Properties

Base field \(\Q(\sqrt{-11}) \)
Label 2.0.11.1-2500.3-h4
Conductor \((50)\)
Conductor norm \( 2500 \)
CM no
base-change yes: 6050.bi4,50.a4
Q-curve yes
Torsion order \( 1 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{-11}) \)

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

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

Weierstrass equation

\( y^2 + x y + y = x^{3} + 549 x - 2202 \)
magma: E := ChangeRing(EllipticCurve([1, 0, 1, 549, -2202]),K);
sage: E = EllipticCurve(K, [1, 0, 1, 549, -2202])
gp (2.8): E = ellinit([1, 0, 1, 549, -2202],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((50)\) = \( \left(2\right) \cdot \left(-a - 1\right)^{2} \cdot \left(a - 2\right)^{2} \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 2500 \) = \( 4 \cdot 5^{4} \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((12800000000)\) = \( \left(2\right)^{15} \cdot \left(-a - 1\right)^{8} \cdot \left(a - 2\right)^{8} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 163840000000000000000 \) = \( 4^{15} \cdot 5^{16} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{46969655}{32768} \)
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: \( 0 \)
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: 1

magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

Torsion subgroup

Structure: Trivial
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]

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\((j)_{-}\)
\( \left(-a - 1\right) \) \(5\) \(1\) \(IV^*\) Additive \(-1\) \(2\) \(8\) \(0\)
\( \left(a - 2\right) \) \(5\) \(1\) \(IV^*\) Additive \(-1\) \(2\) \(8\) \(0\)
\( \left(2\right) \) \(4\) \(15\) \(I_{15}\) Split multiplicative \(-1\) \(1\) \(15\) \(15\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.2
\(5\) 5B.1.4

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 5 and 15.
Its isogeny class 2500.3-h consists of curves linked by isogenies of degrees dividing 15.

Base change

This curve is the base-change of elliptic curves 6050.bi4, 50.a4, defined over \(\Q\), so it is also a \(\Q\)-curve.