# Properties

 Base field $$\Q(\sqrt{29})$$ Label 2.2.29.1-16.1-a1 Conductor $$(4)$$ Conductor norm $$16$$ CM no base-change no Q-curve yes Torsion order $$1$$ Rank not available

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Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\sqrt{29})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 7$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 7)

gp (2.8): K = nfinit(a^2 - a - 7);

## Weierstrass equation

$$y^2 = x^{3} + \left(a - 1\right) x^{2} + \left(-2 a - 1\right) x - 3 a - 7$$
magma: E := ChangeRing(EllipticCurve([0, a - 1, 0, -2*a - 1, -3*a - 7]),K);

sage: E = EllipticCurve(K, [0, a - 1, 0, -2*a - 1, -3*a - 7])

gp (2.8): E = ellinit([0, a - 1, 0, -2*a - 1, -3*a - 7],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(4)$$ = $$\left(2\right)^{2}$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$16$$ = $$4^{2}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(256)$$ = $$\left(2\right)^{8}$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$65536$$ = $$4^{8}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$-58240 a - 127696$$ magma: jInvariant(E);  sage: E.j_invariant()  gp (2.8): 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()

magma: Generators(E); // includes torsion

sage: E.gens()

Regulator: not available

magma: Regulator(Generators(E));

sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: Trivial magma: TorsionSubgroup(E);  sage: E.torsion_subgroup().gens()  gp (2.8): elltors(E)[2] magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp (2.8): elltors(E)[1]

## 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$$(j)_{-}$$
$$\left(2\right)$$ $$4$$ $$1$$ $$IV^*$$ Additive $$1$$ $$2$$ $$8$$ $$0$$

## Galois Representations

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

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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 7.
Its isogeny class 16.1-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 a $$\Q$$-curve.