Properties

Base field \(\Q(\zeta_{7})^+\)
Label 3.3.49.1-49.1-a4
Conductor \((7,2 a^{2} + a - 6)\)
Conductor norm \( 49 \)
CM yes (\(-7\))
base-change yes: 49.a4
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: K = nfinit(a^3 - a^2 - 2*a + 1);
 

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((7,2 a^{2} + a - 6)\) = \( \left(-a^{2} - a + 2\right)^{2} \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 49 \) = \( 7^{2} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((343,343 a,343 a^{2} - 686)\) = \( \left(-a^{2} - a + 2\right)^{9} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 40353607 \) = \( 7^{9} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( -3375 \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp: E.j
 
\( \text{End} (E) \) = \(\Z[(1+\sqrt{-7})/2]\)   ( Complex Multiplication )
magma: HasComplexMultiplication(E);
 
sage: E.has_cm(), E.cm_discriminant()
 
\( \text{ST} (E) \) = $N(\mathrm{U}(1))$

Mordell-Weil group

Rank not available.

magma: Rank(E);
 
sage: E.rank()
 

Regulator: not available

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: gens = E.gens(); gens
 
magma: Regulator(gens);
 
sage: E.regulator_of_points(gens)
 

Torsion subgroup

Structure: \(\Z/2\Z\)
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
Generator: $\left(2 : -1 : 1\right)$
magma: [piT(P) : P in Generators(T)];
 
sage: T.gens()
 
gp: T[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\((j)_{-}\)
\( \left(-a^{2} - a + 2\right) \) \(7\) \(2\) \(III^*\) Additive \(-1\) \(2\) \(9\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(7\) 7B.1.6[3]

For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 7 and 14.
Its isogeny class 49.1-a consists of curves linked by isogenies of degrees dividing 14.

Base change

This curve is the base-change of elliptic curves 49.a4, defined over \(\Q\), so it is also a \(\Q\)-curve.