Properties

Base field \(\Q(\zeta_{7})^+\)
Label 3.3.49.1-71.3-a1
Conductor \((71,a^{2} - 6)\)
Conductor norm \( 71 \)
CM no
base-change no
Q-curve yes
Torsion order \( 2 \)
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

\( y^2 + x y + \left(a^{2} - 1\right) y = x^{3} + \left(a^{2} - 2\right) x^{2} + \left(4 a^{2} - 10 a - 44\right) x - 32 a^{2} - 61 a - 73 \)
magma: E := ChangeRing(EllipticCurve([1, a^2 - 2, a^2 - 1, 4*a^2 - 10*a - 44, -32*a^2 - 61*a - 73]),K);
sage: E = EllipticCurve(K, [1, a^2 - 2, a^2 - 1, 4*a^2 - 10*a - 44, -32*a^2 - 61*a - 73])
gp (2.8): E = ellinit([1, a^2 - 2, a^2 - 1, 4*a^2 - 10*a - 44, -32*a^2 - 61*a - 73],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((71,a^{2} - 6)\) = \( \left(a^{2} - 6\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 71 \) = \( 71 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((1804229351,a + 629662061,a^{2} + 892712459)\) = \( \left(a^{2} - 6\right)^{5} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 1804229351 \) = \( 71^{5} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -\frac{925759129064909928}{1804229351} a^{2} + \frac{385959155459705697}{1804229351} a + \frac{2310445605654755654}{1804229351} \)
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: \(\Z/2\Z\)
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: $\left(3 a^{2} + 2 a - 5 : -2 a^{2} - a + 3 : 1\right)$
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 ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(a^{2} - 6\right) \) \(71\) \(5\) \( I_{5} \) Split multiplicative \(1\) \(5\) \(5\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(5\) 5B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 5 and 10.
Its isogeny class 71.3-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 a \(\Q\)-curve.