Properties

Label 3.3.81.1-17.3-a4
Base field \(\Q(\zeta_{9})^+\)
Conductor norm \( 17 \)
CM no
Base change no
Q-curve no
Torsion order \( 12 \)
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

\({y}^2+\left(a^{2}-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(98a^{2}-19a-307\right){x}-668a^{2}+195a+1990\)
sage: E = EllipticCurve([K([-2,0,1]),K([1,-1,-1]),K([0,1,0]),K([-307,-19,98]),K([1990,195,-668])])
 
gp: E = ellinit([Polrev([-2,0,1]),Polrev([1,-1,-1]),Polrev([0,1,0]),Polrev([-307,-19,98]),Polrev([1990,195,-668])], K);
 
magma: E := EllipticCurve([K![-2,0,1],K![1,-1,-1],K![0,1,0],K![-307,-19,98],K![1990,195,-668]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^2-2a-3)\) = \((a^2-2a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 17 \) = \(17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((2a^2-5a+14)\) = \((a^2-2a-3)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 4913 \) = \(17^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{127953099}{4913} a^{2} - \frac{170514783}{4913} a - \frac{127470024}{4913} \)
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: \(\Z/12\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(4 a^{2} - 2 a - 9 : -21 a^{2} + 7 a + 59 : 1\right)$
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: \( 219.13785712855616235688770900708778259 \)
Tamagawa product: \( 3 \)
Torsion order: \(12\)
Leading coefficient: \( 0.50726355816795407952983265973862912638 \)
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\((j)_{-}\)
\((a^2-2a-3)\) \(17\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

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
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 17.3-a consists of curves linked by isogenies of degrees dividing 12.

Base change

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

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