Properties

Base field \(\Q(\zeta_{7})^+\)
Label 3.3.49.1-91.1-a3
Conductor \((91,4 a + 1)\)
Conductor norm \( 91 \)
CM no
base-change no
Q-curve no
Torsion order \( 16 \)
Rank not available

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Base field \(\Q(\zeta_{7})^+\)

Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 2 x + 1 \); class number \(1\).

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

Weierstrass equation

\( y^2 + \left(a^{2} - 2\right) x y = x^{3} + \left(a^{2} - 3\right) x^{2} - 4 x - 3 a^{2} - a + 9 \)
magma: E := ChangeRing(EllipticCurve([a^2 - 2, a^2 - 3, 0, -4, -3*a^2 - a + 9]),K);
sage: E = EllipticCurve(K, [a^2 - 2, a^2 - 3, 0, -4, -3*a^2 - a + 9])
gp (2.8): E = ellinit([a^2 - 2, a^2 - 3, 0, -4, -3*a^2 - a + 9],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((91,4 a + 1)\) = \( \left(-a^{2} - a + 2\right) \cdot \left(-2 a^{2} + a + 2\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 91 \) = \( 7 \cdot 13 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((199927,7 a + 116914,a^{2} + 4 a + 122413)\) = \( \left(-a^{2} - a + 2\right)^{2} \cdot \left(-2 a^{2} + a + 2\right)^{4} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 1399489 \) = \( 7^{2} \cdot 13^{4} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -\frac{3698907677516}{199927} a^{2} + \frac{293174005427}{28561} a + \frac{8312780816110}{199927} \)
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: \(\Z/2\Z\times\Z/8\Z\)
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]
Generators: $\left(-a + 1 : a^{2} - 2 : 1\right)$,$\left(-2 : a^{2} - 2 : 1\right)$
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\((j)_{-}\)
\( \left(-a^{2} - a + 2\right) \) \(7\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\( \left(-2 a^{2} + a + 2\right) \) \(13\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

Galois Representations

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

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 91.1-a consists of curves linked by isogenies of degrees dividing 16.

Base change

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