Elliptic curve with Cremona label 64a3 (LMFDB label 64.a2)

• Label: 64a3
• Conductor: $64$
• Discriminant: $32768$
• j-invariant: $287496$
• CM: yes ($D=-16$)
• Rank: $0$
• Torsion structure: $\Z/{4}\Z$

Minimal Weierstrass equation

$y^2=x^3-44x+112$

(, )

Mordell-Weil group structure

$\Z/{4}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(6, 8)$	$0$	$4$

Integral points

$\left(4, 0\right)$, $(6,\pm 8)$

Invariants

Conductor:	$N$	=	$64$	=	$2^{6}$
Discriminant:	$\Delta$	=	$32768$	=	$2^{15}$
j-invariant:	$j$	=	$287496$	=	$2^{3} \cdot 3^{3} \cdot 11^{3}$
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z[\sqrt{-4}]$    (potential complex multiplication)
Sato-Tate group:	$\mathrm{ST}(E)$	=	$N(\mathrm{U}(1))$
Faltings height:	$h_{\mathrm{Faltings}}$	≈	$-0.27081215507159155412642690112$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-1.1372461307715231908979670529$
$abc$ quality:	$Q$	≈	$1.172456969504371$
Szpiro ratio:	$\sigma_{m}$	≈	$5.522197059679227$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$
Real period:	$\Omega$	≈	$3.7081493546027438368677006944$
Tamagawa product:	$\prod_{p}c_p$	=	$4$  = $2^{2}$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$4$
Special value:	$L(E,1)$	≈	$0.92703733865068595921692517360$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$    (exact)

BSD formula

$\displaystyle 0.927037339 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 3.708149 \cdot 1.000000 \cdot 4}{4^2} \approx 0.927037339$

Modular invariants

Modular form   64.2.a.a

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

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

Modular degree:	4
$\Gamma_0(N)$-optimal:	no
Manin constant:	1

Local data at primes of bad reduction

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

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$4$	$I_{5}^{*}$	additive	-1	6	15	0

Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$	mod-$\ell$ image	$\ell$-adic image
$2$	2B	16.192.3.545

The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.

$\ell$	Reduction type	Serre weight	Serre conductor
$2$	additive	$4$	$1$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 64a consists of 4 curves linked by isogenies of degrees dividing 4.

Twists

The minimal quadratic twist of this elliptic curve is 32a3, its twist by $-8$.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$2$	$\Q(\sqrt{2})$	$\Z/2\Z \oplus \Z/4\Z$	2.2.8.1-32.1-a6
$4$	$\Q(\zeta_{16})^+$	$\Z/2\Z \oplus \Z/8\Z$	not in database
$8$	8.0.16777216.2	$\Z/4\Z \oplus \Z/4\Z$	not in database
$8$	8.0.4194304.1	$\Z/2\Z \oplus \Z/8\Z$	not in database
$8$	8.2.573308928.1	$\Z/12\Z$	not in database
$8$	8.0.32768000.1	$\Z/20\Z$	not in database
$16$	16.0.18014398509481984.1	$\Z/4\Z \oplus \Z/8\Z$	not in database
$16$	16.8.73786976294838206464.2	$\Z/2\Z \oplus \Z/16\Z$	not in database
$16$	16.0.288230376151711744.2	$\Z/2\Z \oplus \Z/16\Z$	not in database
$16$	16.0.328683126924509184.1	$\Z/3\Z \oplus \Z/12\Z$	not in database
$16$	16.4.16777216000000000000.2	$\Z/20\Z$	not in database
$16$	16.4.5258930030792146944.1	$\Z/2\Z \oplus \Z/12\Z$	not in database
$16$	16.0.17179869184000000.1	$\Z/2\Z \oplus \Z/20\Z$	not in database

We only show fields where the torsion growth is primitive.

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.

$p$-adic regulators

All $p$-adic regulators are identically $1$ since the rank is $0$.