3} + \frac{7310207971123}{3} a^{2} - 182347992864521 a - 62777053162061 \) |
\( \bigl[a^{2} - 3\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 4 a - 2\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a - 1\) , \( \frac{19}{3} a^{3} + \frac{1}{3} a^{2} - 63 a - 29\) , \( 7 a^{3} - 18 a^{2} - 128 a - 55\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-3a-1\right){y}={x}^{3}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-4a-2\right){x}^{2}+\left(\frac{19}{3}a^{3}+\frac{1}{3}a^{2}-63a-29\right){x}+7a^{3}-18a^{2}-128a-55$ |
| 27.2-d2 |
27.2-d |
$2$ |
$3$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{9} \) |
$18.99423$ |
$(-1/3a^3-1/3a^2+3a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$196.1860248$ |
1.393494589 |
\( \frac{18512}{3} a^{3} + \frac{1769}{3} a^{2} - 48583 a - 15892 \) |
\( \bigl[\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a\) , \( -a^{2} + 5\) , \( a\) , \( -a^{2} - a + 9\) , \( \frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 2 a + 5\bigr] \) |
${y}^2+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+5\right){x}^{2}+\left(-a^{2}-a+9\right){x}+\frac{1}{3}a^{3}-\frac{2}{3}a^{2}-2a+5$ |
| 27.2-e1 |
27.2-e |
$1$ |
$1$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{5} \) |
$18.99423$ |
$(-1/3a^3-1/3a^2+3a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1.078952824$ |
$99.31703355$ |
3.044552071 |
\( \frac{6364868}{3} a^{3} + \frac{8738927}{3} a^{2} - 8598468 a - 3255814 \) |
\( \bigl[a^{2} - a - 4\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + a - 2\) , \( a^{2} - a - 4\) , \( -\frac{4}{3} a^{3} + \frac{20}{3} a^{2} - 2 a - 21\) , \( \frac{17}{3} a^{3} + \frac{8}{3} a^{2} - 66 a + 28\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+a-2\right){x}^{2}+\left(-\frac{4}{3}a^{3}+\frac{20}{3}a^{2}-2a-21\right){x}+\frac{17}{3}a^{3}+\frac{8}{3}a^{2}-66a+28$ |
| 27.2-f1 |
27.2-f |
$2$ |
$3$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{3} \) |
$18.99423$ |
$(-1/3a^3-1/3a^2+3a+2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1.962322637$ |
$594.7379778$ |
3.684264377 |
\( \frac{2272}{3} a^{3} + \frac{1171}{3} a^{2} - 5297 a - 3398 \) |
\( \bigl[-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 2\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 1\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 1\) , \( 0\bigr] \) |
${y}^2+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-3\right){x}{y}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a-1\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-2\right){x}^{2}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a-1\right){x}$ |
| 27.2-f2 |
27.2-f |
$2$ |
$3$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{9} \) |
$18.99423$ |
$(-1/3a^3-1/3a^2+3a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$5.886967913$ |
$7.342444170$ |
3.684264377 |
\( -\frac{2659993414}{3} a^{3} + \frac{3059637059}{3} a^{2} + 7384788197 a - 8211399121 \) |
\( \bigl[-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 2\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 1\) , \( -\frac{19}{3} a^{3} - \frac{19}{3} a^{2} + 33 a + 9\) , \( -\frac{67}{3} a^{3} - \frac{124}{3} a^{2} + 59 a + 22\bigr] \) |
${y}^2+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-3\right){x}{y}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a-1\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-2\right){x}^{2}+\left(-\frac{19}{3}a^{3}-\frac{19}{3}a^{2}+33a+9\right){x}-\frac{67}{3}a^{3}-\frac{124}{3}a^{2}+59a+22$ |
*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.
