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
40.1-a1
40.1-a
$1$
$1$
3.3.169.1
$3$
$[3, 0]$
40.1
\( 2^{3} \cdot 5 \)
\( - 2^{3} \cdot 5^{5} \)
$2.14829$
$(-a^2+2a+3), (2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$1$
\( 1 \)
$1$
$14.96312317$
1.151009474
\( -\frac{53626624272}{3125} a^{2} - \frac{177085593453}{6250} a - \frac{20228557469}{3125} \)
\( \bigl[a + 1\) , \( -a^{2} + 2 a + 2\) , \( a^{2} - a - 2\) , \( -a^{2} - 3 a\) , \( -3 a^{2} - 6 a - 3\bigr] \)
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(-a^{2}-3a\right){x}-3a^{2}-6a-3$
40.1-b1
40.1-b
$2$
$3$
3.3.169.1
$3$
$[3, 0]$
40.1
\( 2^{3} \cdot 5 \)
\( - 2^{3} \cdot 5^{3} \)
$2.14829$
$(-a^2+2a+3), (2)$
0
$\Z/3\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B.1.1
$1$
\( 3 \)
$1$
$48.74212197$
1.249797999
\( \frac{4300451}{250} a^{2} - \frac{3024513}{250} a - \frac{11109599}{125} \)
\( \bigl[1\) , \( a^{2} - 3\) , \( a^{2} - a - 3\) , \( 12 a^{2} - 14 a - 42\) , \( -42 a^{2} + 54 a + 154\bigr] \)
${y}^2+{x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(12a^{2}-14a-42\right){x}-42a^{2}+54a+154$
40.1-b2
40.1-b
$2$
$3$
3.3.169.1
$3$
$[3, 0]$
40.1
\( 2^{3} \cdot 5 \)
\( - 2^{9} \cdot 5 \)
$2.14829$
$(-a^2+2a+3), (2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3$
3B.1.2
$9$
\( 1 \)
$1$
$1.805263776$
1.249797999
\( -\frac{590283258329401}{40} a^{2} + \frac{1403221722682083}{40} a + \frac{428609886619833}{40} \)
\( \bigl[1\) , \( a^{2} - 3\) , \( a^{2} - a - 3\) , \( -48 a^{2} + 56 a + 163\) , \( -133 a^{2} + 143 a + 446\bigr] \)
${y}^2+{x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-48a^{2}+56a+163\right){x}-133a^{2}+143a+446$
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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.