Properties

Base field \(\Q(\zeta_{15})^+\)
Label 4.4.1125.1-45.1-b6
Conductor \((15,a^{3} + 2 a^{2} - 3 a - 4)\)
Conductor norm \( 45 \)
CM no
base-change yes: 15.a4,75.b4
Q-curve yes
Torsion order \( 16 \)
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 \(1\).

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

Weierstrass equation

\( y^2 + x y + y = x^{3} + x^{2} - 80 x + 242 \)
magma: E := ChangeRing(EllipticCurve([1, 1, 1, -80, 242]),K);
sage: E = EllipticCurve(K, [1, 1, 1, -80, 242])
gp (2.8): E = ellinit([1, 1, 1, -80, 242],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((15,a^{3} + 2 a^{2} - 3 a - 4)\) = \( \left(-a^{3} + a^{2} + 3 a - 2\right) \cdot \left(-a - 1\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 45 \) = \( 5 \cdot 9 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((15,15 a^{3} - 45 a,15 a,15 a^{2} - 30)\) = \( \left(-a^{3} + a^{2} + 3 a - 2\right)^{2} \cdot \left(-a - 1\right)^{4} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 50625 \) = \( 5^{4} \cdot 9^{2} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{56667352321}{15} \)
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: \(\Z/16\Z\)
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]
Generator: $\left(7 a^{3} + 6 a^{2} - 17 a + 1 : 47 a^{3} + 38 a^{2} - 119 a - 29 : 1\right)$
magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[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\((j)_{-}\)
\( \left(-a - 1\right) \) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(-a^{3} + a^{2} + 3 a - 2\right) \) \(9\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 8, 16 and 32.
Its isogeny class 45.1-b consists of curves linked by isogenies of degrees dividing 32.

Base change

This curve is the base-change of elliptic curves 15.a4, 75.b4, defined over \(\Q\), so it is also a \(\Q\)-curve.