Properties

Label 3.3.316.1-2.2-a1
Base field 3.3.316.1
Conductor \((-a+1)\)
Conductor norm \( 2 \)
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

\({y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(971347319857519a^{2}-514150241464297a-4127391273410075\right){x}+22425156860537519167317a^{2}-11870007338272500196506a-95287643090023949417928\)
sage: E = EllipticCurve([K([-3,1,1]),K([-1,-1,0]),K([-3,0,1]),K([-4127391273410075,-514150241464297,971347319857519]),K([-95287643090023949417928,-11870007338272500196506,22425156860537519167317])])
 
gp: E = ellinit([Pol(Vecrev([-3,1,1])),Pol(Vecrev([-1,-1,0])),Pol(Vecrev([-3,0,1])),Pol(Vecrev([-4127391273410075,-514150241464297,971347319857519])),Pol(Vecrev([-95287643090023949417928,-11870007338272500196506,22425156860537519167317]))], K);
 
magma: E := EllipticCurve([K![-3,1,1],K![-1,-1,0],K![-3,0,1],K![-4127391273410075,-514150241464297,971347319857519],K![-95287643090023949417928,-11870007338272500196506,22425156860537519167317]]);
 

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: \((2a^2-a-7)\) = \((-a+1)^{4}\)
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: \( \frac{200481}{16} a^{2} - \frac{325027}{8} a + \frac{233973}{8} \)
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: $\left(8565099 a^{2} - 4533649 a - 36394310 : 19730591930 a^{2} - 10443729444 a - 83838058011 : 1\right)$ $\left(\frac{27023007}{4} a^{2} - \frac{7151863}{2} a - \frac{114824557}{4} : 7795373 a^{2} - \frac{33009759}{8} a - \frac{264989083}{8} : 1\right)$
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: \( 171.477293083240 \)
Tamagawa product: \( 4 \)
Torsion order: \(8\)
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\((j)_{-}\)
\((-a+1)\) \(2\) \(4\) \(I_{4}\) 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 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.