# Properties

 Base field $$\Q(\sqrt{21})$$ Label 2.2.21.1-2100.1-bl7 Conductor $$(-20 a + 10)$$ Conductor norm $$2100$$ CM no base-change yes: 1470.b4,630.h4 Q-curve yes Torsion order $$4$$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\sqrt{21})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 5$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 5)
gp (2.8): K = nfinit(a^2 - a - 5);

## Weierstrass equation

$$y^2 + \left(a + 1\right) x y + a y = x^{3} + \left(-a + 1\right) x^{2} + \left(-21789 a - 39218\right) x + 2597374 a + 4653447$$
magma: E := ChangeRing(EllipticCurve([a + 1, -a + 1, a, -21789*a - 39218, 2597374*a + 4653447]),K);
sage: E = EllipticCurve(K, [a + 1, -a + 1, a, -21789*a - 39218, 2597374*a + 4653447])
gp (2.8): E = ellinit([a + 1, -a + 1, a, -21789*a - 39218, 2597374*a + 4653447],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-20 a + 10)$$ = $$\left(2\right) \cdot \left(-a + 2\right) \cdot \left(-a\right) \cdot \left(-a + 1\right) \cdot \left(a + 3\right)$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$2100$$ = $$3 \cdot 4 \cdot 5^{2} \cdot 7$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(184570312500)$$ = $$\left(2\right)^{2} \cdot \left(-a + 2\right)^{6} \cdot \left(-a\right)^{12} \cdot \left(-a + 1\right)^{12} \cdot \left(a + 3\right)^{2}$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$34066200256347656250000$$ = $$3^{6} \cdot 4^{2} \cdot 5^{24} \cdot 7^{2}$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$\frac{9150443179640281}{184570312500}$$ 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: $$\Z/2\Z\times\Z/2\Z$$ 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] $\left(-81 a - 139 : 150 a + 272 : 1\right)$,$\left(39 a + 53 : -66 a - 124 : 1\right)$ 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 Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(-a + 2\right)$$ $$3$$ $$2$$ $$I_{6}$$ Non-split multiplicative $$1$$ $$1$$ $$6$$ $$6$$
$$\left(-a\right)$$ $$5$$ $$2$$ $$I_{12}$$ Non-split multiplicative $$1$$ $$1$$ $$12$$ $$12$$
$$\left(-a + 1\right)$$ $$5$$ $$2$$ $$I_{12}$$ Non-split multiplicative $$1$$ $$1$$ $$12$$ $$12$$
$$\left(a + 3\right)$$ $$7$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$\left(2\right)$$ $$4$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2Cs
$$3$$ 3B.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6 and 12.
Its isogeny class 2100.1-bl consists of curves linked by isogenies of degrees dividing 24.

## Base change

This curve is the base-change of elliptic curves 1470.b4, 630.h4, defined over $$\Q$$, so it is also a $$\Q$$-curve.