Properties

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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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

\(\Z/{3}\Z\)

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.

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
\( N \)  =  \( 27 \)  =  \(3^{3}\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
\(\Delta\)  =  \(-27 \)  =  \(-1 \cdot 3^{3} \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
\(j \)  =  \( 0 \)  =  \(0\)
\( \text{End} (E) \)  =  \(\Z[(1+\sqrt{-3})/2]\)   (Complex Multiplication)
\( \text{ST} (E) \)  =  $N(\mathrm{U}(1))$

BSD invariants

magma: Rank(E);
sage: E.rank()
\( r \)  =  \(0\)
magma: Regulator(E);
sage: E.regulator()
\( \text{Reg} \)  =  \(1\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
\( \Omega \)  ≈  \(5.29991625086\)
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
\( \prod_p c_p \)  =  \( 1 \)  = \( 1 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
\( \#E_{\text{tor}} \)  = \(3\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Ш\(_{\text{an}} \)  =   \(1\) (exact)

Modular invariants

Modular form 27.2.1.a

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: 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.

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
3 : curve is not \( \Gamma_0(N) \)-optimal

Special L-value attached to the curve

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 representations of an elliptic curve with Complex Multiplication are non-surjective for all odd primes \( p \). We only show the image for primes \( p\le 37 \).

prime Image of Galois representation
\(3\) Cs.1.1
\(5\) Nn
\(7\) Ns
\(11\) Nn
\(13\) Ns
\(17\) Nn
\(19\) Ns
\(23\) Nn
\(29\) Nn
\(31\) Ns
\(37\) Ns

p-adic data

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 \(0\).

Isogenies

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.