Properties

Base field \(\Q(\sqrt{3}) \)
Label 2.2.12.1-200.1-a1
Conductor \((10 a + 10)\)
Conductor norm \( 200 \)
CM no
base-change yes: 40.a4,720.e4
Q-curve yes
Torsion order \( 4 \)
Rank not available

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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

\( y^2 + \left(a + 1\right) x y = x^{3} + \left(a - 1\right) x^{2} + \left(13 a + 25\right) x - 52 a - 91 \)
magma: E := ChangeRing(EllipticCurve([a + 1, a - 1, 0, 13*a + 25, -52*a - 91]),K);
sage: E = EllipticCurve(K, [a + 1, a - 1, 0, 13*a + 25, -52*a - 91])
gp (2.8): E = ellinit([a + 1, a - 1, 0, 13*a + 25, -52*a - 91],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()
\(\mathfrak{D}\) = \((10000)\) = \( \left(a + 1\right)^{8} \cdot \left(5\right)^{4} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 100000000 \) = \( 2^{8} \cdot 25^{4} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{237276}{625} \)
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: \(\Z/4\Z\)
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: $\left(3 a + 7 : -20 a - 33 : 1\right)$
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 \(-1\) \(3\) \(8\) \(0\)
\( \left(5\right) \) \(25\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

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 40.a4, 720.e4, defined over \(\Q\), so it is also a \(\Q\)-curve.