# Properties

 Base field 5.5.36497.1 Label 5.5.36497.1-39.1-b7 Conductor $$(39,2 a^{4} - 3 a^{3} - 7 a^{2} + 6 a + 2)$$ Conductor norm $$39$$ CM no base-change no Q-curve no Torsion order $$2$$ 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: K = nfinit(a^5 - 2*a^4 - 3*a^3 + 5*a^2 + a - 1);

## Weierstrass equation

$$y^2 + \left(a^{2} - a - 1\right) x y + \left(a^{2} - a - 1\right) y = x^{3} + \left(-a^{2} + a + 2\right) x^{2} + \left(10238 a^{4} - 25455 a^{3} - 18068 a^{2} + 59680 a - 20375\right) x + 897162 a^{4} - 2245513 a^{3} - 1558277 a^{2} + 5265005 a - 1771850$$
magma: E := ChangeRing(EllipticCurve([a^2 - a - 1, -a^2 + a + 2, a^2 - a - 1, 10238*a^4 - 25455*a^3 - 18068*a^2 + 59680*a - 20375, 897162*a^4 - 2245513*a^3 - 1558277*a^2 + 5265005*a - 1771850]),K);

sage: E = EllipticCurve(K, [a^2 - a - 1, -a^2 + a + 2, a^2 - a - 1, 10238*a^4 - 25455*a^3 - 18068*a^2 + 59680*a - 20375, 897162*a^4 - 2245513*a^3 - 1558277*a^2 + 5265005*a - 1771850])

gp: E = ellinit([a^2 - a - 1, -a^2 + a + 2, a^2 - a - 1, 10238*a^4 - 25455*a^3 - 18068*a^2 + 59680*a - 20375, 897162*a^4 - 2245513*a^3 - 1558277*a^2 + 5265005*a - 1771850],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(39,2 a^{4} - 3 a^{3} - 7 a^{2} + 6 a + 2)$$ = $$\left(a^{2} - 1\right) \cdot \left(a^{3} - 3 a\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$39$$ = $$3 \cdot 13$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(1255737557015654093436832343547201,a + 170040169446919936099032081033994,a^{4} - 2 a^{3} - 3 a^{2} + 5 a + 724812017186553703397983614715136,a^{4} - a^{3} - 4 a^{2} + 2 a + 1020525238934952913433544527113438,a^{2} - a + 1201104003246170743075378183277650)$$ = $$\left(a^{2} - 1\right)^{4} \cdot \left(a^{3} - 3 a\right)^{28}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$1255737557015654093436832343547201$$ = $$3^{4} \cdot 13^{28}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{719563077579420768826778357723754883977964219744702}{1255737557015654093436832343547201} a^{4} - \frac{381278393594661046595984821496687550631447920681206}{418579185671884697812277447849067} a^{3} - \frac{876030344833436911152144359572205314535143035049625}{418579185671884697812277447849067} a^{2} + \frac{2519311520172502928352178004640753313415969058351782}{1255737557015654093436832343547201} a + \frac{1753426510580954709523077468627597477978011837927249}{1255737557015654093436832343547201}$$ magma: jInvariant(E);  sage: E.j_invariant()  gp: 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()

Regulator: not available

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

sage: gens = E.gens(); gens

magma: Regulator(gens);

sage: E.regulator_of_points(gens)

## Torsion subgroup

Structure: $$\Z/2\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(-\frac{241}{4} a^{4} + \frac{273}{2} a^{3} + \frac{533}{4} a^{2} - \frac{635}{2} a + \frac{351}{4} : -\frac{115}{8} a^{4} + \frac{229}{8} a^{3} + 24 a^{2} - \frac{369}{8} a + \frac{25}{4} : 1\right)$ magma: [piT(P) : P in Generators(T)];  sage: T.gens()  gp: T[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$$(j)_{-}$$
$$\left(a^{2} - 1\right)$$ $$3$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(a^{3} - 3 a\right)$$ $$13$$ $$2$$ $$I_{28}$$ Non-split multiplicative $$1$$ $$1$$ $$28$$ $$28$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2B
$$7$$ 7B.1.3

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4, 7, 14 and 28.
Its isogeny class 39.1-b consists of curves linked by isogenies of degrees dividing 28.

## Base change

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