Properties

 Label 2.2.17.1-144.4-d2 Base field $$\Q(\sqrt{17})$$ Conductor $$(3 a + 12)$$ Conductor norm $$144$$ CM no Base change no Q-curve no Torsion order $$2$$ Rank $$1$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Base field$$\Q(\sqrt{17})$$

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-4, -1, 1]))

gp: K = nfinit(Pol(Vecrev([-4, -1, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -1, 1]);

Weierstrass equation

$$y^2+axy=x^{3}-x^{2}+\left(6a+8\right)x-33a-52$$
sage: E = EllipticCurve([K([0,1]),K([-1,0]),K([0,0]),K([8,6]),K([-52,-33])])

gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([8,6])),Pol(Vecrev([-52,-33]))], K);

magma: E := EllipticCurve([K![0,1],K![-1,0],K![0,0],K![8,6],K![-52,-33]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(3 a + 12)$$ = $$\left(-a + 2\right)^{4} \cdot \left(3\right)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$144$$ = $$2^{4} \cdot 9$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-69 a + 444)$$ = $$\left(-a + 2\right)^{14} \cdot \left(3\right)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$147456$$ = $$2^{14} \cdot 9$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{59090945}{12} a + \frac{151366337}{12}$$ 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 $\left(8 a + 14 : -61 a - 94 : 1\right)$ Height $$1.83533649527879$$ 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: $\left(a + 1 : -a - 2 : 1\right)$ 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: $$1.83533649527879$$ Period: $$5.02061526101742$$ Tamagawa product: $$2$$  =  $$2\cdot1$$ Torsion order: $$2$$ Leading coefficient: $$2.23484898374840$$ 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$$(j)_{-}$$
$$\left(-a + 2\right)$$ $$2$$ $$2$$ $$I_{6}^*$$ Additive $$-1$$ $$4$$ $$14$$ $$2$$
$$\left(3\right)$$ $$9$$ $$1$$ $$I_{1}$$ Non-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, 4 and 8.
Its isogeny class 144.4-d consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is not the base change of an elliptic curve defined over $$\Q$$. It is not a $$\Q$$-curve.