Properties

Label 3.3.229.1-8.1-a2
Base field 3.3.229.1
Conductor norm \( 8 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 3.3.229.1

Generator \(a\), with minimal polynomial \( x^{3} - 4 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -4, 0, 1]))
 
gp: K = nfinit(Polrev([-1, -4, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -4, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(-78a^{2}+141a+49\right){x}-725a^{2}+1346a+393\)
sage: E = EllipticCurve([K([-2,0,1]),K([4,-1,-1]),K([0,1,0]),K([49,141,-78]),K([393,1346,-725])])
 
gp: E = ellinit([Polrev([-2,0,1]),Polrev([4,-1,-1]),Polrev([0,1,0]),Polrev([49,141,-78]),Polrev([393,1346,-725])], K);
 
magma: E := EllipticCurve([K![-2,0,1],K![4,-1,-1],K![0,1,0],K![49,141,-78],K![393,1346,-725]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2)\) = \((a+1)\cdot(a^2-a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 8 \) = \(2\cdot4\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-320a^2+832)\) = \((a+1)^{15}\cdot(a^2-a-3)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 134217728 \) = \(2^{15}\cdot4^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{5149908848399}{32768} a^{2} - \frac{9582969548701}{32768} a - \frac{2767591890197}{32768} \)
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: $\left(-\frac{7}{4} a^{2} + \frac{17}{4} a + \frac{1}{4} : \frac{13}{8} a^{2} - \frac{31}{8} a - \frac{15}{8} : 1\right)$
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: \( 6.7650905431862827571877351113351568059 \)
Tamagawa product: \( 6 \)  =  \(1\cdot( 2 \cdot 3 )\)
Torsion order: \(2\)
Leading coefficient: \( 0.67057464970504837012570298903524365768 \)
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\((j)_{-}\)
\((a+1)\) \(2\) \(1\) \(I_{15}\) Non-split multiplicative \(1\) \(1\) \(15\) \(15\)
\((a^2-a-3)\) \(4\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)

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
\(5\) 5B.4.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 5, 6, 10, 15 and 30.
Its isogeny class 8.1-a consists of curves linked by isogenies of degrees dividing 30.

Base change

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

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