Properties

Label 3.3.148.1-8.1-a6
Base field 3.3.148.1
Conductor norm \( 8 \)
CM no
Base change no
Q-curve no
Torsion order \( 8 \)
Rank \( 0 \)

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Base field 3.3.148.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -3, -1, 1]))
 
gp: K = nfinit(Polrev([1, -3, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+\left(-23137255028952366700772a^{2}-27072589446997404913200a+10661904569953464468393\right){x}+3529193720251159834920437017851122a^{2}+4129461880751340931496180433614334a-1626291736297668628974848142500566\)
sage: E = EllipticCurve([K([1,1,0]),K([1,1,-1]),K([-1,0,1]),K([10661904569953464468393,-27072589446997404913200,-23137255028952366700772]),K([-1626291736297668628974848142500566,4129461880751340931496180433614334,3529193720251159834920437017851122])])
 
gp: E = ellinit([Polrev([1,1,0]),Polrev([1,1,-1]),Polrev([-1,0,1]),Polrev([10661904569953464468393,-27072589446997404913200,-23137255028952366700772]),Polrev([-1626291736297668628974848142500566,4129461880751340931496180433614334,3529193720251159834920437017851122])], K);
 
magma: E := EllipticCurve([K![1,1,0],K![1,1,-1],K![-1,0,1],K![10661904569953464468393,-27072589446997404913200,-23137255028952366700772],K![-1626291736297668628974848142500566,4129461880751340931496180433614334,3529193720251159834920437017851122]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2)\) = \((a^2-a-2)^{3}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 8 \) = \(2^{3}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((2a-2)\) = \((a^2-a-2)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -16 \) = \(-2^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 297852964 a^{2} + 348495840 a - 137248100 \)
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: \(\Z/8\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(33576656712 a^{2} + 39287592284 a - 15472497027 : 107827492975219 a^{2} + 126167492417068 a - 49688108579886 : 1\right)$
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: \( 223.18826420022044179173853209997750472 \)
Tamagawa product: \( 2 \)
Torsion order: \(8\)
Leading coefficient: \( 0.57331132207744951681123408265720299983 \)
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\((j)_{-}\)
\((a^2-a-2)\) \(2\) \(2\) \(III\) Additive \(-1\) \(3\) \(4\) \(0\)

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 and 8.
Its isogeny class 8.1-a consists of curves linked by isogenies of degrees dividing 8.

Base change

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

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