Properties

Label 3.3.316.1-8.1-a7
Base field 3.3.316.1
Conductor \((2)\)
Conductor norm \( 8 \)
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

\({y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(420286700a^{2}-222464805a-1785857409\right){x}+6903318103765a^{2}-3654040730844a-29333168803431\)
sage: E = EllipticCurve([K([0,1,0]),K([4,-1,-1]),K([0,1,0]),K([-1785857409,-222464805,420286700]),K([-29333168803431,-3654040730844,6903318103765])])
 
gp: E = ellinit([Pol(Vecrev([0,1,0])),Pol(Vecrev([4,-1,-1])),Pol(Vecrev([0,1,0])),Pol(Vecrev([-1785857409,-222464805,420286700])),Pol(Vecrev([-29333168803431,-3654040730844,6903318103765]))], K);
 
magma: E := EllipticCurve([K![0,1,0],K![4,-1,-1],K![0,1,0],K![-1785857409,-222464805,420286700],K![-29333168803431,-3654040730844,6903318103765]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2)\) = \((a)^{2}\cdot(-a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 8 \) = \(2^{2}\cdot2\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-16a^2-64a+80)\) = \((a)^{8}\cdot(-a+1)^{12}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1048576 \) = \(2^{8}\cdot2^{12}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{532560105997425}{4096} a^{2} - \frac{749207151764755}{2048} a + \frac{293646661918461}{2048} \)
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: $\left(-11452 a^{2} + \frac{12135}{2} a + \frac{97341}{2} : \frac{10769}{4} a^{2} - \frac{5727}{4} a - 11452 : 1\right)$
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: \( 2.70256844437289 \)
Tamagawa product: \( 2 \)  =  \(1\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 0.684141088092391 \)
Analytic order of Ш: \( 9 \) (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\((j)_{-}\)
\((a)\) \(2\) \(1\) \(IV^{*}\) Additive \(-1\) \(2\) \(8\) \(0\)
\((-a+1)\) \(2\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)

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
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 8.1-a consists of curves linked by isogenies of degrees dividing 12.

Base change

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