Properties

Label 3.3.733.1-8.1-b3
Base field 3.3.733.1
Conductor norm \( 8 \)
CM no
Base change no
Q-curve no
Torsion order \( 6 \)
Rank \( 0 \)

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Base field 3.3.733.1

Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 7 x + 8 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([8, -7, -1, 1]))
 
gp: K = nfinit(Polrev([8, -7, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -7, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(398163a^{2}+83660a-2735554\right){x}-225568877a^{2}-41410904a+1534686372\)
sage: E = EllipticCurve([K([1,0,0]),K([1,-1,0]),K([-4,0,1]),K([-2735554,83660,398163]),K([1534686372,-41410904,-225568877])])
 
gp: E = ellinit([Polrev([1,0,0]),Polrev([1,-1,0]),Polrev([-4,0,1]),Polrev([-2735554,83660,398163]),Polrev([1534686372,-41410904,-225568877])], K);
 
magma: E := EllipticCurve([K![1,0,0],K![1,-1,0],K![-4,0,1],K![-2735554,83660,398163],K![1534686372,-41410904,-225568877]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2)\) = \((a^2-6)\cdot(a^2+a-3)\)
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: \((8a^2-8a-56)\) = \((a^2-6)^{3}\cdot(a^2+a-3)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 32768 \) = \(2^{3}\cdot4^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 15216 a^{2} - \frac{319193}{64} a - 83228 \)
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: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-180 a^{2} - 145 a + 1507 : -3221 a^{2} + 2007 a + 15370 : 1\right)$
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: \( 192.85042244371909563368473925472005908 \)
Tamagawa product: \( 6 \)  =  \(1\cdot( 2 \cdot 3 )\)
Torsion order: \(6\)
Leading coefficient: \( 1.1871821465214534665762161365038999363 \)
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\((j)_{-}\)
\((a^2-6)\) \(2\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\((a^2+a-3)\) \(4\) \(6\) \(I_{6}\) 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.1.1

Isogenies and isogeny class

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

Base change

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

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