# Properties

 Label 54.a1 Conductor $54$ Discriminant $-90699264$ j-invariant $$-\frac{1167051}{512}$$ CM no Rank $0$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -123, -667]) # or

sage: E = EllipticCurve("54.a1")

gp: E = ellinit([1, -1, 0, -123, -667]) \\ or

gp: E = ellinit("54.a1")

magma: E := EllipticCurve([1, -1, 0, -123, -667]); // or

magma: E := EllipticCurve("54.a1");

$$y^2+xy=x^3-x^2-123x-667$$

trivial

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$54$$ = $$2 \cdot 3^{3}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-90699264$$ = $$-1 \cdot 2^{9} \cdot 3^{11}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{1167051}{512}$$ = $$-1 \cdot 2^{-9} \cdot 3 \cdot 73^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$0$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.70157482532401024962405804753$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$1$$  = $$1\cdot1$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$1$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

magma: ModularForm(E);

$$q - q^{2} + q^{4} + 3q^{5} - q^{7} - q^{8} - 3q^{10} - 3q^{11} - 4q^{13} + q^{14} + q^{16} + 2q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 18 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### Special L-value

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar[2]/factorial(ar[1])

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

$$L(E,1)$$ ≈ $$0.70157482532401024962405804752704343244$$

## Local data

This elliptic curve is semistable. There are 2 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)[5]

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

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$1$$ $$I_{9}$$ Non-split multiplicative 1 1 9 9
$$3$$ $$1$$ $$II^{*}$$ Additive -1 3 11 0

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

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

The mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$3$$ B.1.2

## $p$-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 nonsplit add 1 - 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 54.a consists of 3 curves linked by isogenies of degrees dividing 9.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-3})$$ $$\Z/9\Z$$ 2.0.3.1-324.1-a4 $3$ 3.1.216.1 $$\Z/2\Z$$ Not in database $3$ 3.1.243.1 $$\Z/3\Z$$ Not in database $6$ 6.0.1119744.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.0.177147.2 $$\Z/3\Z \times \Z/9\Z$$ Not in database $6$ 6.0.139968.1 $$\Z/18\Z$$ Not in database $9$ 9.1.198359290368.1 $$\Z/6\Z$$ Not in database $12$ 12.2.1925877696823296.2 $$\Z/4\Z$$ Not in database $12$ 12.0.1253826625536.1 $$\Z/2\Z \times \Z/18\Z$$ Not in database $18$ 18.0.16599265906765726789632.7 $$\Z/27\Z$$ Not in database $18$ 18.0.118039224225889612726272.3 $$\Z/3\Z \times \Z/18\Z$$ Not in database $18$ 18.0.60436082803655481715851264.2 $$\Z/2\Z \times \Z/6\Z$$ Not in database

We only show fields where the torsion growth is primitive.