# Properties

 Base field 5.5.36497.1 Label 5.5.36497.1-37.1-a1 Conductor $$(37,a^{4} - 2 a^{3} - 2 a^{2} + 4 a + 1)$$ Conductor norm $$37$$ CM no base-change no Q-curve no Torsion order $$1$$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field 5.5.36497.1

Generator $$a$$, with minimal polynomial $$x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, 5, -3, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - 2*x^4 - 3*x^3 + 5*x^2 + x - 1)
gp (2.8): K = nfinit(a^5 - 2*a^4 - 3*a^3 + 5*a^2 + a - 1);

## Weierstrass equation

$$y^2 + a x y + \left(2 a^{4} - 3 a^{3} - 6 a^{2} + 7 a + 2\right) y = x^{3} + \left(2 a^{4} - 3 a^{3} - 7 a^{2} + 7 a + 3\right) x^{2} + \left(32 a^{4} - 13 a^{3} - 148 a^{2} - 4 a + 30\right) x + 154 a^{4} - 69 a^{3} - 676 a^{2} - 48 a + 128$$
magma: E := ChangeRing(EllipticCurve([a, 2*a^4 - 3*a^3 - 7*a^2 + 7*a + 3, 2*a^4 - 3*a^3 - 6*a^2 + 7*a + 2, 32*a^4 - 13*a^3 - 148*a^2 - 4*a + 30, 154*a^4 - 69*a^3 - 676*a^2 - 48*a + 128]),K);
sage: E = EllipticCurve(K, [a, 2*a^4 - 3*a^3 - 7*a^2 + 7*a + 3, 2*a^4 - 3*a^3 - 6*a^2 + 7*a + 2, 32*a^4 - 13*a^3 - 148*a^2 - 4*a + 30, 154*a^4 - 69*a^3 - 676*a^2 - 48*a + 128])
gp (2.8): E = ellinit([a, 2*a^4 - 3*a^3 - 7*a^2 + 7*a + 3, 2*a^4 - 3*a^3 - 6*a^2 + 7*a + 2, 32*a^4 - 13*a^3 - 148*a^2 - 4*a + 30, 154*a^4 - 69*a^3 - 676*a^2 - 48*a + 128],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(37,a^{4} - 2 a^{3} - 2 a^{2} + 4 a + 1)$$ = $$\left(a^{4} - 2 a^{3} - 2 a^{2} + 4 a + 1\right)$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$37$$ = $$37$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(50653,a + 11342,a^{4} - 2 a^{3} - 3 a^{2} + 5 a + 34054,a^{4} - a^{3} - 4 a^{2} + 2 a + 9982,a^{2} - a + 6314)$$ = $$\left(a^{4} - 2 a^{3} - 2 a^{2} + 4 a + 1\right)^{3}$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$50653$$ = $$37^{3}$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$-\frac{1272945793794372}{50653} a^{4} + \frac{859591705080800}{50653} a^{3} + \frac{4922533480405137}{50653} a^{2} + \frac{233798144853843}{50653} a - \frac{948239678172729}{50653}$$ 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: Trivial 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]

## 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$$(j)_{-}$$
$$\left(a^{4} - 2 a^{3} - 2 a^{2} + 4 a + 1\right)$$ $$37$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3B.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 37.1-a consists of curves linked by isogenies of degree3.

## Base change

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