Properties

 Label 2.0.11.1-1225.2-a3 Base field $$\Q(\sqrt{-11})$$ Conductor $$(35)$$ Conductor norm $$1225$$ CM no Base change yes: 4235.c3,35.a3 Q-curve yes Torsion order $$3$$ Rank $$0$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Base field$$\Q(\sqrt{-11})$$

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -1, 1]))

gp: K = nfinit(Pol(Vecrev([3, -1, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -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: $$(35)$$ = $$(-a-1)\cdot(a-2)\cdot(7)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$1225$$ = $$5\cdot5\cdot49$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-42875)$$ = $$(-a-1)^{3}\cdot(a-2)^{3}\cdot(7)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1838265625$$ = $$5^{3}\cdot5^{3}\cdot49^{3}$$ 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: $$0$$ 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: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$2.32492560669298$$ Tamagawa product: $$3$$  =  $$1\cdot1\cdot3$$ Torsion order: $$3$$ Leading coefficient: $$0.467327630478181$$ 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$$(j)_{-}$$
$$(-a-1)$$ $$5$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$
$$(a-2)$$ $$5$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$
$$(7)$$ $$49$$ $$3$$ $$I_{3}$$ 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 1225.2-a consists of curves linked by isogenies of degrees dividing 9.

Base change

This curve is the base change of 4235.c3, 35.a3, defined over $$\Q$$, so it is also a $$\Q$$-curve.