Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-3249.1-c1
Conductor \((57)\)
Conductor norm \( 3249 \)
CM no
base-change yes: 57.c4,912.b4
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 (2.8): K = nfinit(i^2 + 1);

Weierstrass equation

\( y^2 + i x y + i y = x^{3} + 9 x - 29 \)
magma: E := ChangeRing(EllipticCurve([i, 0, i, 9, -29]),K);
sage: E = EllipticCurve(K, [i, 0, i, 9, -29])
gp (2.8): E = ellinit([i, 0, i, 9, -29],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((57)\) = \( \left(3\right) \cdot \left(19\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 3249 \) = \( 9 \cdot 361 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((390963)\) = \( \left(3\right) \cdot \left(19\right)^{4} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 152852067369 \) = \( 9 \cdot 361^{4} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{67419143}{390963} \)
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: $\left(\frac{9}{2} : -\frac{11}{4} i - \frac{39}{4} : 1\right)$

Height: 2.6018939134308194

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

Regulator: 2.60189391343

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

Torsion subgroup

Structure: \(\Z/4\Z\)
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]
Generator: $\left(7 : -4 i - 19 : 1\right)$
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\((j)_{-}\)
\( \left(3\right) \) \(9\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\( \left(19\right) \) \(361\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) 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 and 4.
Its isogeny class 3249.1-c consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base-change of elliptic curves 57.c4, 912.b4, defined over \(\Q\), so it is also a \(\Q\)-curve.