# Properties

 Base field $$\Q(\sqrt{17})$$ Label 2.2.17.1-144.5-c1 Conductor $$(-3 a + 15)$$ Conductor norm $$144$$ CM no base-change no Q-curve yes Torsion order $$2$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field $$\Q(\sqrt{17})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 4$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 4)

gp: K = nfinit(a^2 - a - 4);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -1, 1]);

## Weierstrass equation

$$y^2 + \left(a + 1\right) x y + \left(a + 1\right) y = x^{3} - x^{2} + \left(-119 a - 186\right) x - 1469 a - 2294$$
sage: E = EllipticCurve(K, [a + 1, -1, a + 1, -119*a - 186, -1469*a - 2294])

gp: E = ellinit([a + 1, -1, a + 1, -119*a - 186, -1469*a - 2294],K)

magma: E := ChangeRing(EllipticCurve([a + 1, -1, a + 1, -119*a - 186, -1469*a - 2294]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-3 a + 15)$$ = $$\left(-a - 1\right)^{4} \cdot \left(3\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$144$$ = $$2^{4} \cdot 9$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(111 a + 69)$$ = $$\left(-a - 1\right)^{12} \cdot \left(3\right)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$36864$$ = $$2^{12} \cdot 9$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-\frac{396321250}{3} a + 338397375$$ 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: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(\frac{25}{4} a + \frac{39}{4} : -\frac{93}{8} a - \frac{143}{8} : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(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$$(j)_{-}$$
$$\left(-a - 1\right)$$ $$2$$ $$4$$ $$I_{4}^*$$ Additive $$-1$$ $$4$$ $$12$$ $$0$$
$$\left(3\right)$$ $$9$$ $$1$$ $$I_{1}$$ 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, 4, 8 and 16.
Its isogeny class 144.5-c consists of curves linked by isogenies of degrees dividing 16.

## Base change

This curve is not the base-change of an elliptic curve defined over $$\Q$$. It is a $$\Q$$-curve.