Properties

Label 4.4.10512.1-4.1-d4
Base field 4.4.10512.1
Conductor norm \( 4 \)
CM no
Base change no
Q-curve no
Torsion order \( 5 \)
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

\({y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a^{3}-6a-4\right){y}={x}^{3}+\left(a^{3}-6a-6\right){x}^{2}+\left(43a^{3}-48a^{2}-206a-90\right){x}+45a^{3}-287a^{2}+192a+798\)
sage: E = EllipticCurve([K([-1,-4,-1,1]),K([-6,-6,0,1]),K([-4,-6,0,1]),K([-90,-206,-48,43]),K([798,192,-287,45])])
 
gp: E = ellinit([Polrev([-1,-4,-1,1]),Polrev([-6,-6,0,1]),Polrev([-4,-6,0,1]),Polrev([-90,-206,-48,43]),Polrev([798,192,-287,45])], K);
 
magma: E := EllipticCurve([K![-1,-4,-1,1],K![-6,-6,0,1],K![-4,-6,0,1],K![-90,-206,-48,43],K![798,192,-287,45]]);
 

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: \((a^3-a^2-5a-2)\) = \((a^3-a^2-5a)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 4 \) = \(4\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{4312559010154335}{2} a^{3} - 6432420491639788 a^{2} - 4094666867772510 a + \frac{1445659614068901}{2} \)
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: \(\Z/5\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-4 a^{3} + 5 a^{2} + 18 a + 6 : -a^{3} - a^{2} + 10 a + 8 : 1\right)$
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: \( 361.38520368451935333081734456501895792 \)
Tamagawa product: \( 1 \)
Torsion order: \(5\)
Leading coefficient: \( 1.26890816456830 \)
Analytic order of Ш: \( 9 \) (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\((j)_{-}\)
\((a^3-a^2-5a)\) \(4\) \(1\) \(I_{1}\) 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
\(3\) 3B
\(5\) 5B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 5, 9, 15 and 45.
Its isogeny class 4.1-d 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.