Properties

Label 2.0.3.1-100009.7-b2
Base field \(\Q(\sqrt{-3}) \)
Conductor norm \( 100009 \)
CM no
Base change no
Q-curve no
Torsion order \( 3 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{-3}) \)

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

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

Weierstrass equation

\({y}^2+a{y}={x}^{3}+{x}^{2}+\left(-14a+238\right){x}+1384a-1237\)
sage: E = EllipticCurve([K([0,0]),K([1,0]),K([0,1]),K([238,-14]),K([-1237,1384])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([1,0]),Polrev([0,1]),Polrev([238,-14]),Polrev([-1237,1384])], K);
 
magma: E := EllipticCurve([K![0,0],K![1,0],K![0,1],K![238,-14],K![-1237,1384]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((343a-280)\) = \((-3a+1)\cdot(3a-2)\cdot(4a-3)\cdot(-13a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 100009 \) = \(7\cdot7\cdot13\cdot157\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((889365043a-765787288)\) = \((-3a+1)^{9}\cdot(3a-2)^{3}\cdot(4a-3)\cdot(-13a+1)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 696335905851813409 \) = \(7^{9}\cdot7^{3}\cdot13\cdot157^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{3259043377277014016}{2030133836302663} a + \frac{110775144237600768}{2030133836302663} \)
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 $\left(\frac{147}{4} a - 35 : \frac{19}{4} a + \frac{1603}{8} : 1\right)$
Height \(0.69398044560042439843489516208551982955\)
Torsion structure: \(\Z/3\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-21 a + \frac{14}{3} : -\frac{134}{9} a - \frac{455}{9} : 1\right)$
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: \( 0.69398044560042439843489516208551982955 \)
Period: \( 0.44232816376404432510917763403901099246 \)
Tamagawa product: \( 81 \)  =  \(3^{2}\cdot3\cdot1\cdot3\)
Torsion order: \(3\)
Leading coefficient: \( 6.3801912822477944994654782324891242245 \)
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\((j)_{-}\)
\((-3a+1)\) \(7\) \(9\) \(I_{9}\) Split multiplicative \(-1\) \(1\) \(9\) \(9\)
\((3a-2)\) \(7\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((4a-3)\) \(13\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((-13a+1)\) \(157\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(3\) 3B.1.1[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 100009.7-b consists of curves linked by isogenies of degree 3.

Base change

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

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