Properties

 Base field $$\Q(\sqrt{53})$$ Label 2.2.53.1-53.1-a1 Conductor $$(-2 a + 1)$$ Conductor norm $$53$$ CM no base-change yes: 53.a1,2809.a1 Q-curve yes Torsion order $$1$$ Rank not available

Related objects

Show commands for: Magma / SageMath / Pari/GP

Base field $$\Q(\sqrt{53})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 13$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-13, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 13)
gp (2.8): K = nfinit(a^2 - a - 13);

Weierstrass equation

$$y^2 + x y + y = x^{3} - x^{2}$$
magma: E := ChangeRing(EllipticCurve([1, -1, 1, 0, 0]),K);
sage: E = EllipticCurve(K, [1, -1, 1, 0, 0])
gp (2.8): E = ellinit([1, -1, 1, 0, 0],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(-2 a + 1)$$ = $$\left(-2 a + 1\right)$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$53$$ = $$53$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(53)$$ = $$\left(-2 a + 1\right)^{2}$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$2809$$ = $$53^{2}$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$\frac{3375}{53}$$ 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: Trivial 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]

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$$(j)_{-}$$
$$\left(-2 a + 1\right)$$ $$53$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$

Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ .

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 53.1-a consists of this curve only.

Base change

This curve is the base-change of elliptic curves 53.a1, 2809.a1, defined over $$\Q$$, so it is also a $$\Q$$-curve.