# Properties

 Base field $$\Q(\sqrt{29})$$ Label 2.2.29.1-25.1-b4 Conductor $$(5)$$ Conductor norm $$25$$ CM no base-change no Q-curve no Torsion order $$2$$ Rank not available

# Related objects

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

$$y^2 + a x y + \left(a + 1\right) y = x^{3} + \left(57 a - 198\right) x + 397 a - 1296$$
magma: E := ChangeRing(EllipticCurve([a, 0, a + 1, 57*a - 198, 397*a - 1296]),K);

sage: E = EllipticCurve(K, [a, 0, a + 1, 57*a - 198, 397*a - 1296])

gp (2.8): E = ellinit([a, 0, a + 1, 57*a - 198, 397*a - 1296],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(5)$$ = $$\left(-a - 1\right) \cdot \left(-a + 2\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$25$$ = $$5^{2}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(1165 a - 855)$$ = $$\left(-a - 1\right) \cdot \left(-a + 2\right)^{9}$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$9765625$$ = $$5^{10}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$\frac{2164654005908433}{1953125} a + \frac{4746203093393263}{1953125}$$ 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: $$\Z/2\Z$$ 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] $\left(a - 6 : 2 a - 4 : 1\right)$ magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);  sage: E.torsion_subgroup().gens()  gp (2.8): elltors(E)[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$$(j)_{-}$$
$$\left(-a - 1\right)$$ $$5$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$\left(-a + 2\right)$$ $$5$$ $$1$$ $$I_{9}$$ Non-split multiplicative $$1$$ $$1$$ $$9$$ $$9$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2B
$$3$$ 3B.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 25.1-b consists of curves linked by isogenies of degrees dividing 6.

## Base change

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