Properties

Label 4.4.4205.1-23.1-b2
Base field 4.4.4205.1
Conductor norm \( 23 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 4.4.4205.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} - x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, -5, -1, 1]))
 
gp: K = nfinit(Polrev([1, -1, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+{y}={x}^{3}+\left(a^{3}-2a^{2}-3a\right){x}^{2}+\left(-11a^{3}-14a^{2}+a+2\right){x}-63a^{3}-120a^{2}-16a+23\)
sage: E = EllipticCurve([K([0,0,0,0]),K([0,-3,-2,1]),K([1,0,0,0]),K([2,1,-14,-11]),K([23,-16,-120,-63])])
 
gp: E = ellinit([Polrev([0,0,0,0]),Polrev([0,-3,-2,1]),Polrev([1,0,0,0]),Polrev([2,1,-14,-11]),Polrev([23,-16,-120,-63])], K);
 
magma: E := EllipticCurve([K![0,0,0,0],K![0,-3,-2,1],K![1,0,0,0],K![2,1,-14,-11],K![23,-16,-120,-63]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-a^2-4a-4)\) = \((a^3-a^2-4a-4)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 23 \) = \(23\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a^2-3a-1)\) = \((a^3-a^2-4a-4)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -23 \) = \(-23\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{131224616960}{23} a^{3} + \frac{85170323456}{23} a^{2} + \frac{685859135488}{23} a + \frac{372101771264}{23} \)
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: \( 101.11580076687039648462291877583872341 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.55932278699343 \)
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\((j)_{-}\)
\((a^3-a^2-4a-4)\) \(23\) \(1\) \(I_{1}\) Non-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
\(5\) 5B.4.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 23.1-b consists of curves linked by isogenies of degree 5.

Base change

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

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