# Properties

 Base field $$\Q(\sqrt{23})$$ Label 2.2.92.1-200.1-a4 Conductor $$(-10 a + 50)$$ Conductor norm $$200$$ CM no base-change yes: 80.a1,21160.e1 Q-curve yes Torsion order $$4$$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\sqrt{23})$$

Generator $$a$$, with minimal polynomial $$x^{2} - 23$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-23, 0, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 23)

gp: K = nfinit(a^2 - 23);

## Weierstrass equation

$$y^2 + \left(a + 1\right) x y = x^{3} + a x^{2} + \left(-6414 a - 30760\right) x + 584948 a + 2805312$$
magma: E := ChangeRing(EllipticCurve([a + 1, a, 0, -6414*a - 30760, 584948*a + 2805312]),K);

sage: E = EllipticCurve(K, [a + 1, a, 0, -6414*a - 30760, 584948*a + 2805312])

gp: E = ellinit([a + 1, a, 0, -6414*a - 30760, 584948*a + 2805312],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-10 a + 50)$$ = $$\left(-a + 5\right)^{3} \cdot \left(5\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$200$$ = $$2^{3} \cdot 25$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(80)$$ = $$\left(-a + 5\right)^{8} \cdot \left(5\right)$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$6400$$ = $$2^{8} \cdot 25$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{132304644}{5}$$ 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: $$\Z/4\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(12 a + 58 : -84 a - 402 : 1\right)$ 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$$(j)_{-}$$
$$\left(-a + 5\right)$$ $$2$$ $$4$$ $$I_{1}^*$$ Additive $$-1$$ $$3$$ $$8$$ $$0$$
$$\left(5\right)$$ $$25$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

## 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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 200.1-a consists of curves linked by isogenies of degrees dividing 4.

## Base change

This curve is the base-change of elliptic curves 80.a1, 21160.e1, defined over $$\Q$$, so it is also a $$\Q$$-curve.