# Properties

 Base field $$\Q(\sqrt{29})$$ Label 2.2.29.1-20.2-a1 Conductor $$(-2 a + 4)$$ Conductor norm $$20$$ CM no base-change no Q-curve no Torsion order $$1$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

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

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

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

gp: K = nfinit(a^2 - a - 7);

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

## Weierstrass equation

$$y^2 + \left(a + 1\right) x y = x^{3} + \left(142 a - 449\right) x + 1521 a - 4857$$
sage: E = EllipticCurve(K, [a + 1, 0, 0, 142*a - 449, 1521*a - 4857])

gp: E = ellinit([a + 1, 0, 0, 142*a - 449, 1521*a - 4857],K)

magma: E := ChangeRing(EllipticCurve([a + 1, 0, 0, 142*a - 449, 1521*a - 4857]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-2 a + 4)$$ = $$\left(2\right) \cdot \left(-a + 2\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$20$$ = $$4 \cdot 5$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(-512 a + 1024)$$ = $$\left(2\right)^{9} \cdot \left(-a + 2\right)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$1310720$$ = $$4^{9} \cdot 5$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-\frac{19984640951}{640} a - \frac{175841574349}{2560}$$ 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 not available.

sage: E.rank()

magma: Rank(E);

Regulator: not available

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$$(j)_{-}$$
$$\left(-a + 2\right)$$ $$5$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$\left(2\right)$$ $$4$$ $$9$$ $$I_{9}$$ Split multiplicative $$-1$$ $$1$$ $$9$$ $$9$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3B.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 20.2-a consists of curves linked by isogenies of degree 3.

## Base change

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