Properties

 Base field $$\Q(\sqrt{15})$$ Label 2.2.60.1-147.1-k5 Conductor $$(21,7 a)$$ Conductor norm $$147$$ CM no base-change yes: 336.a2,1575.c2 Q-curve yes Torsion order $$8$$ Rank not available

Related objects

Show commands for: Magma / Pari/GP / SageMath

Base field $$\Q(\sqrt{15})$$

Generator $$a$$, with minimal polynomial $$x^{2} - 15$$; class number $$2$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 15)

gp: K = nfinit(a^2 - 15);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-15, 0, 1]);

Weierstrass equation

$$y^2 + a x y + a y = x^{3} - x^{2} - 54 x + 84$$
sage: E = EllipticCurve(K, [a, -1, a, -54, 84])

gp: E = ellinit([a, -1, a, -54, 84],K)

magma: E := ChangeRing(EllipticCurve([a, -1, a, -54, 84]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(21,7 a)$$ = $$\left(3, a\right) \cdot \left(7, a + 1\right) \cdot \left(7, a + 6\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$147$$ = $$3 \cdot 7^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(21609)$$ = $$\left(3, a\right)^{4} \cdot \left(7, a + 1\right)^{4} \cdot \left(7, a + 6\right)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$466948881$$ = $$3^{4} \cdot 7^{8}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{13027640977}{21609}$$ 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: $$\Z/2\Z\times\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(\frac{3}{2} : -\frac{5}{4} a - \frac{21}{4} : 1\right)$,$\left(3 : -2 a : 1\right)$ 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$$(j)_{-}$$
$$\left(3, a\right)$$ $$3$$ $$2$$ $$I_{4}$$ Non-split multiplicative $$1$$ $$1$$ $$4$$ $$4$$
$$\left(7, a + 1\right)$$ $$7$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(7, a + 6\right)$$ $$7$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$

Galois Representations

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

prime Image of Galois Representation
$$2$$ 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 147.1-k consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base-change of elliptic curves 336.a2, 1575.c2, defined over $$\Q$$, so it is also a $$\Q$$-curve.