Properties

Label 2.0.3.1-99372.5-a1
Base field \(\Q(\sqrt{-3}) \)
Conductor norm \( 99372 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 1 \)
Rank \( 0 \)

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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

\({y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-100484{x}-12372091\)
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([1,0]),K([-100484,0]),K([-12372091,0])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([1,0]),Polrev([1,0]),Polrev([-100484,0]),Polrev([-12372091,0])], K);
 
magma: E := EllipticCurve([K![1,0],K![1,0],K![1,0],K![-100484,0],K![-12372091,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-364a+182)\) = \((-2a+1)\cdot(2)\cdot(-3a+1)\cdot(3a-2)\cdot(-4a+1)\cdot(4a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 99372 \) = \(3\cdot4\cdot7\cdot7\cdot13\cdot13\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-745029571313664)\) = \((-2a+1)^{14}\cdot(2)^{17}\cdot(-3a+1)\cdot(3a-2)\cdot(-4a+1)^{5}\cdot(4a-3)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 555069062131821951814673104896 \) = \(3^{14}\cdot4^{17}\cdot7\cdot7\cdot13^{5}\cdot13^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{112205650221491190337}{745029571313664} \)
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: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 0.034096494268966215781985005437744089388 \)
Tamagawa product: \( 34 \)  =  \(2\cdot17\cdot1\cdot1\cdot1\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 1.3386221698334920242003419476857696753 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(2\) \(I_{14}\) Non-split multiplicative \(1\) \(1\) \(14\) \(14\)
\((2)\) \(4\) \(17\) \(I_{17}\) Split multiplicative \(-1\) \(1\) \(17\) \(17\)
\((-3a+1)\) \(7\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((3a-2)\) \(7\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((-4a+1)\) \(13\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)
\((4a-3)\) \(13\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 99372.5-a consists of this curve only.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 546.e1
\(\Q\) 1638.a1