Properties

Label 2.2.92.1-16.1-c1
Base field \(\Q(\sqrt{23}) \)
Conductor norm \( 16 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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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

\({y}^2={x}^{3}+\left(a-1\right){x}^{2}+11{x}+a-2\)
sage: E = EllipticCurve([K([0,0]),K([-1,1]),K([0,0]),K([11,0]),K([-2,1])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([-1,1]),Polrev([0,0]),Polrev([11,0]),Polrev([-2,1])], K);
 
magma: E := EllipticCurve([K![0,0],K![-1,1],K![0,0],K![11,0],K![-2,1]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((4)\) = \((-a+5)^{4}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 16 \) = \(2^{4}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-16)\) = \((-a+5)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 256 \) = \(2^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 12800 a - 60672 \)
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 $\left(a + 7 : 5 a + 24 : 1\right)$
Height \(0.46351453556461598673928884107991996252\)
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: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.46351453556461598673928884107991996252 \)
Period: \( 25.954855879911723843520560114017996879 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 2.5085228516354954042534003754241619425 \)
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\((j)_{-}\)
\((-a+5)\) \(2\) \(1\) \(I_0^{*}\) Additive \(-1\) \(4\) \(8\) \(0\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 16.1-c consists of this curve only.

Base change

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

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