# Properties

 Label 27a4 Conductor 27 Discriminant -243 j-invariant $$-12288000$$ CM yes ($$D=-27$$) Rank 0 Torsion Structure $$\Z/{3}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 1, -30, 63]); // or
magma: E := EllipticCurve("27a4");
sage: E = EllipticCurve([0, 0, 1, -30, 63]) # or
sage: E = EllipticCurve("27a4")
gp: E = ellinit([0, 0, 1, -30, 63]) \\ or
gp: E = ellinit("27a4")

$$y^2 + y = x^{3} - 30 x + 63$$

## Mordell-Weil group structure

$$\Z/{3}\Z$$

## Torsion generators

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

$$\left(3, 0\right)$$

## Integral points

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

$$\left(3, 0\right)$$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$27$$ = $$3^{3}$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$-243$$ = $$-1 \cdot 3^{5}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$-12288000$$ = $$-1 \cdot 2^{15} \cdot 3 \cdot 5^{3}$$ Endomorphism ring: $$\Z[(1+\sqrt{-27})/2]$$ ( 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: $$5.29991625086$$ 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: $$1$$  = $$1$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] Torsion order: $$3$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form27.2.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)

$$q - 2q^{4} - q^{7} + 5q^{13} + 4q^{16} - 7q^{19} + O(q^{20})$$

#### Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
9 . This curve is not $$\Gamma_0(N)$$-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])

$$L(E,1)$$ ≈ $$0.588879583428$$

## 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$$(j)_{-}$$
$$3$$ $$1$$ $$IV$$ Additive -1 3 5 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
$$3$$ B.1.1

For all other primes $$p$$, the image is 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 ss add 0,5 - 0,0 -

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 $$d=$$ 3 and 9.
Its isogeny class 27.a consists of 4 curves linked by isogenies of degrees dividing 27.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.108.1 $$\Z/6\Z$$ Not in database
$$\Q(\zeta_{9})^+$$ $$\Z/9\Z$$ 3.3.81.1-27.1-a3
6 6.0.34992.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database
6.0.177147.2 $$\Z/3\Z \times \Z/3\Z$$ Not in database
6.0.177147.1 $$\Z/9\Z$$ 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.