Properties

Label 2.2.92.1-98.1-d6
Base field \(\Q(\sqrt{23}) \)
Conductor norm \( 98 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{23}) \)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-23, 0, 1]))
 
gp: K = nfinit(Polrev([-23, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-23, 0, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(77610a-372205\right){x}+25987299a-124630708\)
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([1,0]),K([-372205,77610]),K([-124630708,25987299])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([-372205,77610]),Polrev([-124630708,25987299])], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,0],K![1,0],K![-372205,77610],K![-124630708,25987299]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-7a+35)\) = \((-a+5)\cdot(-a+4)\cdot(-a-4)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 98 \) = \(2\cdot7\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((5335785a-13768181)\) = \((-a+5)\cdot(-a+4)^{16}\cdot(-a-4)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -465261027974414 \) = \(-2\cdot7^{16}\cdot7\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{7271608475713622625}{66465861139202} a + \frac{28557847263805318125}{66465861139202} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-104 a + \frac{1995}{4} : 52 a - \frac{1999}{8} : 1\right)$
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: \( 0.57771024539513663366290054851370762802 \)
Tamagawa product: \( 2 \)  =  \(1\cdot2\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 1.9273746130133726203330448615952713309 \)
Analytic order of Ш: \( 64 \) (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\((j)_{-}\)
\((-a+5)\) \(2\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((-a+4)\) \(7\) \(2\) \(I_{16}\) Non-split multiplicative \(1\) \(1\) \(16\) \(16\)
\((-a-4)\) \(7\) \(1\) \(I_{1}\) 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, 8 and 16.
Its isogeny class 98.1-d consists of curves linked by isogenies of degrees dividing 16.

Base change

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

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