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## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -3, 3]) # or

sage: E = EllipticCurve("54.a2")

gp: E = ellinit([1, -1, 0, -3, 3]) \\ or

gp: E = ellinit("54.a2")

magma: E := EllipticCurve([1, -1, 0, -3, 3]); // or

magma: E := EllipticCurve("54.a2");

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

## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

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

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

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

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$54$$ = $$2 \cdot 3^{3}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-54$$ = $$-1 \cdot 2 \cdot 3^{3}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{132651}{2}$$ = $$-1 \cdot 2^{-1} \cdot 3^{3} \cdot 17^{3}$$ 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  magma: RealPeriod(E); Real period: $$6.3141734279160922466165224278$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$1$$  = $$1\cdot1$$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $$3$$ 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)/(2*xy+E.a1*xy+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: 3

#### Special L-value

sage: r = E.rank();

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

gp: ar = ellanalyticrank(E);

gp: ar/factorial(ar)

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)

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

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$3$$ $$1$$ $$II$$ Additive -1 3 3 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.1.1

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.216.1 $$\Z/6\Z$$ Not in database $3$ $$\Q(\zeta_{9})^+$$ $$\Z/9\Z$$ 3.3.81.1-216.1-a1 $6$ 6.0.1119744.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $6$ 6.0.34992.1 $$\Z/3\Z \times \Z/3\Z$$ Not in database $6$ 6.0.314928.1 $$\Z/9\Z$$ Not in database $9$ 9.3.7346640384.1 $$\Z/18\Z$$ Not in database $12$ 12.2.1925877696823296.2 $$\Z/12\Z$$ Not in database $18$ 18.0.364318593289782755328.1 $$\Z/3\Z \times \Z/9\Z$$ Not in database $18$ 18.0.2529990231179046912.1 $$\Z/3\Z \times \Z/9\Z$$ Not in database $18$ 18.0.56860576059859402752.1 $$\Z/3\Z \times \Z/6\Z$$ Not in database $18$ 18.0.41451359947637504606208.1 $$\Z/18\Z$$ Not in database $18$ 18.0.82902719895275009212416.1 $$\Z/2\Z \times \Z/18\Z$$ Not in database

We only show fields where the torsion growth is primitive.