Properties

Label 3.3.316.1-16.1-a3
Base field 3.3.316.1
Conductor \((2a)\)
Conductor norm \( 16 \)
CM no
Base change no
Q-curve no
Torsion order \( 8 \)
Rank \( 0 \)

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Base field 3.3.316.1

Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 4 x + 2 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -4, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([2, -4, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -4, -1, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(5607a^{2}-2969a-23826\right){x}-340349a^{2}+180146a+1446179\)
sage: E = EllipticCurve([K([0,1,0]),K([-4,1,1]),K([0,0,0]),K([-23826,-2969,5607]),K([1446179,180146,-340349])])
 
gp: E = ellinit([Pol(Vecrev([0,1,0])),Pol(Vecrev([-4,1,1])),Pol(Vecrev([0,0,0])),Pol(Vecrev([-23826,-2969,5607])),Pol(Vecrev([1446179,180146,-340349]))], K);
 
magma: E := EllipticCurve([K![0,1,0],K![-4,1,1],K![0,0,0],K![-23826,-2969,5607],K![1446179,180146,-340349]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a)\) = \((a)^{3}\cdot(-a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 16 \) = \(2^{3}\cdot2\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((16)\) = \((a)^{8}\cdot(-a+1)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 4096 \) = \(2^{8}\cdot2^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{29580655}{16} a^{2} + \frac{7825805}{8} a + \frac{62873277}{8} \)
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: \(\Z/2\Z\times\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-17 a^{2} + 8 a + 71 : 76 a^{2} - 40 a - 322 : 1\right)$ $\left(41 a^{2} - 23 a - 176 : -9 a^{2} + 6 a + 41 : 1\right)$
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: \( 235.811223005318 \)
Tamagawa product: \( 8 \)  =  \(2\cdot2^{2}\)
Torsion order: \(8\)
Leading coefficient: \( 1.65817721191157 \)
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\((j)_{-}\)
\((a)\) \(2\) \(2\) \(I_1^{*}\) Additive \(1\) \(3\) \(8\) \(0\)
\((-a+1)\) \(2\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

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 and 4.
Its isogeny class 16.1-a consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.