# Properties

 Base field 4.4.19821.1 Label 4.4.19821.1-57.1-d1 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 $$2$$ Rank not available

# Learn more about

Show commands for: Magma / Pari/GP / SageMath

## Base field 4.4.19821.1

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

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 - 8*x^2 + 6*x + 3)

gp: K = nfinit(a^4 - a^3 - 8*a^2 + 6*a + 3);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 6, -8, -1, 1]);

## Weierstrass equation

$$y^2 + \left(-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 3\right) x y + \left(\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 1\right) y = x^{3} + \left(-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\right) x^{2} + \left(305 a^{3} + 31 a^{2} - 2403 a - 828\right) x - 3302 a^{3} - 350 a^{2} + 26036 a + 8964$$
sage: E = EllipticCurve(K, [-1/3*a^3 + 2/3*a^2 + 2*a - 3, -1/3*a^3 + 2/3*a^2 + 3*a - 3, 1/3*a^3 + 1/3*a^2 - 2*a - 1, 305*a^3 + 31*a^2 - 2403*a - 828, -3302*a^3 - 350*a^2 + 26036*a + 8964])

gp: E = ellinit([-1/3*a^3 + 2/3*a^2 + 2*a - 3, -1/3*a^3 + 2/3*a^2 + 3*a - 3, 1/3*a^3 + 1/3*a^2 - 2*a - 1, 305*a^3 + 31*a^2 - 2403*a - 828, -3302*a^3 - 350*a^2 + 26036*a + 8964],K)

magma: E := ChangeRing(EllipticCurve([-1/3*a^3 + 2/3*a^2 + 2*a - 3, -1/3*a^3 + 2/3*a^2 + 3*a - 3, 1/3*a^3 + 1/3*a^2 - 2*a - 1, 305*a^3 + 31*a^2 - 2403*a - 828, -3302*a^3 - 350*a^2 + 26036*a + 8964]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(57,\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 3 a + 1)$$ = $$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right) \cdot \left(\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 2 a + 5\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$57$$ = $$3 \cdot 19$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(46619598843,129140163 a + 20145865428,\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + 108143056 a + 29553442726,-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 89208332 a + 10003103333)$$ = $$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)^{34} \cdot \left(\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 2 a + 5\right)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$6020462593579631409$$ = $$3^{34} \cdot 19^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{4957694999417395}{46619598843} a^{3} + \frac{9593307021539665}{46619598843} a^{2} - \frac{11878525618289761}{46619598843} a - \frac{5175805466469244}{46619598843}$$ 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); $\left(-\frac{5}{4} a^{3} + \frac{1}{4} a^{2} + \frac{37}{4} a + \frac{5}{2} : -\frac{5}{12} a^{3} - \frac{5}{12} a^{2} + \frac{31}{8} a + \frac{3}{8} : 1\right)$ 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$$(j)_{-}$$
$$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)$$ $$3$$ $$2$$ $$I_{34}$$ Non-split multiplicative $$1$$ $$1$$ $$34$$ $$34$$
$$\left(\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 2 a + 5\right)$$ $$19$$ $$2$$ $$I_{2}$$ 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
$$2$$ 2B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 57.1-d consists of curves linked by isogenies of degree 2.

## Base change

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