Properties

Base field \(\Q(\zeta_{7})^+\)
Label 3.3.49.1-104.1-a2
Conductor \((26,2 a^{2} + 2 a - 6)\)
Conductor norm \( 104 \)
CM no
base-change no
Q-curve no
Torsion order \( 7 \)
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

\( y^2 + a x y + \left(a^{2} - 2\right) y = x^{3} + \left(-a - 1\right) x^{2} \)
magma: E := ChangeRing(EllipticCurve([a, -a - 1, a^2 - 2, 0, 0]),K);
 
sage: E = EllipticCurve(K, [a, -a - 1, a^2 - 2, 0, 0])
 
gp (2.8): E = ellinit([a, -a - 1, a^2 - 2, 0, 0],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((26,2 a^{2} + 2 a - 6)\) = \( \left(2\right) \cdot \left(a^{2} + a - 3\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 104 \) = \( 8 \cdot 13 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((26,2 a + 14,2 a^{2} + 6)\) = \( \left(2\right) \cdot \left(a^{2} + a - 3\right) \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 104 \) = \( 8 \cdot 13 \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{209979}{13} a^{2} - \frac{245189}{26} a - \frac{460974}{13} \)
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: \(\Z/7\Z\)
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: $\left(1 : -a + 1 : 1\right)$
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\((j)_{-}\)
\( \left(a^{2} + a - 3\right) \) \(13\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\( \left(2\right) \) \(8\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

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

prime Image of Galois Representation
\(7\) 7B.1.1[3]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 7.
Its isogeny class 104.1-a consists of curves linked by isogenies of degree 7.

Base change

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