# Properties

 Label 2.2.21.1-25.1-a2 Base field $$\Q(\sqrt{21})$$ Conductor $$(5)$$ Conductor norm $$25$$ CM no Base change no Q-curve yes Torsion order $$2$$ Rank $$0$$

# 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

$$y^2+\left(a+1\right)xy+\left(a+1\right)y=x^{3}+\left(a+1\right)x^{2}+\left(29a-66\right)x-125a+369$$
sage: E = EllipticCurve(K, [a + 1, a + 1, a + 1, 29*a - 66, -125*a + 369])

gp: E = ellinit([a + 1, a + 1, a + 1, 29*a - 66, -125*a + 369],K)

magma: E := ChangeRing(EllipticCurve([a + 1, a + 1, a + 1, 29*a - 66, -125*a + 369]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(5)$$ = $$\left(-a\right) \cdot \left(-a + 1\right)$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$25$$ = $$5^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(35625 a + 381250)$$ = $$\left(-a\right)^{12} \cdot \left(-a + 1\right)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$152587890625$$ = $$5^{16}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{359104782699}{244140625} a - \frac{619846590969}{244140625}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$

## Mordell-Weil group

 Rank: $$0$$ Torsion structure: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(\frac{9}{4} a - \frac{17}{2} : \frac{3}{2} a - \frac{15}{8} : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$4.96153189471116$$ Tamagawa product: $$4$$  =  $$2\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$1.08269502240246$$ Analytic order of Ш: $$1$$ (rounded)

## 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$$(j)_{-}$$
$$\left(-a\right)$$ $$5$$ $$2$$ $$I_{12}$$ Non-split multiplicative $$1$$ $$1$$ $$12$$ $$12$$
$$\left(-a + 1\right)$$ $$5$$ $$2$$ $$I_{4}$$ Non-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
$$2$$ 2B
$$3$$ 3B

## 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 25.1-a consists of curves linked by isogenies of degrees dividing 12.

## Base change

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