Properties

Label 2.2.89.1-4.1-a3
Base field \(\Q(\sqrt{89}) \)
Conductor norm \( 4 \)
CM no
Base change no
Q-curve yes
Torsion order \( 1 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{89}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 22 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-22, -1, 1]))
 
gp: K = nfinit(Polrev([-22, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-22, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(9584a-49997\right){x}+1212619a-6326231\)
sage: E = EllipticCurve([K([1,0]),K([1,-1]),K([0,1]),K([-49997,9584]),K([-6326231,1212619])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([1,-1]),Polrev([0,1]),Polrev([-49997,9584]),Polrev([-6326231,1212619])], K);
 
magma: E := EllipticCurve([K![1,0],K![1,-1],K![0,1],K![-49997,9584],K![-6326231,1212619]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2)\) = \((a+4)\cdot(a-5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 4 \) = \(2\cdot2\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-806464a+4198272)\) = \((a+4)^{30}\cdot(a-5)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 68719476736 \) = \(2^{30}\cdot2^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1224782159965}{1073741824} a - \frac{6388780435367}{1073741824} \)
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: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 2.4757843733063484590685257390952404353 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(1\)
Leading coefficient: \( 1.0497304748230415573126749691242586821 \)
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\((j)_{-}\)
\((a+4)\) \(2\) \(2\) \(I_{30}\) Non-split multiplicative \(1\) \(1\) \(30\) \(30\)
\((a-5)\) \(2\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.2
\(5\) 5B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 5 and 15.
Its isogeny class 4.1-a consists of curves linked by isogenies of degrees dividing 15.

Base change

This elliptic curve is a \(\Q\)-curve.

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