# Properties

 Base field $$\Q(\sqrt{2})$$ Label 2.2.8.1-578.1-d2 Conductor $$(17 a)$$ Conductor norm $$578$$ CM no base-change no Q-curve yes Torsion order $$2$$ Rank $$1$$

# Learn more about

Show commands for: Magma / Pari/GP / SageMath

## Base field $$\Q(\sqrt{2})$$

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

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

gp: K = nfinit(a^2 - 2);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 0, 1]);

## Weierstrass equation

$$y^2 + x y = x^{3} + \left(355 a - 193\right) x - 3135 a + 2315$$
sage: E = EllipticCurve(K, [1, 0, 0, 355*a - 193, -3135*a + 2315])

gp: E = ellinit([1, 0, 0, 355*a - 193, -3135*a + 2315],K)

magma: E := ChangeRing(EllipticCurve([1, 0, 0, 355*a - 193, -3135*a + 2315]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(17 a)$$ = $$\left(a\right) \cdot \left(-3 a - 1\right) \cdot \left(3 a - 1\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$578$$ = $$2 \cdot 17^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(-693514167 a + 2585878942)$$ = $$\left(a\right) \cdot \left(-3 a - 1\right)^{3} \cdot \left(3 a - 1\right)^{12}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$5724846103019631586$$ = $$2 \cdot 17^{15}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-\frac{10008966027980714375}{1165244474459522} a + \frac{7568104598365729000}{582622237229761}$$ 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: $$1$$

sage: E.rank()

magma: Rank(E);

Generator: $\left(-\frac{347}{98} a + \frac{1381}{98} : -\frac{27289}{1372} a - \frac{10025}{686} : 1\right)$

sage: gens = E.gens(); gens

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

Height: 5.623384133658726

sage: [P.height() for P in gens]

magma: [Height(P):P in gens];

Regulator: 5.62338413366

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

## Torsion subgroup

Structure: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(-4 a + \frac{47}{4} : 2 a - \frac{47}{8} : 1\right)$ 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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$\left(-3 a - 1\right)$$ $$17$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$
$$\left(3 a - 1\right)$$ $$17$$ $$2$$ $$I_{12}$$ Non-split multiplicative $$1$$ $$1$$ $$12$$ $$12$$

## 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.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6 and 12.
Its isogeny class 578.1-d consists of curves linked by isogenies of degrees dividing 12.

## Base change

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