Properties

Base field \(\Q(\zeta_{7})^+\)
Label 3.3.49.1-41.2-a2
Conductor \((41,2 a^{2} - 3 a - 4)\)
Conductor norm \( 41 \)
CM no
base-change no
Q-curve yes
Torsion order \( 10 \)
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

\( y^2 + x y + y = x^{3} + \left(a + 1\right) x^{2} + \left(6 a - 5\right) x + 4 a^{2} - 6 a + 2 \)
magma: E := ChangeRing(EllipticCurve([1, a + 1, 1, 6*a - 5, 4*a^2 - 6*a + 2]),K);
sage: E = EllipticCurve(K, [1, a + 1, 1, 6*a - 5, 4*a^2 - 6*a + 2])
gp (2.8): E = ellinit([1, a + 1, 1, 6*a - 5, 4*a^2 - 6*a + 2],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((41,2 a^{2} - 3 a - 4)\) = \( \left(2 a^{2} - 3 a - 4\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 41 \) = \( 41 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((1681,a + 1014,a^{2} + 576)\) = \( \left(2 a^{2} - 3 a - 4\right)^{2} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 1681 \) = \( 41^{2} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{2962060985575}{1681} a^{2} - \frac{6655766653200}{1681} a + \frac{2375528385434}{1681} \)
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: \(\Z/10\Z\)
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: $\left(0 : -2 a + 1 : 1\right)$
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 ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(2 a^{2} - 3 a - 4\right) \) \(41\) \(2\) \( I_{2} \) Split multiplicative \(1\) \(2\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(5\) 5B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 5 and 10.
Its isogeny class 41.2-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.