# Properties

 Base field 4.4.19821.1 Label 4.4.19821.1-57.1-c1 Conductor $$(57,\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 3 a + 1)$$ Conductor norm $$57$$ CM no base-change no Q-curve no Torsion order $$1$$ Rank not available

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## 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

$$y^2 + \left(a^{2} - a - 4\right) x y + \left(a^{2} - a - 4\right) y = x^{3} + \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a\right) x^{2} + \left(\frac{106}{3} a^{3} + \frac{10}{3} a^{2} - 278 a - 102\right) x + 883 a^{3} + 93 a^{2} - 6961 a - 2401$$
magma: E := ChangeRing(EllipticCurve([a^2 - a - 4, -1/3*a^3 - 1/3*a^2 + 3*a, a^2 - a - 4, 106/3*a^3 + 10/3*a^2 - 278*a - 102, 883*a^3 + 93*a^2 - 6961*a - 2401]),K);
sage: E = EllipticCurve(K, [a^2 - a - 4, -1/3*a^3 - 1/3*a^2 + 3*a, a^2 - a - 4, 106/3*a^3 + 10/3*a^2 - 278*a - 102, 883*a^3 + 93*a^2 - 6961*a - 2401])
gp (2.8): E = ellinit([a^2 - a - 4, -1/3*a^3 - 1/3*a^2 + 3*a, a^2 - a - 4, 106/3*a^3 + 10/3*a^2 - 278*a - 102, 883*a^3 + 93*a^2 - 6961*a - 2401],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(57,\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 3 a + 1)$$ = $$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right) \cdot \left(\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 2 a + 5\right)$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$57$$ = $$3 \cdot 19$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(41553,2187 a + 8748,\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + 280 a + 30586,-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 602 a + 34334)$$ = $$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)^{14} \cdot \left(\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 2 a + 5\right)$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$90876411$$ = $$3^{14} \cdot 19$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$-\frac{404963364612649}{4617} a^{3} + \frac{4913008494921869}{41553} a^{2} + \frac{27447645111084058}{41553} a - \frac{31419748638630586}{41553}$$ 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$$(j)_{-}$$
$$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)$$ $$3$$ $$2$$ $$I_{14}$$ Non-split multiplicative $$1$$ $$1$$ $$14$$ $$14$$
$$\left(\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 2 a + 5\right)$$ $$19$$ $$1$$ $$I_{1}$$ 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 57.1-c 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.