Properties

Label 2.2.93.1-124.1-h1
Base field \(\Q(\sqrt{93}) \)
Conductor norm \( 124 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{93}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 23 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-23, -1, 1]))
 
gp: K = nfinit(Polrev([-23, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-23, -1, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-63a-239\right){x}-2815a-12129\)
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([0,0]),K([-239,-63]),K([-12129,-2815])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([1,-1]),Polrev([0,0]),Polrev([-239,-63]),Polrev([-12129,-2815])], K);
 
magma: E := EllipticCurve([K![0,1],K![1,-1],K![0,0],K![-239,-63],K![-12129,-2815]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((6a-34)\) = \((2)\cdot(3a-17)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 124 \) = \(4\cdot31\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-496)\) = \((2)^{4}\cdot(3a-17)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 246016 \) = \(4^{4}\cdot31^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{35937}{496} \)
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: \(1\)
Generator $\left(\frac{63778}{31} a + \frac{276268}{31} : -\frac{284195743}{961} a - \frac{1228312820}{961} : 1\right)$
Height \(9.4455714026700784187277859215457444659\)
Torsion structure: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(10 a + 40 : -115 a - 504 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 9.4455714026700784187277859215457444659 \)
Period: \( 5.0136061226340085667987997688758259214 \)
Tamagawa product: \( 8 \)  =  \(2^{2}\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 4.9106272914561013813185211506769505787 \)
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\((j)_{-}\)
\((2)\) \(4\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((3a-17)\) \(31\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

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 and 4.
Its isogeny class 124.1-h consists of curves linked by isogenies of degrees dividing 4.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 558.c4
\(\Q\) 1922.d4