# Properties

 Base field 3.3.733.1 Label 3.3.733.1-2.1-c1 Conductor $$(2,-a + 2)$$ Conductor norm $$2$$ CM no base-change no Q-curve no Torsion order $$4$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field 3.3.733.1

Generator $$a$$, with minimal polynomial $$x^{3} - x^{2} - 7 x + 8$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 - 7*x + 8)

gp: K = nfinit(a^3 - a^2 - 7*a + 8);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -7, -1, 1]);

## Weierstrass equation

$$y^2 + \left(a^{2} - 5\right) x y + y = x^{3} + \left(-a + 1\right) x^{2} + \left(-1022218105 a^{2} - 1551930198 a + 3247458136\right) x + 33252622286582 a^{2} + 50484087904678 a - 105639391507833$$
sage: E = EllipticCurve(K, [a^2 - 5, -a + 1, 1, -1022218105*a^2 - 1551930198*a + 3247458136, 33252622286582*a^2 + 50484087904678*a - 105639391507833])

gp: E = ellinit([a^2 - 5, -a + 1, 1, -1022218105*a^2 - 1551930198*a + 3247458136, 33252622286582*a^2 + 50484087904678*a - 105639391507833],K)

magma: E := ChangeRing(EllipticCurve([a^2 - 5, -a + 1, 1, -1022218105*a^2 - 1551930198*a + 3247458136, 33252622286582*a^2 + 50484087904678*a - 105639391507833]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(2,-a + 2)$$ = $$\left(a^{2} - 6\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$2$$ = $$2$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(4,a,a^{2} - 4)$$ = $$\left(a^{2} - 6\right)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$4$$ = $$2^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{38239925}{4} a^{2} + \frac{58077475}{4} a - \frac{121417863}{4}$$ 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: $$\Z/2\Z\times\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(\frac{28015}{4} a^{2} + \frac{42533}{4} a - \frac{89005}{4} : -\frac{37573}{8} a^{2} - \frac{57051}{8} a + \frac{119355}{8} : 1\right)$,$\left(\frac{27851}{4} a^{2} + 10571 a - 22121 : -\frac{37353}{8} a^{2} - \frac{56717}{8} a + 14832 : 1\right)$ 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$$(j)_{-}$$
$$\left(a^{2} - 6\right)$$ $$2$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

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

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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 2.1-c consists of curves linked by isogenies of degrees dividing 8.

## Base change

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