# Properties

 Base field 6.6.1229312.1 Label 6.6.1229312.1-392.1-k5 Conductor $$(14,-\frac{1}{2} a^{3} + 4 a)$$ Conductor norm $$392$$ CM no base-change yes: 14.a4,448.g4 Q-curve yes Torsion order $$36$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field 6.6.1229312.1

Generator $$a$$, with minimal polynomial $$x^{6} - 10 x^{4} + 24 x^{2} - 8$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 10*x^4 + 24*x^2 - 8)

gp: K = nfinit(a^6 - 10*a^4 + 24*a^2 - 8);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, 0, 24, 0, -10, 0, 1]);

## Weierstrass equation

$$y^2 + x y + y = x^{3} - 11 x + 12$$
sage: E = EllipticCurve(K, [1, 0, 1, -11, 12])

gp: E = ellinit([1, 0, 1, -11, 12],K)

magma: E := ChangeRing(EllipticCurve([1, 0, 1, -11, 12]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(14,-\frac{1}{2} a^{3} + 4 a)$$ = $$\left(2, -\frac{1}{4} a^{5} + 2 a^{3} - 3 a\right) \cdot \left(7, a + 1\right) \cdot \left(7, a - 1\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$392$$ = $$7^{2} \cdot 8$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(98,49 a^{2} - 196,\frac{49}{2} a^{4} - 147 a^{2} + 98,\frac{49}{2} a^{5} - 196 a^{3} + 294 a,98 a,49 a^{3} - 294 a)$$ = $$\left(2, -\frac{1}{4} a^{5} + 2 a^{3} - 3 a\right)^{2} \cdot \left(7, a + 1\right)^{6} \cdot \left(7, a - 1\right)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$885842380864$$ = $$7^{12} \cdot 8^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{128787625}{98}$$ 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: $$\Z/2\Z\times\Z/18\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(\frac{1}{2} a^{4} - \frac{7}{2} a^{2} + 3 : -\frac{3}{4} a^{4} + 5 a^{2} - 3 : 1\right)$,$\left(\frac{1}{2} a^{5} - 4 a^{3} + 6 a - 1 : -\frac{1}{4} a^{5} + 2 a^{3} - 3 a : 1\right)$ 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$$(j)_{-}$$
$$\left(7, a + 1\right)$$ $$7$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$\left(7, a - 1\right)$$ $$7$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$\left(2, -\frac{1}{4} a^{5} + 2 a^{3} - 3 a\right)$$ $$8$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$

## 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
$$3$$ 3B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 6, 9 and 18.
Its isogeny class 392.1-k consists of curves linked by isogenies of degrees dividing 36.

## Base change

This curve is the base-change of elliptic curves 14.a4, 448.g4, defined over $$\Q$$, so it is also a $$\Q$$-curve.