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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Show commands: Magma / Pari/GP / SageMath

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

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 1, 0, 0])
gp: E = ellinit([0, 0, 1, 0, 0])
magma: E := EllipticCurve([0, 0, 1, 0, 0]);

\(y^2+y=x^3\) Copy content Toggle raw display

Mordell-Weil group structure


Torsion generators

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

\( \left(0, 0\right) \) Copy content Toggle raw display

Integral points

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

\( \left(0, 0\right) \), \( \left(0, -1\right) \) Copy content Toggle raw display


sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
Conductor: \( 27 \)  =  $3^{3}$
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
Discriminant: $-27 $  =  $-1 \cdot 3^{3} $
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
j-invariant: \( 0 \)  =  $0$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z[(1+\sqrt{-3})/2]\) (potential complex multiplication)
Sato-Tate group: $N(\mathrm{U}(1))$
Faltings height: $-1.0464643562610104921410578866\dots$
Stable Faltings height: $-1.3211174284280379149898691958\dots$

BSD invariants

sage: E.rank()
magma: Rank(E);
Analytic rank: $0$
sage: E.regulator()
magma: Regulator(E);
Regulator: $1$
sage: E.period_lattice().omega()
magma: RealPeriod(E);
Real period: $5.2999162508563498719410684990\dots$
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 $
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)
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);
Special value: $ L(E,1) $ ≈ $ 0.58887958342848331910456316655 $

Modular invariants

Modular form   27.2.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);

\( q - 2 q^{4} - q^{7} + 5 q^{13} + 4 q^{16} - 7 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
magma: ModularDegree(E);
Modular degree: 3
$ \Gamma_0(N) $-optimal: no
Manin constant: 3

Local data

This elliptic curve is not semistable. There is only one prime 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$(j)_{-}$
$3$ $1$ $II$ Additive -1 3 3 0

Galois representations

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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$3$ 3Cs.1.1 27.1944.55.31

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 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
$2$ \(\Q(\sqrt{-3}) \) \(\Z/3\Z \times \Z/3\Z\)
$3$ \(\Z/6\Z\) Not in database
$3$ \(\Q(\zeta_{9})^+\) \(\Z/9\Z\)
$6$ 6.0.34992.1 \(\Z/6\Z \times \Z/6\Z\) Not in database
$6$ \(\Q(\zeta_{9})\) \(\Z/3\Z \times \Z/9\Z\) Not in database
$9$ 9.3.918330048.1 \(\Z/18\Z\) Not in database
$12$ 12.2.15045919506432.1 \(\Z/12\Z\) Not in database
$12$ 12.0.241162079949.1 \(\Z/3\Z \times \Z/21\Z\) Not in database
$18$ 18.0.4052555153018976267.1 \(\Z/9\Z \times \Z/9\Z\) Not in database
$18$ 18.0.2529990231179046912.1 \(\Z/6\Z \times \Z/18\Z\) Not in database

We only show fields where the torsion growth is primitive.

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 $(X:Y:Z) = (X_1^3 : Y_1^3 : X_1 Y_1 Z_1)$.