# Properties

 Base field 4.4.19821.1 Label 4.4.19821.1-21.1-a2 Conductor $(21,\frac{2}{3} a^{3} - \frac{1}{3} a^{2} - 5 a - 1)$ Conductor norm $21$ CM no base-change no Q-curve not determined Torsion order $6$ Rank not available

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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

$y^2 + \left(\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a\right) x y + \left(a^{2} - a - 3\right) y = x^{3} + \left(\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a - 2\right) x^{2} + \left(-\frac{26}{3} a^{3} - \frac{44}{3} a^{2} + 25 a + 1\right) x + 47 a^{3} + 84 a^{2} - 120 a - 36$
magma: E := ChangeRing(EllipticCurve([1/3*a^3 + 1/3*a^2 - 3*a, 1/3*a^3 + 1/3*a^2 - 3*a - 2, a^2 - a - 3, -26/3*a^3 - 44/3*a^2 + 25*a + 1, 47*a^3 + 84*a^2 - 120*a - 36]),K);
sage: E = EllipticCurve(K, [1/3*a^3 + 1/3*a^2 - 3*a, 1/3*a^3 + 1/3*a^2 - 3*a - 2, a^2 - a - 3, -26/3*a^3 - 44/3*a^2 + 25*a + 1, 47*a^3 + 84*a^2 - 120*a - 36])
gp (2.8): E = ellinit([1/3*a^3 + 1/3*a^2 - 3*a, 1/3*a^3 + 1/3*a^2 - 3*a - 2, a^2 - a - 3, -26/3*a^3 - 44/3*a^2 + 25*a + 1, 47*a^3 + 84*a^2 - 120*a - 36],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $\mathfrak{N}$ = $(21,\frac{2}{3} a^{3} - \frac{1}{3} a^{2} - 5 a - 1)$ = $\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right) \cdot \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 2 a\right)$ magma: Conductor(E); sage: E.conductor() $N(\mathfrak{N})$ = $21$ = $3 \cdot 7$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $\mathfrak{D}$ = $(1058841,9 a + 1021446,\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + a + 78808,-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 8 a + 973016)$ = $\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)^{4} \cdot \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 2 a\right)^{6}$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $N(\mathfrak{D})$ = $9529569$ = $3^{4} \cdot 7^{6}$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $j$ = $\frac{74079408535072}{1058841} a^{3} + \frac{42308209276609}{352947} a^{2} - \frac{67381326805957}{352947} a - \frac{31682015546507}{1058841}$ 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 rank and generators

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: $\Z/6\Z$ 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] $\left(\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + 1 : 3 : 1\right)$ magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E); sage: E.torsion_subgroup().gens() gp (2.8): elltors(E)[3]

## Local data at primes of bad reduction

magma: LocalInformation(E);
sage: E.local_data()
prime Norm Tamagawa number Kodaira symbol Reduction type ord($\mathfrak{N}$) ord($\mathfrak{D}$) ord$(j)_{-}$
$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)$ 3 $4$ $I_{4}$ Split multiplicative 1 4 4
$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 2 a\right)$ 7 $6$ $I_{6}$ Split multiplicative 1 6 6

## Galois Representations

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

prime Image of Galois Representation
$2$ 2B
$3$ 3B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 21.1-a consists of curves linked by isogenies of degrees dividing 6.

## Base change

This curve is not the base-change of an elliptic curve defined over $\Q$. It has not yet been determined whether or not it is a $\Q$-curve.