Properties

Label 4.4.4205.1-49.2-a2
Base field 4.4.4205.1
Conductor norm \( 49 \)
CM no
Base change yes
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Base field 4.4.4205.1

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

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

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^3+a^2+6a)\) = \((-a^3+2a^2+3a-3)\cdot(a^2-2a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 49 \) = \(7\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((34a^3-105a^2-28a+216)\) = \((-a^3+2a^2+3a-3)^{5}\cdot(a^2-2a-3)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 282475249 \) = \(7^{5}\cdot7^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{173650213}{16807} a^{3} + \frac{173650213}{16807} a^{2} + \frac{1041901278}{16807} a + \frac{583264453}{16807} \)
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: $\left(0 : 0 : 1\right)$
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: \( 411.67676832003757634179099024420667465 \)
Tamagawa product: \( 1 \)  =  \(1\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 1.58713317020868 \)
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\((j)_{-}\)
\((-a^3+2a^2+3a-3)\) \(7\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)
\((a^2-2a-3)\) \(7\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)

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.4.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 5 and 10.
Its isogeny class 49.2-a consists of curves linked by isogenies of degrees dividing 10.

Base change

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

Base field Curve
\(\Q(\sqrt{29}) \) 2.2.29.1-175.6-b2
\(\Q(\sqrt{29}) \) 2.2.29.1-7.1-a2