Properties

Label 2.0.4.1-46818.2-e2
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 46818 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 0 \)

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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

\({y}^2+{x}{y}+i{y}={x}^{3}-{x}^{2}+\left(889i+48\right){x}+6527i+7750\)
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([0,1]),K([48,889]),K([7750,6527])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([0,1]),Polrev([48,889]),Polrev([7750,6527])], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,1],K![48,889],K![7750,6527]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((153i+153)\) = \((i+1)\cdot(3)^{2}\cdot(i+4)\cdot(i-4)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 46818 \) = \(2\cdot9^{2}\cdot17\cdot17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-610553538i+910141920)\) = \((i+1)^{2}\cdot(3)^{8}\cdot(i+4)^{6}\cdot(i-4)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1201133937305603844 \) = \(2^{2}\cdot9^{8}\cdot17^{6}\cdot17^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{9390341072075}{144825414} i - \frac{6456168132412}{217238121} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(0\)
Torsion structure: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(11 i - 12 : -6 i + 6 : 1\right)$ $\left(14 i - \frac{45}{4} : -\frac{15}{2} i + \frac{45}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 0.33945605076348356273798522409668569925 \)
Tamagawa product: \( 32 \)  =  \(2\cdot2^{2}\cdot2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 2.7156484061078685019038817927734855940 \)
Analytic order of Ш: \( 4 \) (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\((j)_{-}\)
\((i+1)\) \(2\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((3)\) \(9\) \(4\) \(I_{2}^{*}\) Additive \(1\) \(2\) \(8\) \(2\)
\((i+4)\) \(17\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\((i-4)\) \(17\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

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.
Its isogeny class 46818.2-e consists of curves linked by isogenies of degrees dividing 4.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.