# Properties

 Base field 4.4.4205.1 Label 4.4.4205.1-49.2-a3 Conductor $$(7,-a^{3} + a^{2} + 6 a)$$ Conductor norm $$49$$ CM no base-change no Q-curve no Torsion order $$2$$ Rank not available

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Show commands for: Magma / Pari/GP / SageMath

## Base field 4.4.4205.1

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

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

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

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

## Weierstrass equation

$$y^2 + x y = x^{3} + \left(a^{3} - a^{2} - 6 a - 2\right) x^{2} - 27 x + 15 a^{3} - 15 a^{2} - 90 a - 5$$
sage: E = EllipticCurve(K, [1, a^3 - a^2 - 6*a - 2, 0, -27, 15*a^3 - 15*a^2 - 90*a - 5])

gp: E = ellinit([1, a^3 - a^2 - 6*a - 2, 0, -27, 15*a^3 - 15*a^2 - 90*a - 5],K)

magma: E := ChangeRing(EllipticCurve([1, a^3 - a^2 - 6*a - 2, 0, -27, 15*a^3 - 15*a^2 - 90*a - 5]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(7,-a^{3} + a^{2} + 6 a)$$ = $$\left(-a^{3} + 2 a^{2} + 3 a - 3\right) \cdot \left(a^{2} - 2 a - 3\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$49$$ = $$7^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(49,49 a,a^{3} - a^{2} + 43 a + 7,-a^{3} + 2 a^{2} + 47 a + 43)$$ = $$\left(-a^{3} + 2 a^{2} + 3 a - 3\right)^{2} \cdot \left(a^{2} - 2 a - 3\right)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$2401$$ = $$7^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{213433415640625}{49} a^{3} - \frac{213433415640625}{49} a^{2} - \frac{1280600493843750}{49} a + \frac{467970351097797}{49}$$ 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: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(-a^{3} + a^{2} + 6 a + \frac{23}{4} : \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - 3 a - \frac{23}{8} : 1\right)$ 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$$(j)_{-}$$
$$\left(-a^{3} + 2 a^{2} + 3 a - 3\right)$$ $$7$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$\left(a^{2} - 2 a - 3\right)$$ $$7$$ $$2$$ $$I_{2}$$ Non-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
$$2$$ 2B
$$5$$ 5B.4.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 5 and 10.
Its isogeny class 49.2-a consists of curves linked by isogenies of degrees dividing 10.

## Base change

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