Properties

Label 4.4.10512.1-4.1-c3
Base field 4.4.10512.1
Conductor norm \( 4 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 4.4.10512.1

Generator \(a\), with minimal polynomial \( x^{4} - 7 x^{2} - 6 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -6, -7, 0, 1]))
 
gp: K = nfinit(Polrev([1, -6, -7, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, -7, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}-5a-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(a^{2}-2a-5\right){x}^{2}+\left(a^{3}+3a^{2}-3a-11\right){x}+a^{3}+4a^{2}+3a-5\)
sage: E = EllipticCurve([K([-4,-5,0,1]),K([-5,-2,1,0]),K([-4,0,1,0]),K([-11,-3,3,1]),K([-5,3,4,1])])
 
gp: E = ellinit([Polrev([-4,-5,0,1]),Polrev([-5,-2,1,0]),Polrev([-4,0,1,0]),Polrev([-11,-3,3,1]),Polrev([-5,3,4,1])], K);
 
magma: E := EllipticCurve([K![-4,-5,0,1],K![-5,-2,1,0],K![-4,0,1,0],K![-11,-3,3,1],K![-5,3,4,1]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-a^2-5a)\) = \((a^3-a^2-5a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 4 \) = \(4\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2a^3+2a^2+10a+4)\) = \((a^3-a^2-5a)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 64 \) = \(4^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{580191}{4} a^{3} + \frac{580191}{4} a^{2} + \frac{2900955}{4} a - 104276 \)
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: \( 157.83506631149255370623146126353258245 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.53943310240620 \)
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\((j)_{-}\)
\((a^3-a^2-5a)\) \(4\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3Cs
\(5\) 5B.4.1

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-c consists of curves linked by isogenies of degrees dividing 45.

Base change

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

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