Properties

Label 3.3.316.1-16.2-a3
Base field 3.3.316.1
Conductor \((a^2-2a-2)\)
Conductor norm \( 16 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
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

\({y}^2+a{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(5718326614a^{2}-3026804944a-24297973164\right){x}+344200023976745a^{2}-182190779570766a-1462554275103174\)
sage: E = EllipticCurve([K([0,1,0]),K([2,-1,-1]),K([-2,0,1]),K([-24297973164,-3026804944,5718326614]),K([-1462554275103174,-182190779570766,344200023976745])])
 
gp: E = ellinit([Pol(Vecrev([0,1,0])),Pol(Vecrev([2,-1,-1])),Pol(Vecrev([-2,0,1])),Pol(Vecrev([-24297973164,-3026804944,5718326614])),Pol(Vecrev([-1462554275103174,-182190779570766,344200023976745]))], K);
 
magma: E := EllipticCurve([K![0,1,0],K![2,-1,-1],K![-2,0,1],K![-24297973164,-3026804944,5718326614],K![-1462554275103174,-182190779570766,344200023976745]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^2-2a-2)\) = \((a)^{4}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 16 \) = \(2^{4}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-7a^2+10a+10)\) = \((a)^{11}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2048 \) = \(2^{11}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -343127540 a^{2} + 181623451 a + 1457997356 \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(21124 a^{2} - \frac{22365}{2} a - \frac{179523}{2} : -\frac{19885}{4} a^{2} + \frac{10531}{4} a + 21125 : 1\right)$
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: \( 23.3981765380198 \)
Tamagawa product: \( 4 \)
Torsion order: \(2\)
Leading coefficient: \( 1.31625026633286 \)
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\((j)_{-}\)
\((a)\) \(2\) \(4\) \(I_3^{*}\) Additive \(-1\) \(4\) \(11\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 8 and 16.
Its isogeny class 16.2-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.