Elliptic curve with LMFDB label 32.a3 (Cremona label 32a2)

• Label: 32.a3
• Conductor: $32$
• Discriminant: $64$
• j-invariant: \( 1728 \)
• CM: yes (\(D=-4\))
• Rank: $0$
• Torsion structure: \(\Z/{2}\Z \times \Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2=x^3-x\)  

Mordell-Weil group structure

\(\Z/{2}\Z \times \Z/{2}\Z\)

Torsion generators

\( \left(0, 0\right) \), \( \left(1, 0\right) \)  

Integral points

\( \left(-1, 0\right) \), \( \left(0, 0\right) \), \( \left(1, 0\right) \)  

Invariants

Conductor:	\( 32 \)	=	\(2^{5}\)
Discriminant:	\(64 \)	=	\(2^{6} \)
j-invariant:	\( 1728 \)	=	\(2^{6} \cdot 3^{3}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z[\sqrt{-1}]\)	(potential complex multiplication)
Sato-Tate group:	$N(\mathrm{U}(1))$
Faltings height:	\(-0.96395933563153686354365902258\dots\)
Stable Faltings height:	\(-1.3105329259115095182522750833\dots\)

BSD invariants

Analytic rank:	\(0\)
Regulator:	\(1\)
Real period:	\(5.2441151085842396209296791798\dots\)
Tamagawa product:	\( 2 \)  = \( 2 \)
Torsion order:	\(4\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form   32.2.a.a

\( q - 2q^{5} - 3q^{9} + 6q^{13} + 2q^{17} + O(q^{20}) \)  

For more coefficients, see the Downloads section to the right.

Modular degree:	2
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	2

Special L-value

\( L(E,1) \) ≈ \( 0.65551438857302995261620989747275764966 \)

Local data

This elliptic curve is not semistable. There is only one prime of bad reduction:

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(2\)	\(III\)	Additive	-1	5	6	0

Galois representations

The mod \( p \) Galois representation has maximal image for all primes \( p < 1000 \) except those listed.

prime	Image of Galois representation
\(2\)	Cs

For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -1 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -1 }{p}\right)=-1\).

$p$-adic data

$p$-adic regulators

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$	2
Reduction type	add
$\lambda$-invariant(s)	-
$\mu$-invariant(s)	-

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 32.a consists of 2 curves linked by isogenies of degrees dividing 4.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$2$	\(\Q(\sqrt{-1}) \)	\(\Z/2\Z \times \Z/4\Z\)	2.0.4.1-64.1-CMa1
$2$	\(\Q(\sqrt{2}) \)	\(\Z/2\Z \times \Z/4\Z\)	2.2.8.1-32.1-a4
$4$	\(\Q(\zeta_{8})\)	\(\Z/4\Z \times \Z/4\Z\)	Not in database
$8$	8.0.4194304.1	\(\Z/4\Z \times \Z/8\Z\)	Not in database
$8$	8.4.67108864.1	\(\Z/2\Z \times \Z/8\Z\)	Not in database
$8$	8.2.143327232.1	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$8$	8.0.8192000.1	\(\Z/2\Z \times \Z/20\Z\)	Not in database
$16$	16.0.18014398509481984.1	\(\Z/8\Z \times \Z/8\Z\)	Not in database
$16$	16.0.20542695432781824.1	\(\Z/6\Z \times \Z/12\Z\)	Not in database
$16$	16.4.1048576000000000000.1	\(\Z/2\Z \times \Z/10\Z\)	Not in database
$16$	16.4.5258930030792146944.1	\(\Z/2\Z \times \Z/12\Z\)	Not in database
$16$	16.0.17179869184000000.1	\(\Z/4\Z \times \Z/20\Z\)	Not in database

We only show fields where the torsion growth is primitive.