# Properties

 Base field 5.5.65657.1 Label 5.5.65657.1-25.1-c1 Conductor $$(25,-a^{3} + a^{2} + 3 a - 2)$$ Conductor norm $$25$$ CM no base-change no Q-curve no Torsion order $$1$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field 5.5.65657.1

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

sage: x = polygen(QQ); K.<a> = NumberField(x^5 - x^4 - 5*x^3 + 2*x^2 + 5*x + 1)

gp: K = nfinit(a^5 - a^4 - 5*a^3 + 2*a^2 + 5*a + 1);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -5, -1, 1]);

## Weierstrass equation

$$y^2 + \left(-a^{4} + 2 a^{3} + 4 a^{2} - 5 a - 3\right) x y + \left(a^{3} - 4 a - 1\right) y = x^{3} + \left(-a^{4} + 2 a^{3} + 3 a^{2} - 4 a - 2\right) x^{2} + \left(1015 a^{4} - 1849 a^{3} - 3633 a^{2} + 4940 a + 991\right) x - 23261 a^{4} + 41576 a^{3} + 81774 a^{2} - 113086 a - 22399$$
sage: E = EllipticCurve(K, [-a^4 + 2*a^3 + 4*a^2 - 5*a - 3, -a^4 + 2*a^3 + 3*a^2 - 4*a - 2, a^3 - 4*a - 1, 1015*a^4 - 1849*a^3 - 3633*a^2 + 4940*a + 991, -23261*a^4 + 41576*a^3 + 81774*a^2 - 113086*a - 22399])

gp: E = ellinit([-a^4 + 2*a^3 + 4*a^2 - 5*a - 3, -a^4 + 2*a^3 + 3*a^2 - 4*a - 2, a^3 - 4*a - 1, 1015*a^4 - 1849*a^3 - 3633*a^2 + 4940*a + 991, -23261*a^4 + 41576*a^3 + 81774*a^2 - 113086*a - 22399],K)

magma: E := ChangeRing(EllipticCurve([-a^4 + 2*a^3 + 4*a^2 - 5*a - 3, -a^4 + 2*a^3 + 3*a^2 - 4*a - 2, a^3 - 4*a - 1, 1015*a^4 - 1849*a^3 - 3633*a^2 + 4940*a + 991, -23261*a^4 + 41576*a^3 + 81774*a^2 - 113086*a - 22399]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(25,-a^{3} + a^{2} + 3 a - 2)$$ = $$\left(-a^{2} + a + 2\right)^{2}$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$25$$ = $$5^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(1953125,a + 913693,a^{4} - a^{3} - 4 a^{2} + 2 a + 1627699,-a^{4} + a^{3} + 5 a^{2} - 3 a + 450984,-a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 853586)$$ = $$\left(-a^{2} + a + 2\right)^{9}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$1953125$$ = $$5^{9}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{9227278873526453906244312610219036327836}{125} a^{4} + \frac{3459009377990835094505717267240191015816}{125} a^{3} - \frac{41380713886380859304809564621687978514293}{125} a^{2} - \frac{38438451731868504360194666541225269140904}{125} a - \frac{6711393728551335750914883813930912798473}{125}$$ 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: Trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(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$$(j)_{-}$$
$$\left(-a^{2} + a + 2\right)$$ $$5$$ $$4$$ $$I_{3}^*$$ Additive $$1$$ $$2$$ $$9$$ $$3$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3B
$$5$$ 5B.4.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3, 5 and 15.
Its isogeny class 25.1-c consists of curves linked by isogenies of degrees dividing 15.

## Base change

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