Properties

Label 3.3.316.1-32.1-c5
Base field 3.3.316.1
Conductor \((-2a^2+8)\)
Conductor norm \( 32 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

Related objects

Downloads

Learn more

Show commands for: Magma / Pari/GP / SageMath

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

\({y}^2+a{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{2}+a-2\right){x}^{2}+\left(486273921a^{2}-257393114a-2066246710\right){x}-8535646743838a^{2}+4518059339552a+36269162590988\)
sage: E = EllipticCurve([K([0,1,0]),K([-2,1,1]),K([-2,0,1]),K([-2066246710,-257393114,486273921]),K([36269162590988,4518059339552,-8535646743838])])
 
gp: E = ellinit([Pol(Vecrev([0,1,0])),Pol(Vecrev([-2,1,1])),Pol(Vecrev([-2,0,1])),Pol(Vecrev([-2066246710,-257393114,486273921])),Pol(Vecrev([36269162590988,4518059339552,-8535646743838]))], K);
 
magma: E := EllipticCurve([K![0,1,0],K![-2,1,1],K![-2,0,1],K![-2066246710,-257393114,486273921],K![36269162590988,4518059339552,-8535646743838]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^2+8)\) = \((a)^{4}\cdot(-a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 32 \) = \(2^{4}\cdot2\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2a^2+12a+12)\) = \((a)^{12}\cdot(-a+1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -8192 \) = \(2^{12}\cdot2\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{10154621765056003}{2} a^{2} + 2687505089603077 a + 21574207961612782 \)
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: \(1\)
Generator $\left(-6131 a^{2} + 3293 a + 26138 : 1769 a^{2} - 318 a - 6396 : 1\right)$
Height \(0.264164657112299\)
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: $\left(-\frac{24637}{4} a^{2} + 3255 a + 26162 : \frac{11613}{8} a^{2} - \frac{1525}{2} a - \frac{24633}{4} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.264164657112299 \)
Period: \( 30.3125634354532 \)
Tamagawa product: \( 4 \)  =  \(2^{2}\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 1.35137254263725 \)
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\((j)_{-}\)
\((a)\) \(2\) \(4\) \(I_4^{*}\) Additive \(1\) \(4\) \(12\) \(0\)
\((-a+1)\) \(2\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

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 32.1-c 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.