Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-6400.2-e5
Conductor \((80)\)
Conductor norm \( 6400 \)
CM no
base-change yes: 320.f2,320.a2
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 (2.8): K = nfinit(i^2 + 1);
 

Weierstrass equation

\( y^2 = x^{3} - i x^{2} + 36 x - 140 i \)
magma: E := ChangeRing(EllipticCurve([0, -i, 0, 36, -140*i]),K);
 
sage: E = EllipticCurve(K, [0, -i, 0, 36, -140*i])
 
gp (2.8): E = ellinit([0, -i, 0, 36, -140*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()
 
\(\mathfrak{D}\) = \((4000000)\) = \( \left(i + 1\right)^{16} \cdot \left(-i - 2\right)^{6} \cdot \left(2 i + 1\right)^{6} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 16000000000000 \) = \( 2^{16} \cdot 5^{12} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( -\frac{20720464}{15625} \)
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: \( 1 \)
magma: Rank(E);
 
sage: E.rank()
 

Generator: $\left(3 i - 5 : 15 i + 5 : 1\right)$

Height: 0.13781230466972083

magma: Generators(E); // includes torsion
 
sage: E.gens()
 

Regulator: 0.13781230467

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

Torsion subgroup

Structure: \(\Z/2\Z\times\Z/2\Z\)
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]
 
Generators: $\left(4 i - 2 : 0 : 1\right)$,$\left(-7 i : 0 : 1\right)$
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\((j)_{-}\)
\( \left(i + 1\right) \) \(2\) \(4\) \(I_{4}^*\) Additive \(1\) \(8\) \(16\) \(0\)
\( \left(-i - 2\right) \) \(5\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\( \left(2 i + 1\right) \) \(5\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) 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.f2, 320.a2, defined over \(\Q\), so it is also a \(\Q\)-curve.