Properties

 Base field $$\Q(\sqrt{5}, \sqrt{17})$$ Label 4.4.7225.1-236.3-a2 Conductor $$(118,\frac{7}{6} a^{3} - \frac{1}{2} a^{2} - \frac{77}{6} a + 9)$$ Conductor norm $$236$$ CM no base-change no Q-curve no Torsion order $$2$$ Rank not available

Related objects

Show commands for: Magma / SageMath / Pari/GP

Base field $$\Q(\sqrt{5}, \sqrt{17})$$

Generator $$a$$, with minimal polynomial $$x^{4} - 11 x^{2} + 9$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 0, -11, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 - 11*x^2 + 9)
gp (2.8): K = nfinit(a^4 - 11*a^2 + 9);

Weierstrass equation

$$y^2 + \left(-\frac{1}{6} a^{3} + \frac{1}{2} a^{2} + \frac{11}{6} a - 2\right) x y + \left(-\frac{1}{6} a^{3} + \frac{7}{3} a + \frac{3}{2}\right) y = x^{3} + \left(\frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{11}{6} a + 3\right) x^{2} + \left(\frac{2}{3} a^{3} - a^{2} - \frac{22}{3} a + 8\right) x - a^{3} - 6 a^{2} - 6 a + 9$$
magma: E := ChangeRing(EllipticCurve([-1/6*a^3 + 1/2*a^2 + 11/6*a - 2, 1/6*a^3 - 1/2*a^2 - 11/6*a + 3, -1/6*a^3 + 7/3*a + 3/2, 2/3*a^3 - a^2 - 22/3*a + 8, -a^3 - 6*a^2 - 6*a + 9]),K);
sage: E = EllipticCurve(K, [-1/6*a^3 + 1/2*a^2 + 11/6*a - 2, 1/6*a^3 - 1/2*a^2 - 11/6*a + 3, -1/6*a^3 + 7/3*a + 3/2, 2/3*a^3 - a^2 - 22/3*a + 8, -a^3 - 6*a^2 - 6*a + 9])
gp (2.8): E = ellinit([-1/6*a^3 + 1/2*a^2 + 11/6*a - 2, 1/6*a^3 - 1/2*a^2 - 11/6*a + 3, -1/6*a^3 + 7/3*a + 3/2, 2/3*a^3 - a^2 - 22/3*a + 8, -a^3 - 6*a^2 - 6*a + 9],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(118,\frac{7}{6} a^{3} - \frac{1}{2} a^{2} - \frac{77}{6} a + 9)$$ = $$\left(2, -\frac{1}{6} a^{3} + \frac{7}{3} a + \frac{1}{2}\right) \cdot \left(59, \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{47}{2}\right)$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$236$$ = $$4 \cdot 59$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(24234722,\frac{1}{3} a^{3} - \frac{8}{3} a + 5570257,-\frac{1}{6} a^{3} + \frac{7}{3} a + \frac{39033897}{2},\frac{1}{6} a^{3} + \frac{1}{2} a^{2} - \frac{11}{6} a + 3374161)$$ = $$\left(2, -\frac{1}{6} a^{3} + \frac{7}{3} a + \frac{1}{2}\right) \cdot \left(59, \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{47}{2}\right)^{4}$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$48469444$$ = $$4 \cdot 59^{4}$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$\frac{35867214803}{48469444} a^{3} - \frac{15888032143}{48469444} a^{2} - \frac{455784216367}{48469444} a + \frac{114223634126}{12117361}$$ magma: jInvariant(E); sage: E.j_invariant() gp (2.8): E.j $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) magma: HasComplexMultiplication(E); sage: E.has_cm(), E.cm_discriminant() $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: not available

magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

Torsion subgroup

Structure: $$\Z/2\Z$$ magma: TorsionSubgroup(E); sage: E.torsion_subgroup().gens() gp (2.8): elltors(E)[2] magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp (2.8): elltors(E)[1] $\left(-\frac{1}{24} a^{3} + \frac{3}{8} a^{2} + \frac{29}{24} a - 2 : -\frac{5}{16} a^{3} - \frac{1}{8} a^{2} + \frac{3}{2} a - \frac{45}{16} : 1\right)$ magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E); sage: E.torsion_subgroup().gens() gp (2.8): elltors(E)[3]

Local data at primes of bad reduction

magma: LocalInformation(E);
sage: E.local_data()
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(59, \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{47}{2}\right)$$ $$59$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(2, -\frac{1}{6} a^{3} + \frac{7}{3} a + \frac{1}{2}\right)$$ $$4$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ 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 and 4.
Its isogeny class 236.3-a consists of curves linked by isogenies of degrees dividing 4.

Base change

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