Properties

 Base field 4.4.4205.1 Label 4.4.4205.1-49.4-c1 Conductor $$(49,2 a^{3} - 3 a^{2} - 7 a)$$ Conductor norm $$49$$ CM no base-change no Q-curve no Torsion order $$1$$ Rank not available

Related objects

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Base field 4.4.4205.1

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

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 - 5*x^2 - x + 1)

gp: K = nfinit(a^4 - a^3 - 5*a^2 - a + 1);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -5, -1, 1]);

Weierstrass equation

$$y^2 + \left(a^{3} - a^{2} - 4 a - 1\right) x y + \left(a + 1\right) y = x^{3} + \left(-2 a^{3} + 3 a^{2} + 8 a + 1\right) x^{2} + \left(66 a^{3} - 37 a^{2} - 364 a - 233\right) x - 457 a^{3} + 318 a^{2} + 2384 a + 1238$$
sage: E = EllipticCurve(K, [a^3 - a^2 - 4*a - 1, -2*a^3 + 3*a^2 + 8*a + 1, a + 1, 66*a^3 - 37*a^2 - 364*a - 233, -457*a^3 + 318*a^2 + 2384*a + 1238])

gp: E = ellinit([a^3 - a^2 - 4*a - 1, -2*a^3 + 3*a^2 + 8*a + 1, a + 1, 66*a^3 - 37*a^2 - 364*a - 233, -457*a^3 + 318*a^2 + 2384*a + 1238],K)

magma: E := ChangeRing(EllipticCurve([a^3 - a^2 - 4*a - 1, -2*a^3 + 3*a^2 + 8*a + 1, a + 1, 66*a^3 - 37*a^2 - 364*a - 233, -457*a^3 + 318*a^2 + 2384*a + 1238]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(49,2 a^{3} - 3 a^{2} - 7 a)$$ = $$\left(-a^{3} + 2 a^{2} + 3 a - 3\right)^{2}$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$49$$ = $$7^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(5764801,a + 2899171,a^{3} - a^{2} - 5 a + 3123108,-a^{3} + 2 a^{2} + 3 a + 5274931)$$ = $$\left(-a^{3} + 2 a^{2} + 3 a - 3\right)^{8}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$5764801$$ = $$7^{8}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-156521478494 a^{3} + 257550722470 a^{2} + 616417035615 a - 241680788762$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.

sage: E.rank()

magma: Rank(E);

Regulator: not available

sage: gens = E.gens(); gens

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

Torsion subgroup

Structure: Trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T);

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$$(j)_{-}$$
$$\left(-a^{3} + 2 a^{2} + 3 a - 3\right)$$ $$7$$ $$1$$ $$IV^*$$ Additive $$1$$ $$2$$ $$8$$ $$0$$

Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ except those listed.

prime Image of Galois Representation
$$7$$ 7B.6.3

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 7.
Its isogeny class 49.4-c consists of curves linked by isogenies of degree 7.

Base change

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