# Properties

 Label 2.2.44.1-392.1-c1 Base field $$\Q(\sqrt{11})$$ Conductor $$(14a+42)$$ Conductor norm $$392$$ CM no Base change yes: 56.a4,13552.p4 Q-curve yes Torsion order $$4$$ Rank $$0$$

# Learn more

Show commands for: Magma / Pari/GP / SageMath

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

Generator $$a$$, with minimal polynomial $$x^{2} - 11$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-11, 0, 1]))

gp: K = nfinit(Pol(Vecrev([-11, 0, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11, 0, 1]);

## Weierstrass equation

$${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(18a+61\right){x}+342a+1134$$
sage: E = EllipticCurve([K([1,1]),K([0,1]),K([0,0]),K([61,18]),K([1134,342])])

gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([0,1])),Pol(Vecrev([0,0])),Pol(Vecrev([61,18])),Pol(Vecrev([1134,342]))], K);

magma: E := EllipticCurve([K![1,1],K![0,1],K![0,0],K![61,18],K![1134,342]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(14a+42)$$ = $$(a+3)^{3}\cdot(a+2)\cdot(a-2)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$392$$ = $$2^{3}\cdot7\cdot7$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-28)$$ = $$(a+3)^{4}\cdot(a+2)\cdot(a-2)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$784$$ = $$2^{4}\cdot7\cdot7$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{432}{7}$$ 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: $$\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(a + 4 : -12 a - 39 : 1\right)$ 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: $$24.4747121231892$$ Tamagawa product: $$2$$  =  $$2\cdot1\cdot1$$ Torsion order: $$4$$ Leading coefficient: $$1.84485084010411$$ Analytic order of Ш: $$4$$ (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$$(j)_{-}$$
$$(a+3)$$ $$2$$ $$2$$ $$III$$ Additive $$1$$ $$3$$ $$4$$ $$0$$
$$(a+2)$$ $$7$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$
$$(a-2)$$ $$7$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 392.1-c consists of curves linked by isogenies of degrees dividing 4.

## Base change

This curve is the base change of elliptic curves 56.a4, 13552.p4, defined over $$\Q$$, so it is also a $$\Q$$-curve.