# Properties

 Base field $$\Q(\sqrt{-7})$$ Label 2.0.7.1-6300.2-e2 Conductor $$(-60 a + 30)$$ Conductor norm $$6300$$ CM no base-change yes: 1470.q6,210.c6 Q-curve yes Torsion order $$8$$ Rank $$0$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\sqrt{-7})$$

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

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

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

gp: K = nfinit(a^2 - a + 2);

## Weierstrass equation

$$y^2 + x y + y = x^{3} + x^{2} + 10 x - 13$$
magma: E := ChangeRing(EllipticCurve([1, 1, 1, 10, -13]),K);

sage: E = EllipticCurve(K, [1, 1, 1, 10, -13])

gp: E = ellinit([1, 1, 1, 10, -13],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-60 a + 30)$$ = $$\left(a\right) \cdot \left(-a + 1\right) \cdot \left(3\right) \cdot \left(5\right) \cdot \left(-2 a + 1\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$6300$$ = $$2^{2} \cdot 7 \cdot 9 \cdot 25$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(188160)$$ = $$\left(a\right)^{8} \cdot \left(-a + 1\right)^{8} \cdot \left(3\right) \cdot \left(5\right) \cdot \left(-2 a + 1\right)^{4}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$35404185600$$ = $$2^{16} \cdot 7^{4} \cdot 9 \cdot 25$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{109902239}{188160}$$ 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: $$0$$

magma: Rank(E);

sage: E.rank()

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: $$\Z/8\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(-2 a - 1 : 4 a - 5 : 1\right)$ magma: [piT(P) : P in Generators(T)];  sage: T.gens()  gp: T[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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$8$$ $$I_{8}$$ Split multiplicative $$-1$$ $$1$$ $$8$$ $$8$$
$$\left(-a + 1\right)$$ $$2$$ $$8$$ $$I_{8}$$ Split multiplicative $$-1$$ $$1$$ $$8$$ $$8$$
$$\left(-2 a + 1\right)$$ $$7$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(3\right)$$ $$9$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$\left(5\right)$$ $$25$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4 and 8.
Its isogeny class 6300.2-e consists of curves linked by isogenies of degrees dividing 8.

## Base change

This curve is the base-change of elliptic curves 1470.q6, 210.c6, defined over $$\Q$$, so it is also a $$\Q$$-curve.