# Properties

 Base field 4.4.18625.1 Label 4.4.18625.1-11.1-a2 Conductor $$(11,-\frac{1}{6} a^{3} + \frac{7}{3} a + \frac{11}{6})$$ Conductor norm $$11$$ CM no base-change no Q-curve no Torsion order $$2$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field 4.4.18625.1

Generator $$a$$, with minimal polynomial $$x^{4} - x^{3} - 14 x^{2} + 9 x + 41$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 - 14*x^2 + 9*x + 41)

gp: K = nfinit(a^4 - a^3 - 14*a^2 + 9*a + 41);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, 9, -14, -1, 1]);

## Weierstrass equation

$$y^2 + \left(\frac{1}{6} a^{3} + a^{2} - \frac{1}{3} a - \frac{35}{6}\right) x y + \left(a + 1\right) y = x^{3} + \left(\frac{1}{6} a^{3} - \frac{4}{3} a + \frac{1}{6}\right) x^{2} + \left(\frac{47}{6} a^{3} + 17 a^{2} - \frac{155}{3} a - \frac{547}{6}\right) x + \frac{611}{3} a^{3} + 320 a^{2} - \frac{6034}{3} a - \frac{9808}{3}$$
sage: E = EllipticCurve(K, [1/6*a^3 + a^2 - 1/3*a - 35/6, 1/6*a^3 - 4/3*a + 1/6, a + 1, 47/6*a^3 + 17*a^2 - 155/3*a - 547/6, 611/3*a^3 + 320*a^2 - 6034/3*a - 9808/3])

gp: E = ellinit([1/6*a^3 + a^2 - 1/3*a - 35/6, 1/6*a^3 - 4/3*a + 1/6, a + 1, 47/6*a^3 + 17*a^2 - 155/3*a - 547/6, 611/3*a^3 + 320*a^2 - 6034/3*a - 9808/3],K)

magma: E := ChangeRing(EllipticCurve([1/6*a^3 + a^2 - 1/3*a - 35/6, 1/6*a^3 - 4/3*a + 1/6, a + 1, 47/6*a^3 + 17*a^2 - 155/3*a - 547/6, 611/3*a^3 + 320*a^2 - 6034/3*a - 9808/3]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(11,-\frac{1}{6} a^{3} + \frac{7}{3} a + \frac{11}{6})$$ = $$\left(\frac{1}{6} a^{3} - \frac{7}{3} a - \frac{11}{6}\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$11$$ = $$11$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(14641,\frac{1}{6} a^{3} - \frac{4}{3} a + \frac{24175}{6},a + 12237,a^{2} + 3979)$$ = $$\left(\frac{1}{6} a^{3} - \frac{7}{3} a - \frac{11}{6}\right)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$14641$$ = $$11^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-\frac{1060007393}{87846} a^{3} + \frac{746018190}{14641} a^{2} + \frac{411434192}{43923} a - \frac{13361296451}{87846}$$ 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); $\left(-\frac{29}{24} a^{3} - \frac{7}{4} a^{2} + \frac{67}{6} a + \frac{379}{24} : \frac{55}{24} a^{3} + \frac{41}{8} a^{2} - \frac{437}{24} a - \frac{439}{12} : 1\right)$ 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$$(j)_{-}$$
$$\left(\frac{1}{6} a^{3} - \frac{7}{3} a - \frac{11}{6}\right)$$ $$11$$ $$4$$ $$I_{4}$$ 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
$$2$$ 2B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 11.1-a consists of curves linked by isogenies of degrees dividing 4.

## Base change

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