Properties

Label 2.0.3.1-15876.3-c3
Base field \(\Q(\sqrt{-3}) \)
Conductor norm \( 15876 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{-3}) \)

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

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

Weierstrass equation

\({y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(167a-277\right){x}+1305a-1549\)
sage: E = EllipticCurve([K([0,1]),K([0,1]),K([0,0]),K([-277,167]),K([-1549,1305])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([0,1]),Polrev([0,0]),Polrev([-277,167]),Polrev([-1549,1305])], K);
 
magma: E := EllipticCurve([K![0,1],K![0,1],K![0,0],K![-277,167],K![-1549,1305]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-54a-90)\) = \((-2a+1)^{4}\cdot(2)\cdot(3a-2)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 15876 \) = \(3^{4}\cdot4\cdot7^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-4635648a+3488256)\) = \((-2a+1)^{4}\cdot(2)^{9}\cdot(3a-2)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 17486835351552 \) = \(3^{4}\cdot4^{9}\cdot7^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{457190997}{1792} a + \frac{1524555639}{3584} \)
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: 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: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 0.74901443890309325100481878604922253144 \)
Tamagawa product: \( 2 \)  =  \(1\cdot1\cdot2\)
Torsion order: \(1\)
Leading coefficient: \( 1.7297747517104695560204272570106898189 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(1\) \(II\) Additive \(1\) \(4\) \(4\) \(0\)
\((2)\) \(4\) \(1\) \(I_{9}\) Non-split multiplicative \(1\) \(1\) \(9\) \(9\)
\((3a-2)\) \(7\) \(2\) \(I_{1}^{*}\) Additive \(-1\) \(2\) \(7\) \(1\)

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[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
Its isogeny class 15876.3-c consists of curves linked by isogenies of degrees dividing 9.

Base change

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

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