Properties

Label 3.3.316.1-2.2-a5
Base field 3.3.316.1
Conductor \((-a+1)\)
Conductor norm \( 2 \)
CM no
Base change no
Q-curve no
Torsion order \( 16 \)
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

\({y}^2+\left(a^{2}+a-3\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(30965253611602296633a^{2}-16390422144515520442a-131575714394521753905\right){x}-112172935746444460857621319378a^{2}+59374994732320022090359795265a+476638568561218377704701696552\)
sage: E = EllipticCurve([K([-3,1,1]),K([2,0,-1]),K([0,1,0]),K([-131575714394521753905,-16390422144515520442,30965253611602296633]),K([476638568561218377704701696552,59374994732320022090359795265,-112172935746444460857621319378])])
 
gp: E = ellinit([Pol(Vecrev([-3,1,1])),Pol(Vecrev([2,0,-1])),Pol(Vecrev([0,1,0])),Pol(Vecrev([-131575714394521753905,-16390422144515520442,30965253611602296633])),Pol(Vecrev([476638568561218377704701696552,59374994732320022090359795265,-112172935746444460857621319378]))], K);
 
magma: E := EllipticCurve([K![-3,1,1],K![2,0,-1],K![0,1,0],K![-131575714394521753905,-16390422144515520442,30965253611602296633],K![476638568561218377704701696552,59374994732320022090359795265,-112172935746444460857621319378]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a+1)\) = \((-a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 2 \) = \(2\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-a^2-4a+5)\) = \((-a+1)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 256 \) = \(2^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{130050481}{256} a^{2} + \frac{87514925}{128} a - \frac{54929731}{128} \)
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: \(\Z/2\Z\times\Z/8\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(1995639652 a^{2} - 1056325156 a - 8479753345 : -232448746868924 a^{2} + 123038975747893 a + 987707393358160 : 1\right)$ $\left(-\frac{3900305081}{4} a^{2} + \frac{1032248073}{2} a + \frac{8286472215}{2} : -\frac{9001021479}{8} a^{2} + \frac{2382194953}{4} a + \frac{19123302629}{4} : 1\right)$
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: \( 342.954586166480 \)
Tamagawa product: \( 8 \)
Torsion order: \(16\)
Leading coefficient: \( 0.602896961660936 \)
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\((j)_{-}\)
\((-a+1)\) \(2\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)

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, 4 and 8.
Its isogeny class 2.2-a consists of curves linked by isogenies of degrees dividing 16.

Base change

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