Properties

Label 4.4.9248.1-8.3-a4
Base field 4.4.9248.1
Conductor norm \( 8 \)
CM no
Base change yes
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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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

\({y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-11a^{2}+7\right){x}-36a^{2}+15\)
sage: E = EllipticCurve([K([0,1,0,0]),K([-3,0,1,0]),K([0,1,0,0]),K([7,0,-11,0]),K([15,0,-36,0])])
 
gp: E = ellinit([Polrev([0,1,0,0]),Polrev([-3,0,1,0]),Polrev([0,1,0,0]),Polrev([7,0,-11,0]),Polrev([15,0,-36,0])], K);
 
magma: E := EllipticCurve([K![0,1,0,0],K![-3,0,1,0],K![0,1,0,0],K![7,0,-11,0],K![15,0,-36,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-4a)\) = \((a)^{3}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 8 \) = \(2^{3}\)
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: \(0\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{5}{4} a^{2} + 1 : \frac{5}{8} a^{3} - a : 1\right)$
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: \( 80.525338566467246316349385232324277085 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 1.67470626346152 \)
Analytic order of Ш: \( 4 \) (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\((j)_{-}\)
\((a)\) \(2\) \(2\) \(I_{1}^{*}\) Additive \(1\) \(3\) \(8\) \(0\)

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 8.3-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
\(\Q(\sqrt{17}) \) 2.2.17.1-8.4-a6
\(\Q(\sqrt{17}) \) a curve with conductor norm 1024 (not in the database)