Properties

 Label 4.4.725.1-3169.1-b2 Base field 4.4.725.1 Conductor norm $$3169$$ CM no Base change no Q-curve no Torsion order $$2$$ Rank $$1$$

Related objects

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Base field4.4.725.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 1, -3, -1, 1]))

gp: K = nfinit(Pol(Vecrev([1, 1, -3, -1, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -3, -1, 1]);

Weierstrass equation

$${y}^2+\left(a^{3}-a^{2}-2a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+3a\right){x}^{2}+\left(-a^{2}+a+1\right){x}$$
sage: E = EllipticCurve([K([1,-2,-1,1]),K([0,3,0,-1]),K([0,1,0,0]),K([1,1,-1,0]),K([0,0,0,0])])

gp: E = ellinit([Pol(Vecrev([1,-2,-1,1])),Pol(Vecrev([0,3,0,-1])),Pol(Vecrev([0,1,0,0])),Pol(Vecrev([1,1,-1,0])),Pol(Vecrev([0,0,0,0]))], K);

magma: E := EllipticCurve([K![1,-2,-1,1],K![0,3,0,-1],K![0,1,0,0],K![1,1,-1,0],K![0,0,0,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(-a^3-3a^2+8)$$ = $$(-a^3-3a^2+8)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$3169$$ = $$3169$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-5a^3+7a^2+12a-1)$$ = $$(-a^3-3a^2+8)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$3169$$ = $$3169$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{2653932}{3169} a^{3} + \frac{5992969}{3169} a^{2} + \frac{3082469}{3169} a + \frac{6495478}{3169}$$ 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 $\left(a^{3} - a^{2} - 2 a + 1 : -a : 1\right)$ Height $$0.0385713298566881$$ 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: $\left(a^{2} - a - 1 : -a : 1\right)$ 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: $$0.0385713298566881$$ Period: $$1961.86685885729$$ Tamagawa product: $$1$$ Torsion order: $$2$$ Leading coefficient: $$2.81038060887703$$ 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$$(j)_{-}$$
$$(-a^3-3a^2+8)$$ $$3169$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

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.
Its isogeny class 3169.1-b consists of curves linked by isogenies of degree 2.

Base change

This elliptic curve is not a $$\Q$$-curve.

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