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
39.1-a1
39.1-a
$8$
$28$
5.5.36497.1
$5$
$[5, 0]$
39.1
\( 3 \cdot 13 \)
\( - 3 \cdot 13^{7} \)
$24.62485$
$(a^2-1), (a^3-3a)$
0
$\Z/4\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 7$
2B , 7B.6.3
$1$
\( 7 \)
$1$
$416.4504504$
0.953702256
\( \frac{2355906526241759285697134}{188245551} a^{4} - \frac{2787719689830134456132135}{62748517} a^{3} + \frac{1984187604735214198024472}{62748517} a^{2} + \frac{2481423171042881641939612}{188245551} a - \frac{1509608417930564685187136}{188245551} \)
\( \bigl[a^{4} - 2 a^{3} - 2 a^{2} + 4 a\) , \( -a^{4} + 2 a^{3} + 4 a^{2} - 7 a - 3\) , \( 2 a^{4} - 3 a^{3} - 7 a^{2} + 7 a + 3\) , \( -10 a^{4} - 442 a^{3} + 162 a^{2} + 997 a - 537\) , \( 1117 a^{4} + 6433 a^{3} - 3401 a^{2} - 13458 a + 5199\bigr] \)
${y}^2+\left(a^{4}-2a^{3}-2a^{2}+4a\right){x}{y}+\left(2a^{4}-3a^{3}-7a^{2}+7a+3\right){y}={x}^{3}+\left(-a^{4}+2a^{3}+4a^{2}-7a-3\right){x}^{2}+\left(-10a^{4}-442a^{3}+162a^{2}+997a-537\right){x}+1117a^{4}+6433a^{3}-3401a^{2}-13458a+5199$
39.1-b1
39.1-b
$8$
$28$
5.5.36497.1
$5$
$[5, 0]$
39.1
\( 3 \cdot 13 \)
\( - 3 \cdot 13^{7} \)
$24.62485$
$(a^2-1), (a^3-3a)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 7$
2B , 7B.1.3
$2401$
\( 1 \)
$1$
$0.477173882$
1.49927138
\( \frac{2355906526241759285697134}{188245551} a^{4} - \frac{2787719689830134456132135}{62748517} a^{3} + \frac{1984187604735214198024472}{62748517} a^{2} + \frac{2481423171042881641939612}{188245551} a - \frac{1509608417930564685187136}{188245551} \)
\( \bigl[2 a^{4} - 3 a^{3} - 6 a^{2} + 7 a + 1\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 2 a^{4} - 3 a^{3} - 6 a^{2} + 7 a + 2\) , \( -324 a^{4} + 554 a^{3} + 1092 a^{2} - 1175 a - 803\) , \( -4619 a^{4} + 7541 a^{3} + 16214 a^{2} - 16147 a - 11168\bigr] \)
${y}^2+\left(2a^{4}-3a^{3}-6a^{2}+7a+1\right){x}{y}+\left(2a^{4}-3a^{3}-6a^{2}+7a+2\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(-324a^{4}+554a^{3}+1092a^{2}-1175a-803\right){x}-4619a^{4}+7541a^{3}+16214a^{2}-16147a-11168$
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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.