Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-6400.2-e6
Conductor \((80)\)
Conductor norm \( 6400 \)
CM no
base-change yes: 320.f4,320.a4
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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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

\( y^2 = x^{3} - i x^{2} - 4 x + 4 i \)
magma: E := ChangeRing(EllipticCurve([0, -i, 0, -4, 4*i]),K);
 
sage: E = EllipticCurve(K, [0, -i, 0, -4, 4*i])
 
gp: E = ellinit([0, -i, 0, -4, 4*i],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((80)\) = \( \left(i + 1\right)^{8} \cdot \left(-i - 2\right) \cdot \left(2 i + 1\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 6400 \) = \( 2^{8} \cdot 5^{2} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((6400)\) = \( \left(i + 1\right)^{16} \cdot \left(-i - 2\right)^{2} \cdot \left(2 i + 1\right)^{2} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 40960000 \) = \( 2^{16} \cdot 5^{4} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( \frac{21296}{25} \)
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: $\left(-i - 1 : i + 3 : 1\right)$

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

Height: 0.4134369140091624

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

Regulator: 0.413436914009

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

Torsion subgroup

Structure: \(\Z/2\Z\times\Z/2\Z\)
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
Generators: $\left(-2 : 0 : 1\right)$,$\left(i : 0 : 1\right)$
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\((j)_{-}\)
\( \left(i + 1\right) \) \(2\) \(4\) \(I_{4}^*\) Additive \(1\) \(8\) \(16\) \(0\)
\( \left(-i - 2\right) \) \(5\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\( \left(2 i + 1\right) \) \(5\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2Cs
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 6400.2-e consists of curves linked by isogenies of degrees dividing 12.

Base change

This curve is the base-change of elliptic curves 320.f4, 320.a4, defined over \(\Q\), so it is also a \(\Q\)-curve.