Properties

Label 3.3.81.1-51.1-a5
Base field \(\Q(\zeta_{9})^+\)
Conductor norm \( 51 \)
CM no
Base change no
Q-curve no
Torsion order \( 8 \)
Rank \( 0 \)

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

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

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

Weierstrass equation

\({y}^2+\left(a^{2}+a-1\right){x}{y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(-3a^{2}+17a+7\right){x}+17a^{2}-13a-7\)
sage: E = EllipticCurve([K([-1,1,1]),K([-1,1,1]),K([0,0,0]),K([7,17,-3]),K([-7,-13,17])])
 
gp: E = ellinit([Polrev([-1,1,1]),Polrev([-1,1,1]),Polrev([0,0,0]),Polrev([7,17,-3]),Polrev([-7,-13,17])], K);
 
magma: E := EllipticCurve([K![-1,1,1],K![-1,1,1],K![0,0,0],K![7,17,-3],K![-7,-13,17]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-3a^2+2a+4)\) = \((-a^2+1)\cdot(a^2-2a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 51 \) = \(3\cdot17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((9a^2-6a-12)\) = \((-a^2+1)^{4}\cdot(a^2-2a-3)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1377 \) = \(3^{4}\cdot17\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{57227017}{153} a^{2} - \frac{97672967}{153} a - \frac{40772771}{153} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(0\)
Torsion structure: \(\Z/8\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-a^{2} + 1 : -2 a^{2} + 5 a + 3 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 188.98585662041989995365341835254009681 \)
Tamagawa product: \( 2 \)  =  \(2\cdot1\)
Torsion order: \(8\)
Leading coefficient: \( 0.65620089104312465261685214705743089170 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

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

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 51.1-a consists of curves linked by isogenies of degrees dividing 8.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.