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

Related objects


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Show commands: Magma / Oscar / PariGP / SageMath

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

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+y=x^3\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+yz^2=x^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3+16\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
sage: E = EllipticCurve([0, 0, 1, 0, 0])
gp: E = ellinit([0, 0, 1, 0, 0])
magma: E := EllipticCurve([0, 0, 1, 0, 0]);
oscar: E = EllipticCurve([0, 0, 1, 0, 0])
sage: E.short_weierstrass_model()
magma: WeierstrassModel(E);
oscar: short_weierstrass_model(E)

Mordell-Weil group structure


magma: MordellWeilGroup(E);

Torsion generators

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

comment: Torsion subgroup
sage: E.torsion_subgroup().gens()
gp: elltors(E)
magma: TorsionSubgroup(E);
oscar: torsion_structure(E)

Integral points

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

comment: Integral points
sage: E.integral_points()
magma: IntegralPoints(E);


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

BSD invariants

Analytic rank: $0$
sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
Regulator: $1$
comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
Real period: $5.2999162508563498719410684990\dots$
comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
Tamagawa product: $ 1 $
comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
Torsion order: $3$
comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
Analytic order of Ш: $1$ (exact)
comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
Special value: $ L(E,1) $ ≈ $ 0.58887958342848331910456316655 $
comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

BSD formula

$\displaystyle 0.588879583 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 5.299916 \cdot 1.000000 \cdot 1}{3^2} \approx 0.588879583$

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;
Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();
omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();
assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
/* self-contained Magma code snippet for the BSD formula (checks rank, computes analyiic sha) */
E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;
sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);
reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);
assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);

Modular invariants

Modular form   27.2.a.a

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

comment: q-expansion of modular form
sage: E.q_eigenform(20)
\\ actual modular form, use for small N
[mf,F] = mffromell(E)
\\ or just the series
magma: ModularForm(E);

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

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

Local data

This elliptic curve is not semistable. There is only one prime of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$3$ $1$ $II$ Additive -1 3 3 0

comment: Local data
sage: E.local_data()
gp: ellglobalred(E)[5]
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]

Galois representations

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

comment: mod p Galois image
sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];


gp: ellisomat(E)

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.


This elliptic curve is its own minimal quadratic twist.

This elliptic curve is its own minimal sextic twist.

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
$2$ \(\Q(\sqrt{-3}) \) \(\Z/3\Z \oplus \Z/3\Z\)
$3$ \(\Z/6\Z\) Not in database
$3$ \(\Q(\zeta_{9})^+\) \(\Z/9\Z\)
$6$ 6.0.34992.1 \(\Z/6\Z \oplus \Z/6\Z\) Not in database
$6$ \(\Q(\zeta_{9})\) \(\Z/3\Z \oplus \Z/9\Z\) 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 \(\Z/3\Z \oplus \Z/21\Z\) Not in database
$18$ 18.0.4052555153018976267.1 \(\Z/9\Z \oplus \Z/9\Z\) Not in database
$18$ 18.0.2529990231179046912.1 \(\Z/6\Z \oplus \Z/18\Z\) Not in database

We only show fields where the torsion growth is primitive.

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.

$p$-adic regulators

All $p$-adic regulators are identically $1$ since the rank is $0$.

Additional information

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