Properties

Label 2.0.4.1-50625.3-f2
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 50625 \)
CM yes (\(-3\))
Base change yes
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

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

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, 1]))
 
gp: K = nfinit(Polrev([1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
 

Weierstrass equation

\({y}^2+{y}={x}^{3}-4219\)
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([1,0]),K([0,0]),K([-4219,0])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([1,0]),Polrev([0,0]),Polrev([-4219,0])], K);
 
magma: E := EllipticCurve([K![0,0],K![0,0],K![1,0],K![0,0],K![-4219,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((225)\) = \((-i-2)^{2}\cdot(2i+1)^{2}\cdot(3)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 50625 \) = \(5^{2}\cdot5^{2}\cdot9^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-7688671875)\) = \((-i-2)^{8}\cdot(2i+1)^{8}\cdot(3)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 59115675201416015625 \) = \(5^{8}\cdot5^{8}\cdot9^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 0 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z[(1+\sqrt{-3})/2]\) (potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $N(\mathrm{U}(1))$

Mordell-Weil group

Rank: \(1\)
Generator $\left(\frac{39}{4} : \frac{459}{8} i - \frac{1}{2} : 1\right)$
Height \(3.0224353582450591452607257416663459501\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 3.0224353582450591452607257416663459501 \)
Period: \( 0.31613030502258025158853715196151958922 \)
Tamagawa product: \( 2 \)  =  \(1\cdot1\cdot2\)
Torsion order: \(1\)
Leading coefficient: \( 3.8219336468521686525343055438940322977 \)
Analytic order of Ш: \( 1 \) (rounded)

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\((j)_{-}\)
\((-i-2)\) \(5\) \(1\) \(IV^{*}\) Additive \(-1\) \(2\) \(8\) \(0\)
\((2i+1)\) \(5\) \(1\) \(IV^{*}\) Additive \(-1\) \(2\) \(8\) \(0\)
\((3)\) \(9\) \(2\) \(III^{*}\) Additive \(1\) \(2\) \(9\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.2
\(5\) 5Ns.2.1

For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 50625.3-f consists of curves linked by isogenies of degree 3.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 225.d1
\(\Q\) 3600.b2