# Properties

 Base field $$\Q(\sqrt{-7})$$ Label 2.0.7.1-28672.7-o1 Conductor $$(-128 a + 64)$$ Conductor norm $$28672$$ CM no base-change yes: 448.g2,3136.e2 Q-curve yes Torsion order $$4$$ Rank not available

# 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 (2.8): K = nfinit(a^2 - a + 2);

## Weierstrass equation

$$y^2 = x^{3} - x^{2} - 10913 x - 436447$$
magma: E := ChangeRing(EllipticCurve([0, -1, 0, -10913, -436447]),K);

sage: E = EllipticCurve(K, [0, -1, 0, -10913, -436447])

gp (2.8): E = ellinit([0, -1, 0, -10913, -436447],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-128 a + 64)$$ = $$\left(a\right)^{6} \cdot \left(-a + 1\right)^{6} \cdot \left(-2 a + 1\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$28672$$ = $$2^{12} \cdot 7$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(481036337152)$$ = $$\left(a\right)^{36} \cdot \left(-a + 1\right)^{36} \cdot \left(-2 a + 1\right)^{2}$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$231395957660612615471104$$ = $$2^{72} \cdot 7^{2}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$-\frac{548347731625}{1835008}$$ 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: $$\Z/2\Z\times\Z/2\Z$$ 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] $\left(-2 a - 59 : 0 : 1\right)$,$\left(2 a - 61 : 0 : 1\right)$ 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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$4$$ $$I_{26}^*$$ Additive $$1$$ $$6$$ $$36$$ $$18$$
$$\left(-a + 1\right)$$ $$2$$ $$4$$ $$I_{26}^*$$ Additive $$1$$ $$6$$ $$36$$ $$18$$
$$\left(-2 a + 1\right)$$ $$7$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2Cs
$$3$$ 3B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 6, 9 and 18.
Its isogeny class 28672.7-o consists of curves linked by isogenies of degrees dividing 36.

## Base change

This curve is the base-change of elliptic curves 448.g2, 3136.e2, defined over $$\Q$$, so it is also a $$\Q$$-curve.