Properties

Label 54.a2
Conductor $54$
Discriminant $-54$
j-invariant \( -\frac{132651}{2} \)
CM no
Rank $0$
Torsion structure \(\Z/{3}\Z\)

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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)[1]
 
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[1]
 
magma: RealPeriod(E);
 
Real period: \(6.3141734279160922466165224278\)
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: \(3\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form   54.2.a.a

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);
 

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

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

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[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\((j)_{-}\)
\(2\) \(1\) \(I_{1}\) 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

$p$ 2 3
Reduction type nonsplit add
$\lambda$-invariant(s) 1 -
$\mu$-invariant(s) 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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 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.