# Properties

 Label 2.2.56.1-175.1-f3 Base field $$\Q(\sqrt{14})$$ Conductor $$(-10a+35)$$ Conductor norm $$175$$ CM no Base change yes: 35.a3,15680.ba3 Q-curve yes Torsion order $$3$$ Rank $$1$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{14})$$

Generator $$a$$, with minimal polynomial $$x^{2} - 14$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, 0, 1]))

gp: K = nfinit(Pol(Vecrev([-14, 0, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, 0, 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: $$(-10a+35)$$ = $$(-a+3)\cdot(-a-3)\cdot(-2a+7)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$175$$ = $$5\cdot5\cdot7$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-42875)$$ = $$(-a+3)^{3}\cdot(-a-3)^{3}\cdot(-2a+7)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1838265625$$ = $$5^{3}\cdot5^{3}\cdot7^{6}$$ 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 $\left(\frac{15}{2} : -\frac{25}{4} a - \frac{1}{2} : 1\right)$ Height $$2.93933365281779$$ 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: $$2.93933365281779$$ Period: $$4.44675789091952$$ Tamagawa product: $$6$$  =  $$1\cdot1\cdot( 2 \cdot 3 )$$ Torsion order: $$3$$ Leading coefficient: $$2.32882628625389$$ 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$$(j)_{-}$$
$$(-a+3)$$ $$5$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$
$$(-a-3)$$ $$5$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$
$$(-2a+7)$$ $$7$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$

## 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 175.1-f consists of curves linked by isogenies of degrees dividing 9.

## Base change

This curve is the base change of elliptic curves 35.a3, 15680.ba3, defined over $$\Q$$, so it is also a $$\Q$$-curve.