# Properties

 Base field $$\Q(\sqrt{21})$$ Label 2.2.21.1-2100.1-bl10 Conductor $$(-20 a + 10)$$ Conductor norm $$2100$$ CM no base-change no Q-curve yes Torsion order $$2$$ 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: K = nfinit(a^2 - a - 5);

## Weierstrass equation

$$y^2 + a x y + \left(a + 1\right) y = x^{3} + x^{2} + \left(-1583 a + 1394\right) x - 7772729 a + 21756901$$
magma: E := ChangeRing(EllipticCurve([a, 1, a + 1, -1583*a + 1394, -7772729*a + 21756901]),K);

sage: E = EllipticCurve(K, [a, 1, a + 1, -1583*a + 1394, -7772729*a + 21756901])

gp: E = ellinit([a, 1, a + 1, -1583*a + 1394, -7772729*a + 21756901],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() $$\mathfrak{D}$$ = $$(-461973562500 a - 414282843750)$$ = $$\left(2\right) \cdot \left(-a + 2\right)^{3} \cdot \left(-a\right)^{6} \cdot \left(-a + 1\right)^{24} \cdot \left(a + 3\right)$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$704079866409301757812500$$ = $$3^{3} \cdot 4 \cdot 5^{30} \cdot 7$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{1953737205973088123082005533}{3755092620849609375} a + \frac{1399882285712874819002585911}{1502037048339843750}$$ magma: jInvariant(E);  sage: E.j_invariant()  gp: 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()

Regulator: not available

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

sage: gens = E.gens(); gens

magma: Regulator(gens);

sage: E.regulator_of_points(gens)

## Torsion subgroup

Structure: $$\Z/2\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(\frac{311}{4} a - \frac{745}{4} : \frac{215}{4} a - \frac{1559}{8} : 1\right)$ magma: [piT(P) : P in Generators(T)];  sage: T.gens()  gp: T[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$$(j)_{-}$$
$$\left(-a + 2\right)$$ $$3$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$
$$\left(-a\right)$$ $$5$$ $$2$$ $$I_{6}$$ Non-split multiplicative $$1$$ $$1$$ $$6$$ $$6$$
$$\left(-a + 1\right)$$ $$5$$ $$2$$ $$I_{24}$$ Non-split multiplicative $$1$$ $$1$$ $$24$$ $$24$$
$$\left(a + 3\right)$$ $$7$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$\left(2\right)$$ $$4$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

## 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.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6, 8, 12 and 24.
Its isogeny class 2100.1-bl consists of curves linked by isogenies of degrees dividing 24.

## Base change

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