| 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 |
| 26.4-a1 |
26.4-a |
$4$ |
$6$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
26.4 |
\( 2 \cdot 13 \) |
\( - 2^{6} \cdot 13^{6} \) |
$12.91321$ |
$(a), (a^2+a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.235921893$ |
$129.3290458$ |
3.807339941 |
\( \frac{13179827674651809978387}{38614472} a^{3} + \frac{28149201874182484162433}{38614472} a^{2} - \frac{5778658371471464319445}{38614472} a - \frac{12341938383390744470007}{38614472} \) |
\( \bigl[a^{3} - 4 a + 1\) , \( a^{3} - 5 a + 1\) , \( a^{2} + a - 3\) , \( -82 a^{3} - 185 a^{2} + 21 a + 87\) , \( 1241 a^{3} + 2655 a^{2} - 538 a - 1169\bigr] \) |
${y}^2+\left(a^{3}-4a+1\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{3}-5a+1\right){x}^{2}+\left(-82a^{3}-185a^{2}+21a+87\right){x}+1241a^{3}+2655a^{2}-538a-1169$ |
| 26.4-a2 |
26.4-a |
$4$ |
$6$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
26.4 |
\( 2 \cdot 13 \) |
\( 2^{12} \cdot 13^{3} \) |
$12.91321$ |
$(a), (a^2+a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.117960946$ |
$258.6580917$ |
3.807339941 |
\( -\frac{1703646625047}{70304} a^{3} + \frac{4889056993005}{140608} a^{2} + \frac{388971174577}{70304} a - \frac{1669374838571}{140608} \) |
\( \bigl[a^{3} - 4 a + 1\) , \( -a^{2} + a + 4\) , \( a^{3} - 4 a\) , \( -13 a^{3} - 20 a^{2} + 81 a + 46\) , \( -13 a^{3} + 76 a^{2} - 42 a - 130\bigr] \) |
${y}^2+\left(a^{3}-4a+1\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(-13a^{3}-20a^{2}+81a+46\right){x}-13a^{3}+76a^{2}-42a-130$ |
| 26.4-a3 |
26.4-a |
$4$ |
$6$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
26.4 |
\( 2 \cdot 13 \) |
\( 2^{4} \cdot 13 \) |
$12.91321$ |
$(a), (a^2+a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.039320315$ |
$2327.922825$ |
3.807339941 |
\( \frac{17731}{26} a^{3} + \frac{3693}{52} a^{2} - \frac{91941}{26} a + \frac{28501}{52} \) |
\( \bigl[a + 1\) , \( a^{3} - 4 a\) , \( a^{3} - 4 a\) , \( -a^{2} + 2 a + 2\) , \( a^{2} - 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a^{3}-4a\right){x}^{2}+\left(-a^{2}+2a+2\right){x}+a^{2}-2$ |
| 26.4-a4 |
26.4-a |
$4$ |
$6$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
26.4 |
\( 2 \cdot 13 \) |
\( - 2^{2} \cdot 13^{2} \) |
$12.91321$ |
$(a), (a^2+a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.078640631$ |
$1163.961412$ |
3.807339941 |
\( -\frac{305812261}{338} a^{3} - \frac{188599157}{338} a^{2} + \frac{1396429107}{338} a + \frac{918916579}{338} \) |
\( \bigl[a^{3} - 4 a + 1\) , \( a^{3} - 5 a + 1\) , \( a^{2} + a - 3\) , \( 3 a^{3} - 5 a^{2} - 19 a + 12\) , \( 3 a^{3} + a^{2} - 9 a + 5\bigr] \) |
${y}^2+\left(a^{3}-4a+1\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{3}-5a+1\right){x}^{2}+\left(3a^{3}-5a^{2}-19a+12\right){x}+3a^{3}+a^{2}-9a+5$ |
| 26.4-b1 |
26.4-b |
$4$ |
$6$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
26.4 |
\( 2 \cdot 13 \) |
\( - 2^{6} \cdot 13^{6} \) |
$12.91321$ |
$(a), (a^2+a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.048583304$ |
$14.24381837$ |
1.863745901 |
\( \frac{13179827674651809978387}{38614472} a^{3} + \frac{28149201874182484162433}{38614472} a^{2} - \frac{5778658371471464319445}{38614472} a - \frac{12341938383390744470007}{38614472} \) |
\( \bigl[a + 1\) , \( a^{3} + a^{2} - 5 a - 4\) , \( 1\) , \( 92 a^{3} + 43 a^{2} - 427 a - 275\) , \( 353 a^{3} + 162 a^{2} - 1654 a - 1064\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a^{3}+a^{2}-5a-4\right){x}^{2}+\left(92a^{3}+43a^{2}-427a-275\right){x}+353a^{3}+162a^{2}-1654a-1064$ |
| 26.4-b2 |
26.4-b |
$4$ |
$6$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
26.4 |
\( 2 \cdot 13 \) |
\( 2^{12} \cdot 13^{3} \) |
$12.91321$ |
$(a), (a^2+a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$2.097166608$ |
$14.24381837$ |
1.863745901 |
\( -\frac{1703646625047}{70304} a^{3} + \frac{4889056993005}{140608} a^{2} + \frac{388971174577}{70304} a - \frac{1669374838571}{140608} \) |
\( \bigl[a + 1\) , \( -a^{2} - a + 3\) , \( a\) , \( -530 a^{3} + 349 a^{2} + 2418 a - 1598\) , \( -11558 a^{3} + 7652 a^{2} + 52724 a - 34910\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(-530a^{3}+349a^{2}+2418a-1598\right){x}-11558a^{3}+7652a^{2}+52724a-34910$ |
| 26.4-b3 |
26.4-b |
$4$ |
$6$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
26.4 |
\( 2 \cdot 13 \) |
\( 2^{4} \cdot 13 \) |
$12.91321$ |
$(a), (a^2+a-3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$0.699055536$ |
$1153.749288$ |
1.863745901 |
\( \frac{17731}{26} a^{3} + \frac{3693}{52} a^{2} - \frac{91941}{26} a + \frac{28501}{52} \) |
\( \bigl[a + 1\) , \( -a^{2} - a + 3\) , \( a\) , \( -15 a^{3} + 9 a^{2} + 68 a - 43\) , \( 16 a^{3} - 11 a^{2} - 73 a + 49\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(-15a^{3}+9a^{2}+68a-43\right){x}+16a^{3}-11a^{2}-73a+49$ |
| 26.4-b4 |
26.4-b |
$4$ |
$6$ |
4.4.9248.1 |
$4$ |
$[4, 0]$ |
26.4 |
\( 2 \cdot 13 \) |
\( - 2^{2} \cdot 13^{2} \) |
$12.91321$ |
$(a), (a^2+a-3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$0.349527768$ |
$1153.749288$ |
1.863745901 |
\( -\frac{305812261}{338} a^{3} - \frac{188599157}{338} a^{2} + \frac{1396429107}{338} a + \frac{918916579}{338} \) |
\( \bigl[a + 1\) , \( a^{3} + a^{2} - 5 a - 4\) , \( 1\) , \( 37 a^{3} + 23 a^{2} - 167 a - 110\) , \( -290 a^{3} - 191 a^{2} + 1321 a + 874\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a^{3}+a^{2}-5a-4\right){x}^{2}+\left(37a^{3}+23a^{2}-167a-110\right){x}-290a^{3}-191a^{2}+1321a+874$ |
*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.
