Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-3249.1-c4
Conductor \((57)\)
Conductor norm \( 3249 \)
CM no
base-change yes: 912.b1,57.c1
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

\( y^2 + x y + y = x^{3} - 102 x + 385 \)
magma: E := ChangeRing(EllipticCurve([1, 0, 1, -102, 385]),K);
 
sage: E = EllipticCurve(K, [1, 0, 1, -102, 385])
 
gp (2.8): E = ellinit([1, 0, 1, -102, 385],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()
 
\(\mathfrak{D}\) = \((1539)\) = \( \left(3\right)^{4} \cdot \left(19\right) \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 2368521 \) = \( 9^{4} \cdot 361 \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{115714886617}{1539} \)
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: $\left(\frac{103}{18} : \frac{5}{108} i - \frac{121}{36} : 1\right)$

Height: 2.601893913430819

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: $\left(5 : -6 : 1\right)$
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\((j)_{-}\)
\( \left(3\right) \) \(9\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(19\right) \) \(361\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

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 912.b1, 57.c1, defined over \(\Q\), so it is also a \(\Q\)-curve.