# Properties

 Base field $$\Q(\sqrt{7})$$ Label 2.2.28.1-578.1-a1 Conductor $$(17 a + 51)$$ Conductor norm $$578$$ CM no base-change yes: 34.a4,13328.e4 Q-curve yes Torsion order $$6$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

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

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

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

gp: K = nfinit(a^2 - 7);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, 0, 1]);

## Weierstrass equation

$$y^2 + x y = x^{3} - 3 x + 1$$
sage: E = EllipticCurve(K, [1, 0, 0, -3, 1])

gp: E = ellinit([1, 0, 0, -3, 1],K)

magma: E := ChangeRing(EllipticCurve([1, 0, 0, -3, 1]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(17 a + 51)$$ = $$\left(a + 3\right) \cdot \left(17\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$578$$ = $$2 \cdot 289$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(1088)$$ = $$\left(a + 3\right)^{12} \cdot \left(17\right)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$1183744$$ = $$2^{12} \cdot 289$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{3048625}{1088}$$ 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: $$\Z/6\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(0 : 1 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(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$$(j)_{-}$$
$$\left(a + 3\right)$$ $$2$$ $$12$$ $$I_{12}$$ Split multiplicative $$-1$$ $$1$$ $$12$$ $$12$$
$$\left(17\right)$$ $$289$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2B
$$3$$ 3B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 578.1-a consists of curves linked by isogenies of degrees dividing 6.

## Base change

This curve is the base-change of elliptic curves 34.a4, 13328.e4, defined over $$\Q$$, so it is also a $$\Q$$-curve.