Properties

Base field \(\Q(\sqrt{-3}) \)
Label 2.0.3.1-49.1-CMa1
Conductor \((3 a - 8)\)
Conductor norm \( 49 \)
CM yes (\(-3\))
base-change no
Q-curve yes
Torsion order \( 7 \)
Rank not available

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Base field \(\Q(\sqrt{-3}) \)

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

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

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((3 a - 8)\) = \( \left(-3 a + 1\right)^{2} \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 49 \) = \( 7^{2} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((3 a - 8)\) = \( \left(-3 a + 1\right)^{2} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 49 \) = \( 7^{2} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( 0 \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp (2.8): E.j
 
\( \text{End} (E) \) = \(\Z[(1+\sqrt{-3})/2]\)   ( Complex Multiplication )
magma: HasComplexMultiplication(E);
 
sage: E.has_cm(), E.cm_discriminant()
 
\( \text{ST} (E) \) = $\mathrm{U}(1)$

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: $\left(-1 : -a : 1\right)$
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\((j)_{-}\)
\( \left(-3 a + 1\right) \) \(7\) \(1\) \(II\) Additive \(-1\) \(2\) \(2\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(7\) 7Cs.1.1

For all other primes \(p\), the image is a Borel subgroup if \(p=3\), a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has no rational isogenies other than endomorphisms. Its isogeny class 49.1-CMa consists of this curve only.

Base change

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