# Properties

 Base field $$\Q(\sqrt{5})$$ Label 2.2.5.1-41.2-a1 Conductor $$(\phi - 7)$$ Conductor norm $$41$$ CM no base-change no Q-curve no Torsion order $$1$$ Rank $$0$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field $$\Q(\sqrt{5})$$

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

sage: x = polygen(QQ); K.<phi> = NumberField(x^2 - x - 1)

gp: K = nfinit(phi^2 - phi - 1);

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

## Weierstrass equation

$$y^2 + \left(\phi + 1\right) y = x^{3} + \left(\phi - 1\right) x^{2} + \left(-10 \phi - 30\right) x - 32 \phi - 82$$
sage: E = EllipticCurve(K, [0, phi - 1, phi + 1, -10*phi - 30, -32*phi - 82])

gp: E = ellinit([0, phi - 1, phi + 1, -10*phi - 30, -32*phi - 82],K)

magma: E := ChangeRing(EllipticCurve([0, phi - 1, phi + 1, -10*phi - 30, -32*phi - 82]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(\phi - 7)$$ = $$\left(\phi - 7\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$41$$ = $$41$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(404979 \phi - 101219)$$ = $$\left(\phi - 7\right)^{7}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$194754273881$$ = $$41^{7}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-\frac{7215644871110656}{194754273881} \phi - \frac{4677283260284928}{194754273881}$$ 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: $$0$$

sage: E.rank()

magma: Rank(E);

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: Trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(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$$(j)_{-}$$
$$\left(\phi - 7\right)$$ $$41$$ $$1$$ $$I_{7}$$ Non-split multiplicative $$1$$ $$1$$ $$7$$ $$7$$

## Galois Representations

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

prime Image of Galois Representation
$$7$$ 7B.1.3

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 7.
Its isogeny class 41.2-a consists of curves linked by isogenies of degree 7.

## Base change

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