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

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

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

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 13)

gp: K = nfinit(a^2 - a - 13);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-13, -1, 1]);

Weierstrass equation

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

gp: E = ellinit([1, -1, 1, 0, 0],K)

magma: E := ChangeRing(EllipticCurve([1, -1, 1, 0, 0]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

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

Galois Representations

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

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.