Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-72.1-a4
Conductor \((6 i + 6)\)
Conductor norm \( 72 \)
CM no
base-change yes: 24.a3,48.a3
Q-curve yes
Torsion order \( 8 \)
Rank \( 0 \)

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

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

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

Weierstrass equation

\( y^2 + \left(i + 1\right) x y + \left(i + 1\right) y = x^{3} + \left(-i + 6\right) x - 5 i \)
sage: E = EllipticCurve(K, [i + 1, 0, i + 1, -i + 6, -5*i])
 
gp: E = ellinit([i + 1, 0, i + 1, -i + 6, -5*i],K)
 
magma: E := ChangeRing(EllipticCurve([i + 1, 0, i + 1, -i + 6, -5*i]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((6 i + 6)\) = \( \left(i + 1\right)^{3} \cdot \left(3\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 72 \) = \( 2^{3} \cdot 9 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((1296)\) = \( \left(i + 1\right)^{8} \cdot \left(3\right)^{4} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 1679616 \) = \( 2^{8} \cdot 9^{4} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( \frac{1556068}{81} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \( 0 \)

sage: E.rank()
 
magma: Rank(E);
 

Regulator: 1

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

Torsion subgroup

Structure: \(\Z/2\Z\times\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Generators: $\left(0 : i - 2 : 1\right)$,$\left(i : -i : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
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 \(1\) \(3\) \(8\) \(0\)
\( \left(3\right) \) \(9\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 72.1-a consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base-change of elliptic curves 24.a3, 48.a3, defined over \(\Q\), so it is also a \(\Q\)-curve.