# Properties

 Base field 4.4.19821.1 Label 4.4.19821.1-39.1-b1 Conductor $$(39,a^{3} - a^{2} - 7 a + 6)$$ Conductor norm $$39$$ CM no base-change no Q-curve no Torsion order $$1$$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field 4.4.19821.1

Generator $$a$$, with minimal polynomial $$x^{4} - x^{3} - 8 x^{2} + 6 x + 3$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 6, -8, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 - 8*x^2 + 6*x + 3)
gp (2.8): K = nfinit(a^4 - a^3 - 8*a^2 + 6*a + 3);

## Weierstrass equation

$$y^2 + \left(\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a\right) y = x^{3} + \left(a^{2} - 5\right) x^{2} + \left(\frac{17}{3} a^{3} - \frac{61}{3} a^{2} + 8 a + 13\right) x + \frac{112}{3} a^{3} - \frac{410}{3} a^{2} + 68 a + 38$$
magma: E := ChangeRing(EllipticCurve([0, a^2 - 5, 1/3*a^3 + 1/3*a^2 - 3*a, 17/3*a^3 - 61/3*a^2 + 8*a + 13, 112/3*a^3 - 410/3*a^2 + 68*a + 38]),K);
sage: E = EllipticCurve(K, [0, a^2 - 5, 1/3*a^3 + 1/3*a^2 - 3*a, 17/3*a^3 - 61/3*a^2 + 8*a + 13, 112/3*a^3 - 410/3*a^2 + 68*a + 38])
gp (2.8): E = ellinit([0, a^2 - 5, 1/3*a^3 + 1/3*a^2 - 3*a, 17/3*a^3 - 61/3*a^2 + 8*a + 13, 112/3*a^3 - 410/3*a^2 + 68*a + 38],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(39,a^{3} - a^{2} - 7 a + 6)$$ = $$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right) \cdot \left(-\frac{2}{3} a^{3} + \frac{1}{3} a^{2} + 5 a\right)$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$39$$ = $$3 \cdot 13$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(351,9 a + 81,\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + a + 238,-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 8 a + 152)$$ = $$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)^{5} \cdot \left(-\frac{2}{3} a^{3} + \frac{1}{3} a^{2} + 5 a\right)$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$3159$$ = $$3^{5} \cdot 13$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$-\frac{3002368}{351} a^{3} + \frac{4784128}{351} a^{2} + \frac{610304}{351} a - \frac{966656}{117}$$ magma: jInvariant(E); sage: E.j_invariant() gp (2.8): E.j $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) magma: HasComplexMultiplication(E); sage: E.has_cm(), E.cm_discriminant() $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank not available.
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: not available

magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: Trivial magma: TorsionSubgroup(E); sage: E.torsion_subgroup().gens() gp (2.8): elltors(E)[2] magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp (2.8): elltors(E)[1]

## Local data at primes of bad reduction

magma: LocalInformation(E);
sage: E.local_data()
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)$$ $$3$$ $$5$$ $$I_{5}$$ Split multiplicative $$-1$$ $$1$$ $$5$$ $$5$$
$$\left(-\frac{2}{3} a^{3} + \frac{1}{3} a^{2} + 5 a\right)$$ $$13$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ .

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 39.1-b consists of this curve only.

## Base change

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