Properties

Base field \(\Q(\sqrt{3}) \)
Label 2.2.12.1-1024.1-k7
Conductor \((32)\)
Conductor norm \( 1024 \)
CM yes (\(-48\))
base-change no
Q-curve yes
Torsion order \( 2 \)
Rank not available

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

Generator \(a\), with minimal polynomial \( x^{2} - 3 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 3)
 
gp (2.8): K = nfinit(a^2 - 3);
 

Weierstrass equation

\( y^2 = x^{3} - a x^{2} + \left(10 a - 59\right) x - 49 a + 2 \)
magma: E := ChangeRing(EllipticCurve([0, -a, 0, 10*a - 59, -49*a + 2]),K);
 
sage: E = EllipticCurve(K, [0, -a, 0, 10*a - 59, -49*a + 2])
 
gp (2.8): E = ellinit([0, -a, 0, 10*a - 59, -49*a + 2],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((32)\) = \( \left(a + 1\right)^{10} \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 1024 \) = \( 2^{10} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((8192)\) = \( \left(a + 1\right)^{26} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 67108864 \) = \( 2^{26} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( 818626500 a + 1417905000 \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp (2.8): E.j
 
\( \text{End} (E) \) = \(\Z[\sqrt{-12}]\)   ( 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()
 
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: $\left(-3 a + 2 : 0 : 1\right)$
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\((j)_{-}\)
\( \left(a + 1\right) \) \(2\) \(4\) \(I_{12}^*\) Additive \(-1\) \(10\) \(26\) \(0\)

Galois Representations

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

The image is a Borel subgroup if \(p\in \{ 2, 3\}\), the normalizer of a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 1024.1-k consists of curves linked by isogenies of degrees dividing 12.

Base change

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