Properties

 Base field $$\Q(\zeta_{15})^+$$ Label 4.4.1125.1-145.3-a7 Conductor $$(145,-a^{3} - a^{2} + 3 a - 2)$$ Conductor norm $$145$$ CM no base-change no Q-curve no Torsion order $$4$$ Rank not available

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Base field $$\Q(\zeta_{15})^+$$

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

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

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -4, -1, 1]);

Weierstrass equation

$$y^2 + \left(a + 1\right) x y + \left(a^{3} - 3 a + 1\right) y = x^{3} + \left(-a^{3} + a^{2} + 4 a - 2\right) x^{2} + \left(2 a^{3} + a^{2} - 6 a\right) x + 7 a^{3} + 3 a^{2} - 24 a - 6$$
sage: E = EllipticCurve(K, [a + 1, -a^3 + a^2 + 4*a - 2, a^3 - 3*a + 1, 2*a^3 + a^2 - 6*a, 7*a^3 + 3*a^2 - 24*a - 6])

gp: E = ellinit([a + 1, -a^3 + a^2 + 4*a - 2, a^3 - 3*a + 1, 2*a^3 + a^2 - 6*a, 7*a^3 + 3*a^2 - 24*a - 6],K)

magma: E := ChangeRing(EllipticCurve([a + 1, -a^3 + a^2 + 4*a - 2, a^3 - 3*a + 1, 2*a^3 + a^2 - 6*a, 7*a^3 + 3*a^2 - 24*a - 6]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(145,-a^{3} - a^{2} + 3 a - 2)$$ = $$\left(-a - 1\right) \cdot \left(-a^{3} + a^{2} + 4 a - 2\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$145$$ = $$5 \cdot 29$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(145,a^{3} - 3 a + 53,a + 131,a^{2} + 94)$$ = $$\left(-a - 1\right) \cdot \left(-a^{3} + a^{2} + 4 a - 2\right)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$145$$ = $$5 \cdot 29$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-\frac{451821}{145} a^{3} - \frac{571893}{145} a^{2} + \frac{1829142}{145} a + \frac{1322804}{145}$$ 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: $$\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(a^{3} - 4 a : -a^{3} + 4 a : 1\right)$ 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$$(j)_{-}$$
$$\left(-a - 1\right)$$ $$5$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$\left(-a^{3} + a^{2} + 4 a - 2\right)$$ $$29$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

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
$$3$$ 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6 and 12.
Its isogeny class 145.3-a consists of curves linked by isogenies of degrees dividing 12.

Base change

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