Properties

 Label 2.2.53.1-1225.1-a3 Base field $$\Q(\sqrt{53})$$ Conductor $$(35)$$ Conductor norm $$1225$$ CM no Base change yes: 35.a3,98315.e3 Q-curve yes Torsion order $$3$$ Rank $$1$$

Related objects

Show commands: Magma / Pari/GP / SageMath

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

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-13, -1, 1]))

gp: K = nfinit(Pol(Vecrev([-13, -1, 1])));

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

Weierstrass equation

$${y}^2+{y}={x}^{3}+{x}^{2}+9{x}+1$$
sage: E = EllipticCurve([K([0,0]),K([1,0]),K([1,0]),K([9,0]),K([1,0])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([9,0])),Pol(Vecrev([1,0]))], K);

magma: E := EllipticCurve([K![0,0],K![1,0],K![1,0],K![9,0],K![1,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(35)$$ = $$(-a-2)\cdot(-a+3)\cdot(5)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$1225$$ = $$7\cdot7\cdot25$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-42875)$$ = $$(-a-2)^{3}\cdot(-a+3)^{3}\cdot(5)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1838265625$$ = $$7^{3}\cdot7^{3}\cdot25^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{71991296}{42875}$$ 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 $\left(8 : 7 a - 4 : 1\right)$ Height $$0.998860286761859$$ Torsion structure: $$\Z/3\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(1 : -4 : 1\right)$ 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: $$0.998860286761859$$ Period: $$4.44675789091952$$ Tamagawa product: $$27$$  =  $$3\cdot3\cdot3$$ Torsion order: $$3$$ Leading coefficient: $$3.66067814604655$$ 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$$(j)_{-}$$
$$(-a-2)$$ $$7$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$
$$(-a+3)$$ $$7$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$
$$(5)$$ $$25$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$

Galois Representations

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

prime Image of Galois Representation
$$3$$ 3Cs.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 1225.1-a consists of curves linked by isogenies of degrees dividing 9.

Base change

This curve is the base change of 35.a3, 98315.e3, defined over $$\Q$$, so it is also a $$\Q$$-curve.