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 $1$
$1$
4.4.19821.1 $4$
$[4, 0]$
39.1
\( 3 \cdot 13 \)
\( - 3^{9} \cdot 13 \)
$19.88769$
$(-1/3a^3-1/3a^2+3a+2), (-2/3a^3+1/3a^2+5a)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
$1$
\( 1 \)
$1$
$165.0993253$
1.172688099
\( \frac{111853568}{3159} a^{3} + \frac{12132352}{3159} a^{2} - \frac{882774016}{3159} a - \frac{100966400}{1053} \)
\( \bigl[0\) , \( -1\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 1\) , \( \frac{4}{3} a^{3} - \frac{14}{3} a^{2} + 2 a + 2\) , \( -\frac{4}{3} a^{3} + \frac{14}{3} a^{2} - 3 a - 2\bigr] \) ${y}^2+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a-1\right){y}={x}^{3}-{x}^{2}+\left(\frac{4}{3}a^{3}-\frac{14}{3}a^{2}+2a+2\right){x}-\frac{4}{3}a^{3}+\frac{14}{3}a^{2}-3a-2$
39.1-b1 39.1-b $1$
$1$
4.4.19821.1 $4$
$[4, 0]$
39.1
\( 3 \cdot 13 \)
\( - 3^{5} \cdot 13 \)
$19.88769$
$(-1/3a^3-1/3a^2+3a+2), (-2/3a^3+1/3a^2+5a)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
$1$
\( 5 \)
$0.257413700$
$156.3609513$
5.717776524
\( -\frac{3002368}{351} a^{3} + \frac{4784128}{351} a^{2} + \frac{610304}{351} a - \frac{966656}{117} \)
\( \bigl[0\) , \( a^{2} - 5\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a\) , \( \frac{17}{3} a^{3} - \frac{61}{3} a^{2} + 8 a + 13\) , \( \frac{112}{3} a^{3} - \frac{410}{3} a^{2} + 68 a + 38\bigr] \) ${y}^2+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-3a\right){y}={x}^{3}+\left(a^{2}-5\right){x}^{2}+\left(\frac{17}{3}a^{3}-\frac{61}{3}a^{2}+8a+13\right){x}+\frac{112}{3}a^{3}-\frac{410}{3}a^{2}+68a+38$
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Pari/GP
SageMath
Magma
Oscar
CSV
*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.
