Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-5000.3-a8
Conductor \((50 i + 50)\)
Conductor norm \( 5000 \)
CM no
base-change yes: 200.c1,400.e1
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

Base field \(\Q(\sqrt{-1}) \)

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

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

Weierstrass equation

\( y^2 + \left(i + 1\right) x y + \left(i + 1\right) y = x^{3} + i x^{2} + \left(-i + 668\right) x + 6990 i \)
magma: E := ChangeRing(EllipticCurve([i + 1, i, i + 1, -i + 668, 6990*i]),K);
 
sage: E = EllipticCurve(K, [i + 1, i, i + 1, -i + 668, 6990*i])
 
gp: E = ellinit([i + 1, i, i + 1, -i + 668, 6990*i],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((50 i + 50)\) = \( \left(i + 1\right)^{3} \cdot \left(-i - 2\right)^{2} \cdot \left(2 i + 1\right)^{2} \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 5000 \) = \( 2^{3} \cdot 5^{4} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((1250000)\) = \( \left(i + 1\right)^{8} \cdot \left(-i - 2\right)^{7} \cdot \left(2 i + 1\right)^{7} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 1562500000000 \) = \( 2^{8} \cdot 5^{14} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( \frac{132304644}{5} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp: 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: \( 1 \)

magma: Rank(E);
 
sage: E.rank()
 

Generator: $\left(-15 i : 9 i - 6 : 1\right)$

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: gens = E.gens(); gens
 

Height: 0.9333935029996572

magma: [Height(P):P in gens];
 
sage: [P.height() for P in gens]
 

Regulator: 0.933393503

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

Torsion subgroup

Structure: \(\Z/4\Z\)
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
Generator: $\left(-18 i : -4 i - 22 : 1\right)$
magma: [piT(P) : P in Generators(T)];
 
sage: T.gens()
 
gp: T[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\((j)_{-}\)
\( \left(i + 1\right) \) \(2\) \(2\) \(I_{1}^*\) Additive \(1\) \(3\) \(8\) \(0\)
\( \left(-i - 2\right) \) \(5\) \(4\) \(I_{1}^*\) Additive \(1\) \(2\) \(7\) \(1\)
\( \left(2 i + 1\right) \) \(5\) \(4\) \(I_{1}^*\) Additive \(1\) \(2\) \(7\) \(1\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) 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, 4, 8 and 16.
Its isogeny class 5000.3-a consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is the base-change of elliptic curves 200.c1, 400.e1, defined over \(\Q\), so it is also a \(\Q\)-curve.