Properties

Label 4.4.2048.1-1.1-a1
Base field \(\Q(\zeta_{16})^+\)
Conductor norm \( 1 \)
CM no
Base change no
Q-curve yes
Torsion order \( 10 \)
Rank \( 0 \)

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Base field \(\Q(\zeta_{16})^+\)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, -4, 0, 1]))
 
gp: K = nfinit(Polrev([2, 0, -4, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, -4, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+{y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(a^{3}-4a\right){x}-22a^{3}-17a^{2}+75a+58\)
sage: E = EllipticCurve([K([-2,-3,1,1]),K([3,-1,-1,0]),K([1,0,0,0]),K([0,-4,0,1]),K([58,75,-17,-22])])
 
gp: E = ellinit([Polrev([-2,-3,1,1]),Polrev([3,-1,-1,0]),Polrev([1,0,0,0]),Polrev([0,-4,0,1]),Polrev([58,75,-17,-22])], K);
 
magma: E := EllipticCurve([K![-2,-3,1,1],K![3,-1,-1,0],K![1,0,0,0],K![0,-4,0,1],K![58,75,-17,-22]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1)\) = \((1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1 \) = 1
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1)\) = \((1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1 \) = 1
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 5699182696 a^{3} - 4361830336 a^{2} - 19458081104 a + 14892491272 \)
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: \(\Z/10\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-6 a^{3} - 3 a^{2} + 19 a + 13 : -35 a^{3} - 31 a^{2} + 124 a + 97 : 1\right)$
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: \( 1018.9179720664743532628681269508193745 \)
Tamagawa product: \( 1 \)
Torsion order: \(10\)
Leading coefficient: \( 0.225151189850328 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
No primes of bad reduction.

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
\(5\) 5B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 5, 8, 10, 20 and 40.
Its isogeny class 1.1-a consists of curves linked by isogenies of degrees dividing 40.

Base change

This elliptic curve is a \(\Q\)-curve.

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