Properties

Label 4.4.18097.1-4.1-b2
Base field 4.4.18097.1
Conductor norm \( 4 \)
CM no
Base change no
Q-curve no
Torsion order \( 5 \)
Rank \( 0 \)

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Base field 4.4.18097.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 7 x^{2} + 6 x + 4 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([4, 6, -7, -1, 1]))
 
gp: K = nfinit(Polrev([4, 6, -7, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 6, -7, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(-2a^{2}+5\right){x}+\frac{11}{2}a^{3}-\frac{5}{2}a^{2}-\frac{61}{2}a-12\)
sage: E = EllipticCurve([K([-2,-5/2,1/2,1/2]),K([3,-1,-1,0]),K([0,-4,0,1]),K([5,0,-2,0]),K([-12,-61/2,-5/2,11/2])])
 
gp: E = ellinit([Polrev([-2,-5/2,1/2,1/2]),Polrev([3,-1,-1,0]),Polrev([0,-4,0,1]),Polrev([5,0,-2,0]),Polrev([-12,-61/2,-5/2,11/2])], K);
 
magma: E := EllipticCurve([K![-2,-5/2,1/2,1/2],K![3,-1,-1,0],K![0,-4,0,1],K![5,0,-2,0],K![-12,-61/2,-5/2,11/2]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a)\) = \((a)\)
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: \((a)\) = \((a)\)
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: \( \frac{25621}{4} a^{3} + \frac{38575}{4} a^{2} - \frac{77051}{4} a - \frac{16681}{2} \)
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: $\left(-\frac{1}{2} a^{3} + \frac{1}{2} a^{2} + \frac{5}{2} a : -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{5}{2} a : 1\right)$
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: \( 644.20658299503525886700560197404435635 \)
Tamagawa product: \( 1 \)
Torsion order: \(5\)
Leading coefficient: \( 0.191549867801905 \)
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\((j)_{-}\)
\((a)\) \(4\) \(1\) \(I_{1}\) 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
\(5\) 5B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 4.1-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.