Properties

 Label 144.a3 Conductor 144 Discriminant -432 j-invariant $$0$$ CM yes ($$D=-3$$) Rank 0 Torsion Structure $$\Z/{2}\Z$$

Learn more about

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 0, 0, -1]); // or
magma: E := EllipticCurve("144a1");
sage: E = EllipticCurve([0, 0, 0, 0, -1]) # or
sage: E = EllipticCurve("144a1")
gp: E = ellinit([0, 0, 0, 0, -1]) \\ or
gp: E = ellinit("144a1")

$$y^2 = x^{3} - 1$$

Mordell-Weil group structure

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

Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

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

Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

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

Note: only one of each pair $\pm P$ is listed.

Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$144$$ = $$2^{4} \cdot 3^{2}$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$-432$$ = $$-1 \cdot 2^{4} \cdot 3^{3}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$0$$ = $$0$$ Endomorphism ring: $$\Z[(1+\sqrt{-3})/2]$$ ( Complex Multiplication) Sato-Tate Group: $N(\mathrm{U}(1))$

BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$0$$ magma: Regulator(E); sage: E.regulator() Regulator: $$1$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$2.42865064789$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$2$$  = $$1\cdot2$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] Torsion order: $$2$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

Modular invariants

Modular form144.2.a.a

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

$$q + 4q^{7} + 2q^{13} - 8q^{19} + O(q^{20})$$

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

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
4 . This curve is $$\Gamma_0(N)$$-optimal.

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
sage: r = E.rank();
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: ar = ellanalyticrank(E);
gp: ar[2]/factorial(ar[1])

$$L(E,1)$$ ≈ $$1.21432532394$$

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)[5]
prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$1$$ $$II$$ Additive -1 4 4 0
$$3$$ $$2$$ $$III$$ Additive 1 2 3 0

Galois representations

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]

The mod $$p$$ Galois representation has maximal image for all primes $$p$$ except those listed.

prime Image of Galois representation
$$2$$ B

For all other primes $$p$$, the image is a Borel subgroup if $$p=3$$, the normalizer of a split Cartan subgroup if $$\left(\frac{ -3 }{p}\right)=+1$$ or the normalizer of a nonsplit Cartan subgroup if $$\left(\frac{ -3 }{p}\right)=-1$$.

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

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

Iwasawa invariants

$p$ Reduction type 2 3 add add - - - -

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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, 3 and 6.
Its isogeny class 144.a consists of 4 curves linked by isogenies of degrees dividing 6.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-3})$$ $$\Z/2\Z \times \Z/2\Z$$ 2.0.3.1-2304.1-CMa1
$$\Q(\sqrt{-1})$$ $$\Z/6\Z$$ 2.0.4.1-324.1-a1
4 4.2.1728.1 $$\Z/4\Z$$ Not in database
$$\Q(\zeta_{12})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
6 6.2.559872.1 $$\Z/6\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.