Properties

Base field \(\Q(\sqrt{3}) \)
Label 2.2.12.1-200.1-a4
Conductor \((10 a + 10)\)
Conductor norm \( 200 \)
CM no
base-change yes: 720.e1,40.a1
Q-curve yes
Torsion order \( 2 \)
Rank not available

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

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

\( y^2 + \left(a + 1\right) x y = x^{3} + \left(a - 1\right) x^{2} + \left(-107 a - 185\right) x - 892 a - 1545 \)
magma: E := ChangeRing(EllipticCurve([a + 1, a - 1, 0, -107*a - 185, -892*a - 1545]),K);
sage: E = EllipticCurve(K, [a + 1, a - 1, 0, -107*a - 185, -892*a - 1545])
gp (2.8): E = ellinit([a + 1, a - 1, 0, -107*a - 185, -892*a - 1545],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((10 a + 10)\) = \( \left(a + 1\right)^{3} \cdot \left(5\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 200 \) = \( 2^{3} \cdot 25 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((80)\) = \( \left(a + 1\right)^{8} \cdot \left(5\right) \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 6400 \) = \( 2^{8} \cdot 25 \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{132304644}{5} \)
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: \(\Z/2\Z\)
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: $\left(-\frac{7}{2} a - 6 : \frac{19}{4} a + \frac{33}{4} : 1\right)$
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\((j)_{-}\)
\( \left(a + 1\right) \) \(2\) \(4\) \(I_{1}^*\) Additive \(-1\) \(3\) \(8\) \(0\)
\( \left(5\right) \) \(25\) \(1\) \(I_{1}\) 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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 200.1-a consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base-change of elliptic curves 720.e1, 40.a1, defined over \(\Q\), so it is also a \(\Q\)-curve.