Properties

Base field \(\Q(\sqrt{3}) \)
Label 2.2.12.1-588.1-h3
Conductor \((14 a)\)
Conductor norm \( 588 \)
CM no
base-change yes: 84.b3,1008.g3
Q-curve yes
Torsion order \( 6 \)
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

\( y^2 + \left(a + 1\right) x y + \left(a + 1\right) y = x^{3} - a x^{2} + \left(27 a - 51\right) x + 52 a - 92 \)
magma: E := ChangeRing(EllipticCurve([a + 1, -a, a + 1, 27*a - 51, 52*a - 92]),K);
sage: E = EllipticCurve(K, [a + 1, -a, a + 1, 27*a - 51, 52*a - 92])
gp (2.8): E = ellinit([a + 1, -a, a + 1, 27*a - 51, 52*a - 92],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((14 a)\) = \( \left(a + 1\right)^{2} \cdot \left(a\right) \cdot \left(7\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 588 \) = \( 2^{2} \cdot 3 \cdot 49 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((20412)\) = \( \left(a + 1\right)^{4} \cdot \left(a\right)^{12} \cdot \left(7\right) \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 416649744 \) = \( 2^{4} \cdot 3^{12} \cdot 49 \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{9826000}{5103} \)
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: \(\Z/6\Z\)
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: $\left(2 a - 4 : -4 a + 6 : 1\right)$
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\((j)_{-}\)
\( \left(a + 1\right) \) \(2\) \(3\) \(IV\) Additive \(-1\) \(2\) \(4\) \(0\)
\( \left(a\right) \) \(3\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)
\( \left(7\right) \) \(49\) \(1\) \(I_{1}\) 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
\(2\) 2B
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 588.1-h consists of curves linked by isogenies of degrees dividing 6.

Base change

This curve is the base-change of elliptic curves 84.b3, 1008.g3, defined over \(\Q\), so it is also a \(\Q\)-curve.