Properties

Label 4.4.4205.1-49.4-c1
Base field 4.4.4205.1
Conductor norm \( 49 \)
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

\({y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-2a^{3}+3a^{2}+8a+1\right){x}^{2}+\left(66a^{3}-37a^{2}-364a-233\right){x}-457a^{3}+318a^{2}+2384a+1238\)
sage: E = EllipticCurve([K([-1,-4,-1,1]),K([1,8,3,-2]),K([1,1,0,0]),K([-233,-364,-37,66]),K([1238,2384,318,-457])])
 
gp: E = ellinit([Polrev([-1,-4,-1,1]),Polrev([1,8,3,-2]),Polrev([1,1,0,0]),Polrev([-233,-364,-37,66]),Polrev([1238,2384,318,-457])], K);
 
magma: E := EllipticCurve([K![-1,-4,-1,1],K![1,8,3,-2],K![1,1,0,0],K![-233,-364,-37,66],K![1238,2384,318,-457]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a^3-3a^2-7a)\) = \((-a^3+2a^2+3a-3)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 49 \) = \(7^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((9a^3-26a^2-49a+50)\) = \((-a^3+2a^2+3a-3)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -5764801 \) = \(-7^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -156521478494 a^{3} + 257550722470 a^{2} + 616417035615 a - 241680788762 \)
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: \( 85.381048763681785110016819080392937606 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.31667468293667 \)
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\((j)_{-}\)
\((-a^3+2a^2+3a-3)\) \(7\) \(1\) \(IV^{*}\) Additive \(1\) \(2\) \(8\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(7\) 7B.6.3

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 7.
Its isogeny class 49.4-c consists of curves linked by isogenies of degree 7.

Base change

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

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