Properties

Label 4.4.2777.1-16.2-a3
Base field 4.4.2777.1
Conductor norm \( 16 \)
CM no
Base change no
Q-curve no
Torsion order \( 8 \)
Rank \( 0 \)

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Base field 4.4.2777.1

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

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

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^3+2a^2+3a-2)\) = \((-a)^{4}\)
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: \((4a^3-3a^2-8a-4)\) = \((-a)^{11}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -2048 \) = \(-2^{11}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -601 a^{3} + 1330 a^{2} + 1931 a - 2180 \)
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: \(\Z/8\Z\)
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: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 928.54695747252963414533593442969054515 \)
Tamagawa product: \( 4 \)
Torsion order: \(8\)
Leading coefficient: \( 1.10127542686603 \)
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\((j)_{-}\)
\((-a)\) \(2\) \(4\) \(I_{3}^{*}\) Additive \(1\) \(4\) \(11\) \(0\)

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 and 8.
Its isogeny class 16.2-a consists of curves linked by isogenies of degrees dividing 8.

Base change

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

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