# Properties

 Label 64.a3 Conductor $64$ Discriminant $4096$ j-invariant $$1728$$ CM yes ($$D=-4$$) Rank $0$ Torsion structure $$\Z/{2}\Z \oplus \Z/{2}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2=x^3-4x$$ y^2=x^3-4x (homogenize, simplify) $$y^2z=x^3-4xz^2$$ y^2z=x^3-4xz^2 (dehomogenize, simplify) $$y^2=x^3-4x$$ y^2=x^3-4x (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, 0, 0, -4, 0])

gp: E = ellinit([0, 0, 0, -4, 0])

magma: E := EllipticCurve([0, 0, 0, -4, 0]);

oscar: E = EllipticCurve([0, 0, 0, -4, 0])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z/{2}\Z \oplus \Z/{2}\Z$$

magma: MordellWeilGroup(E);

## Torsion generators

$$\left(0, 0\right)$$, $$\left(2, 0\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(-2, 0\right)$$, $$\left(0, 0\right)$$, $$\left(2, 0\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$64$$ = $2^{6}$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $4096$ = $2^{12}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$1728$$ = $2^{6} \cdot 3^{3}$ comment: j-invariant  sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E);  oscar: j_invariant(E) Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z[\sqrt{-1}]$$ (potential complex multiplication) sage: E.has_cm()  magma: HasComplexMultiplication(E); Sato-Tate group: $N(\mathrm{U}(1))$ Faltings height: $-0.61738574535156420883504296185\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-1.3105329259115095182522750833\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $\dots$ Szpiro ratio: $3.792481250360578\dots$

## BSD invariants

 Analytic rank: $0$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $1$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $3.7081493546027438368677006944\dots$ comment: Real Period  sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Tamagawa product: $4$  = $2^{2}$ comment: Tamagawa numbers  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E);  oscar: tamagawa_numbers(E) Torsion order: $4$ comment: Torsion order  sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E));  oscar: prod(torsion_structure(E)[1]) Analytic order of Ш: $1$ ( exact) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L(E,1)$ ≈ $0.92703733865068595921692517360$ comment: Special L-value  r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: [r,L1r] = ellanalyticrank(E); L1r/r!  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

## BSD formula

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

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)

E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;

Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();

omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();

assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))

/* self-contained Magma code snippet for the BSD formula (checks rank, computes analyiic sha) */

E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;

sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);

reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);

assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);

## Modular invariants

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

comment: q-expansion of modular form

sage: E.q_eigenform(20)

\\ actual modular form, use for small N

[mf,F] = mffromell(E)

Ser(mfcoefs(mf,20),q)

\\ or just the series

Ser(ellan(E,20),q)*q

magma: ModularForm(E);

Modular degree: 2
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: yes
Manin constant: 1
comment: Manin constant

magma: ManinConstant(E);

## 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$(j)_{-}$
$2$ $4$ $I_{2}^{*}$ Additive -1 6 12 0

comment: Local data

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]

## 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
$2$ 2Cs 16.384.9.462

comment: mod p Galois image

sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

## Isogenies

gp: ellisomat(E)

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

## Twists

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

The minimal quartic twist of this elliptic curve is 32.a3, its quartic twist by $4$.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \oplus \Z/4\Z$$ 2.2.8.1-32.1-a4 $4$ $$\Q(\zeta_{8})$$ $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.4194304.1 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $8$ 8.4.67108864.1 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.2.573308928.1 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $8$ 8.0.32768000.1 $$\Z/2\Z \oplus \Z/10\Z$$ Not in database $16$ 16.0.18014398509481984.1 $$\Z/8\Z \oplus \Z/8\Z$$ Not in database $16$ 16.0.328683126924509184.1 $$\Z/6\Z \oplus \Z/6\Z$$ Not in database $16$ 16.4.16777216000000000000.2 $$\Z/2\Z \oplus \Z/10\Z$$ Not in database $16$ 16.0.17179869184000000.1 $$\Z/4\Z \oplus \Z/20\Z$$ Not in database $16$ 16.4.5258930030792146944.1 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database

We only show fields where the torsion growth is primitive.

## Iwasawa invariants

$p$ 2 add - -

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 $0$.