Properties

Label 4.4.9248.1-16.4-a2
Base field 4.4.9248.1
Conductor norm \( 16 \)
CM no
Base change yes
Q-curve no
Torsion order \( 4 \)
Rank \( 1 \)

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Base field 4.4.9248.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, -5, 0, 1]))
 
gp: K = nfinit(Polrev([2, 0, -5, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, -5, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+a{y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(3a^{3}-7a^{2}-5a+7\right){x}-8a^{3}+16a^{2}+6a-9\)
sage: E = EllipticCurve([K([-2,-3,1,1]),K([-4,1,1,0]),K([0,1,0,0]),K([7,-5,-7,3]),K([-9,6,16,-8])])
 
gp: E = ellinit([Polrev([-2,-3,1,1]),Polrev([-4,1,1,0]),Polrev([0,1,0,0]),Polrev([7,-5,-7,3]),Polrev([-9,6,16,-8])], K);
 
magma: E := EllipticCurve([K![-2,-3,1,1],K![-4,1,1,0],K![0,1,0,0],K![7,-5,-7,3],K![-9,6,16,-8]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^2-2)\) = \((a)^{4}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 16 \) = \(2^{4}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a^2+2)\) = \((a)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 256 \) = \(2^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 2701312025 a^{2} - 1184382322 \)
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 $\left(-a + \frac{5}{2} : -\frac{1}{4} a^{2} - \frac{7}{4} a + \frac{3}{2} : 1\right)$
Height \(1.0795985847676796968437743783623142895\)
Torsion structure: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-a + 2 : -a + 1 : 1\right)$
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: \( 1.0795985847676796968437743783623142895 \)
Period: \( 578.08617425241345608543203103052401011 \)
Tamagawa product: \( 2 \)
Torsion order: \(4\)
Leading coefficient: \( 3.24489750201335 \)
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\((j)_{-}\)
\((a)\) \(2\) \(2\) \(I_0^{*}\) Additive \(-1\) \(4\) \(8\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 8 and 16.
Its isogeny class 16.4-a consists of curves linked by isogenies of degrees dividing 16.

Base change

This elliptic curve is not a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q(\sqrt{17}) \) 2.2.17.1-256.1-f6
\(\Q(\sqrt{17}) \) 2.2.17.1-64.7-a6