Properties

Label 4.4.15317.1-4.1-a11
Base field 4.4.15317.1
Conductor norm \( 4 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 1 \)

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Base field 4.4.15317.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 5, -4, -2, 1]))
 
gp: K = nfinit(Polrev([2, 5, -4, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 5, -4, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-2a+5\right){x}^{2}+\left(138a^{3}-549a^{2}+341a+188\right){x}+2457a^{3}-9401a^{2}+5861a+3045\)
sage: E = EllipticCurve([K([1,-4,-1,1]),K([5,-2,-2,1]),K([-1,-5,0,1]),K([188,341,-549,138]),K([3045,5861,-9401,2457])])
 
gp: E = ellinit([Polrev([1,-4,-1,1]),Polrev([5,-2,-2,1]),Polrev([-1,-5,0,1]),Polrev([188,341,-549,138]),Polrev([3045,5861,-9401,2457])], K);
 
magma: E := EllipticCurve([K![1,-4,-1,1],K![5,-2,-2,1],K![-1,-5,0,1],K![188,341,-549,138],K![3045,5861,-9401,2457]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^2-a-4)\) = \((-a)\cdot(a-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 4 \) = \(2\cdot2\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((292a^3-973a^2-627a+2762)\) = \((-a)^{2}\cdot(a-1)^{36}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 274877906944 \) = \(2^{2}\cdot2^{36}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{25244062543252502649437931}{68719476736} a^{3} - \frac{58811778546102253618393565}{68719476736} a^{2} - \frac{81584408494612303988948299}{68719476736} a + \frac{76560454679459868202267199}{34359738368} \)
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: \(1\)
Generator $\left(2 a^{3} - \frac{31}{4} a^{2} + \frac{3}{4} a + \frac{21}{4} : \frac{25}{4} a^{3} - \frac{135}{8} a^{2} - \frac{1}{8} a + \frac{99}{8} : 1\right)$
Height \(2.5691624343281407373827256376538691091\)
Torsion structure: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(\frac{5}{4} a^{3} - \frac{27}{4} a^{2} + \frac{23}{4} a - \frac{1}{2} : \frac{9}{8} a^{3} - \frac{7}{8} a^{2} - \frac{25}{8} a + \frac{9}{4} : 1\right)$ $\left(a^{3} - 5 a^{2} + 2 a + 1 : a^{3} - 3 a - 1 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 2.5691624343281407373827256376538691091 \)
Period: \( 13.055766083325680657798104451444772571 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 2.43921216239754 \)
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\((j)_{-}\)
\((-a)\) \(2\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((a-1)\) \(2\) \(2\) \(I_{36}\) Non-split multiplicative \(1\) \(1\) \(36\) \(36\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 4.1-a consists of curves linked by isogenies of degrees dividing 24.

Base change

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

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