# Properties

 Base field $$\Q(\zeta_{7})^+$$ Label 3.3.49.1-27.1-a1 Conductor $$(0,3)$$ Conductor norm $$27$$ CM no base-change yes: 147.c1 Q-curve yes Torsion order $$1$$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\zeta_{7})^+$$

Generator $$a$$, with minimal polynomial $$x^{3} - x^{2} - 2 x + 1$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 - 2*x + 1)

gp: K = nfinit(a^3 - a^2 - 2*a + 1);

## Weierstrass equation

$$y^2 + a y = x^{3} - a x^{2} + \left(652 a^{2} - 391 a - 1564\right) x + 10528 a^{2} - 5979 a - 24046$$
magma: E := ChangeRing(EllipticCurve([0, -a, a, 652*a^2 - 391*a - 1564, 10528*a^2 - 5979*a - 24046]),K);

sage: E = EllipticCurve(K, [0, -a, a, 652*a^2 - 391*a - 1564, 10528*a^2 - 5979*a - 24046])

gp: E = ellinit([0, -a, a, 652*a^2 - 391*a - 1564, 10528*a^2 - 5979*a - 24046],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(0,3)$$ = $$\left(3\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$27$$ = $$27$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(1594323,1594323 a,1594323 a^{2} - 3188646)$$ = $$\left(3\right)^{13}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$4052555153018976267$$ = $$27^{13}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$-\frac{1713910976512}{1594323}$$ 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: Trivial magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]

## 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$$(j)_{-}$$
$$\left(3\right)$$ $$27$$ $$13$$ $$I_{13}$$ Split multiplicative $$-1$$ $$1$$ $$13$$ $$13$$

## Galois Representations

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

prime Image of Galois Representation
$$13$$ 13B.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 13.
Its isogeny class 27.1-a consists of curves linked by isogenies of degree 13.

## Base change

This curve is the base-change of elliptic curves 147.c1, defined over $$\Q$$, so it is also a $$\Q$$-curve.