Properties

Base field \(\Q(\zeta_{9})^+\)
Label 3.3.81.1-51.3-a3
Conductor \((51,a^{2} - 3 a - 4)\)
Conductor norm \( 51 \)
CM no
base-change no
Q-curve no
Torsion order \( 4 \)
Rank not available

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Base field \(\Q(\zeta_{9})^+\)

Generator \(a\), with minimal polynomial \( x^{3} - 3 x - 1 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^3 - 3*x - 1)
 
gp (2.8): K = nfinit(a^3 - 3*a - 1);
 

Weierstrass equation

\( y^2 + \left(a + 1\right) x y + \left(a^{2} - 2\right) y = x^{3} + \left(-a^{2} + a + 1\right) x^{2} + \left(-48 a^{2} - 38 a - 12\right) x - 255 a^{2} - 344 a - 86 \)
magma: E := ChangeRing(EllipticCurve([a + 1, -a^2 + a + 1, a^2 - 2, -48*a^2 - 38*a - 12, -255*a^2 - 344*a - 86]),K);
 
sage: E = EllipticCurve(K, [a + 1, -a^2 + a + 1, a^2 - 2, -48*a^2 - 38*a - 12, -255*a^2 - 344*a - 86])
 
gp (2.8): E = ellinit([a + 1, -a^2 + a + 1, a^2 - 2, -48*a^2 - 38*a - 12, -255*a^2 - 344*a - 86],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((51,a^{2} - 3 a - 4)\) = \( \left(-a^{2} + 1\right) \cdot \left(a^{2} + a - 3\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 51 \) = \( 3 \cdot 17 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((60886809,243 a + 47997117,243 a^{2} + 38042622)\) = \( \left(-a^{2} + 1\right)^{16} \cdot \left(a^{2} + a - 3\right)^{4} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 3595305184641 \) = \( 3^{16} \cdot 17^{4} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{479060149170145}{60886809} a^{2} - \frac{736214157382250}{60886809} a - \frac{304538278611719}{60886809} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp (2.8): E.j
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
magma: HasComplexMultiplication(E);
 
sage: E.has_cm(), E.cm_discriminant()
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.
magma: Rank(E);
 
sage: E.rank()
 
magma: Generators(E); // includes torsion
 
sage: E.gens()
 

Regulator: not available

magma: Regulator(Generators(E));
 
sage: E.regulator_of_points(E.gens())
 

Torsion subgroup

Structure: \(\Z/2\Z\times\Z/2\Z\)
magma: TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[2]
 
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp (2.8): elltors(E)[1]
 
Generators: $\left(a^{2} + 6 a + 3 : -4 a^{2} - 6 a - 1 : 1\right)$,$\left(a^{2} - 6 a - 1 : 2 a^{2} + 2 a + 1 : 1\right)$
magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[3]
 

Local data at primes of bad reduction

magma: LocalInformation(E);
 
sage: E.local_data()
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-a^{2} + 1\right) \) \(3\) \(2\) \(I_{16}\) Non-split multiplicative \(1\) \(1\) \(16\) \(16\)
\( \left(a^{2} + a - 3\right) \) \(17\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 51.3-a consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is not the base-change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.