# Properties

 Base field $$\Q(\sqrt{21})$$ Label 2.2.21.1-2100.1-n5 Conductor $$(-20 a + 10)$$ Conductor norm $$2100$$ CM no base-change yes: 210.b6,4410.bi6 Q-curve yes Torsion order $$24$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

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

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

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 5)

gp: K = nfinit(a^2 - a - 5);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -1, 1]);

## Weierstrass equation

$$y^2 + x y + y = x^{3} - 578 x + 2756$$
sage: E = EllipticCurve(K, [1, 0, 1, -578, 2756])

gp: E = ellinit([1, 0, 1, -578, 2756],K)

magma: E := ChangeRing(EllipticCurve([1, 0, 1, -578, 2756]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-20 a + 10)$$ = $$\left(2\right) \cdot \left(-a + 2\right) \cdot \left(-a\right) \cdot \left(-a + 1\right) \cdot \left(a + 3\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$2100$$ = $$3 \cdot 4 \cdot 5^{2} \cdot 7$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(8930250000)$$ = $$\left(2\right)^{4} \cdot \left(-a + 2\right)^{12} \cdot \left(-a\right)^{6} \cdot \left(-a + 1\right)^{6} \cdot \left(a + 3\right)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$79749365062500000000$$ = $$3^{12} \cdot 4^{4} \cdot 5^{12} \cdot 7^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{21302308926361}{8930250000}$$ 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: $$\Z/2\Z\times\Z/12\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(-10 a + 5 : 10 a + 47 : 1\right)$,$\left(5 : -3 : 1\right)$ 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$$(j)_{-}$$
$$\left(-a + 2\right)$$ $$3$$ $$12$$ $$I_{12}$$ Split multiplicative $$-1$$ $$1$$ $$12$$ $$12$$
$$\left(-a\right)$$ $$5$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$\left(-a + 1\right)$$ $$5$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$\left(a + 3\right)$$ $$7$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(2\right)$$ $$4$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ except those listed.

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

## 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-n consists of curves linked by isogenies of degrees dividing 24.

## Base change

This curve is the base-change of elliptic curves 210.b6, 4410.bi6, defined over $$\Q$$, so it is also a $$\Q$$-curve.