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
29.2-a1
29.2-a
$4$
$4$
4.4.2225.1
$4$
$[4, 0]$
29.2
\( 29 \)
\( - 29^{4} \)
$6.42101$
$(a^2-5)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 2 \)
$1$
$144.4501657$
1.531168694
\( -\frac{344097316783618017}{707281} a^{3} + \frac{948538063540097546}{707281} a^{2} + \frac{54286274905112797}{707281} a - \frac{783553811115449441}{707281} \)
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - 3\) , \( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{5}{2} a\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - 2\) , \( \frac{25}{2} a^{3} + \frac{5}{2} a^{2} - \frac{115}{2} a - 42\) , \( -\frac{31}{2} a^{3} - \frac{3}{2} a^{2} + \frac{155}{2} a + 56\bigr] \)
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-3\right){x}{y}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-2\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{5}{2}a\right){x}^{2}+\left(\frac{25}{2}a^{3}+\frac{5}{2}a^{2}-\frac{115}{2}a-42\right){x}-\frac{31}{2}a^{3}-\frac{3}{2}a^{2}+\frac{155}{2}a+56$
29.2-a2
29.2-a
$4$
$4$
4.4.2225.1
$4$
$[4, 0]$
29.2
\( 29 \)
\( 29^{2} \)
$6.42101$
$(a^2-5)$
0
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2Cs
$1$
\( 2 \)
$1$
$577.8006629$
1.531168694
\( -\frac{176483079}{841} a^{3} + \frac{478215419}{841} a^{2} + \frac{40825114}{841} a - \frac{381703735}{841} \)
\( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - 3\) , \( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{5}{2} a\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - 2\) , \( 5 a^{3} - 25 a - 17\) , \( 13 a^{3} + 2 a^{2} - 62 a - 45\bigr] \)
${y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-3\right){x}{y}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-2\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{5}{2}a\right){x}^{2}+\left(5a^{3}-25a-17\right){x}+13a^{3}+2a^{2}-62a-45$
29.2-a3
29.2-a
$4$
$4$
4.4.2225.1
$4$
$[4, 0]$
29.2
\( 29 \)
\( 29 \)
$6.42101$
$(a^2-5)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 1 \)
$1$
$288.9003314$
1.531168694
\( \frac{82412968}{29} a^{3} - \frac{122524715}{29} a^{2} - \frac{272020154}{29} a + \frac{346907209}{29} \)
\( \bigl[a^{2} - 3\) , \( -\frac{1}{2} a^{3} + \frac{1}{2} a^{2} + \frac{3}{2} a + 1\) , \( a + 1\) , \( -3 a^{2} + 4 a + 5\) , \( \frac{7}{2} a^{3} - \frac{21}{2} a^{2} + \frac{1}{2} a + 8\bigr] \)
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}+\frac{3}{2}a+1\right){x}^{2}+\left(-3a^{2}+4a+5\right){x}+\frac{7}{2}a^{3}-\frac{21}{2}a^{2}+\frac{1}{2}a+8$
29.2-a4
29.2-a
$4$
$4$
4.4.2225.1
$4$
$[4, 0]$
29.2
\( 29 \)
\( -29 \)
$6.42101$
$(a^2-5)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 1 \)
$1$
$288.9003314$
1.531168694
\( -\frac{3095615715}{29} a^{3} - \frac{428869366}{29} a^{2} + \frac{14989828975}{29} a + \frac{10875578157}{29} \)
\( \bigl[\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a\) , \( -a^{2} + 2\) , \( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a\) , \( 3 a^{3} - a^{2} - 13 a - 6\) , \( \frac{9}{2} a^{3} + \frac{1}{2} a^{2} - \frac{43}{2} a - 16\bigr] \)
${y}^2+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{3}{2}a\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(3a^{3}-a^{2}-13a-6\right){x}+\frac{9}{2}a^{3}+\frac{1}{2}a^{2}-\frac{43}{2}a-16$
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Pari/GP
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*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.