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

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This is a model for the curve $XY(X+Y)=Z^3$.

Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 1, 0, 0]); // or
magma: E := EllipticCurve("27a3");
sage: E = EllipticCurve([0, 0, 1, 0, 0]) # or
sage: E = EllipticCurve("27a3")
gp: E = ellinit([0, 0, 1, 0, 0]) \\ or
gp: E = ellinit("27a3")

\( y^2 + y = x^{3} \)

Mordell-Weil group structure


Torsion generators

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

\( \left(0, 0\right) \)

Integral points

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

\( \left(0, 0\right) \)

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


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: \(-27 \)  =  \(-1 \cdot 3^{3} \)
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()
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 form 27.2.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)

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

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

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 3
\( \Gamma_0(N) \)-optimal: no
Manin constant: not computed

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\((j)_{-}\)
\(3\) \(1\) \( II \) Additive -1 3 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
\(3\) Cs.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

$p$ 2 3
Reduction type ss add
$\lambda$-invariant(s) 0,5 -
$\mu$-invariant(s) 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.


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
2 \(\Q(\sqrt{-3}) \) \(\Z/3\Z \times \Z/3\Z\)
3 \(\Q(\zeta_{9})^+\) \(\Z/9\Z\) \(\Z/6\Z\) Not in database
6 \(\Q(\zeta_{9})\) \(\Z/3\Z \times \Z/9\Z\) Not in database
6.0.34992.1 \(\Z/6\Z \times \Z/6\Z\) 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.

Additional information

An explicit birational isomorphism from $XY(X+Y)=Z^3$ to $y^2+y = x^3$ is $(x,y) = (Y/X,Z/X)$. The $3$-isogeny from the Fermat cubic $X_1^3 + Y_1^3 = Z_1^3$ is simply $(X:Y:Z) = (X_1^3 : Y_1^3 : X_1 Y_1 Z_1)$.