# Properties

 Base field 4.4.13888.1 Label 4.4.13888.1-28.2-b2 Conductor $$(14,\frac{2}{3} a^{3} - \frac{7}{3} a^{2} - \frac{5}{3} a + 5)$$ Conductor norm $$28$$ CM no base-change no Q-curve no Torsion order $$1$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field 4.4.13888.1

Generator $$a$$, with minimal polynomial $$x^{4} - 2 x^{3} - 7 x^{2} + 6 x + 9$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - 2*x^3 - 7*x^2 + 6*x + 9)

gp: K = nfinit(a^4 - 2*a^3 - 7*a^2 + 6*a + 9);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 6, -7, -2, 1]);

## Weierstrass equation

$$y^2 + \left(\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - \frac{1}{3} a + 1\right) x y + \left(\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{4}{3} a - 2\right) y = x^{3} + \left(-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + \frac{1}{3} a\right) x^{2} + \left(\frac{10}{3} a^{3} - \frac{32}{3} a^{2} - \frac{61}{3} a + 42\right) x + \frac{541}{3} a^{3} - \frac{1574}{3} a^{2} - \frac{2383}{3} a + 1802$$
sage: E = EllipticCurve(K, [1/3*a^3 - 2/3*a^2 - 1/3*a + 1, -1/3*a^3 + 2/3*a^2 + 1/3*a, 1/3*a^3 + 1/3*a^2 - 4/3*a - 2, 10/3*a^3 - 32/3*a^2 - 61/3*a + 42, 541/3*a^3 - 1574/3*a^2 - 2383/3*a + 1802])

gp: E = ellinit([1/3*a^3 - 2/3*a^2 - 1/3*a + 1, -1/3*a^3 + 2/3*a^2 + 1/3*a, 1/3*a^3 + 1/3*a^2 - 4/3*a - 2, 10/3*a^3 - 32/3*a^2 - 61/3*a + 42, 541/3*a^3 - 1574/3*a^2 - 2383/3*a + 1802],K)

magma: E := ChangeRing(EllipticCurve([1/3*a^3 - 2/3*a^2 - 1/3*a + 1, -1/3*a^3 + 2/3*a^2 + 1/3*a, 1/3*a^3 + 1/3*a^2 - 4/3*a - 2, 10/3*a^3 - 32/3*a^2 - 61/3*a + 42, 541/3*a^3 - 1574/3*a^2 - 2383/3*a + 1802]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(14,\frac{2}{3} a^{3} - \frac{7}{3} a^{2} - \frac{5}{3} a + 5)$$ = $$\left(\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - \frac{4}{3} a + 1\right) \cdot \left(\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - 1\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$28$$ = $$4 \cdot 7$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(686,\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - \frac{4}{3} a + 579,686 a,a^{2} + 577 a - 3)$$ = $$\left(\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - \frac{4}{3} a + 1\right) \cdot \left(\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - 1\right)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$470596$$ = $$4 \cdot 7^{6}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{557101}{2058} a^{3} + \frac{132200}{1029} a^{2} - \frac{2506865}{1029} a - \frac{1440367}{686}$$ 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: Trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(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$$(j)_{-}$$
$$\left(\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - 1\right)$$ $$7$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$\left(\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - \frac{4}{3} a + 1\right)$$ $$4$$ $$1$$ $$I_{1}$$ Non-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
$$3$$ 3B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 28.2-b consists of curves linked by isogenies of degree 3.

## Base change

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