Properties

Label 2.0.3.1-108300.2-i2
Base field \(\Q(\sqrt{-3}) \)
Conductor \((-380 a + 190)\)
Conductor norm \( 108300 \)
CM no
Base change yes: 570.k4,1710.j4
Q-curve yes
Torsion order \( 6 \)
Rank \( 0 \)

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

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

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x + 1)
 
gp: K = nfinit(a^2 - a + 1);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
 

Weierstrass equation

\(y^2+\left(a+1\right)xy=x^{3}-ax^{2}+\left(85489a-85489\right)x+8420985\)
sage: E = EllipticCurve(K, [a + 1, -a, 0, 85489*a - 85489, 8420985])
 
gp: E = ellinit([a + 1, -a, 0, 85489*a - 85489, 8420985],K)
 
magma: E := ChangeRing(EllipticCurve([a + 1, -a, 0, 85489*a - 85489, 8420985]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-380 a + 190)\) = \( \left(2\right) \cdot \left(-2 a + 1\right) \cdot \left(5\right) \cdot \left(-5 a + 3\right) \cdot \left(-5 a + 2\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 108300 \) = \( 3 \cdot 4 \cdot 19^{2} \cdot 25 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((70568821500000000)\) = \( \left(2\right)^{8} \cdot \left(-2 a + 1\right)^{2} \cdot \left(5\right)^{9} \cdot \left(-5 a + 3\right)^{6} \cdot \left(-5 a + 2\right)^{6} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 4979958567898862250000000000000000 \) = \( 3^{2} \cdot 4^{8} \cdot 19^{12} \cdot 25^{9} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{69096190760262356111}{70568821500000000} \)
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: $\left(1002 a : 36998 a - 18499 : 1\right)$
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: \( 0.0215028098516952 \)
Tamagawa product: \( 5184 \)  =  \(2\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot2^{3}\cdot3^{2}\)
Torsion order: \(6\)
Leading coefficient: \(3.57542008018835\)
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\((j)_{-}\)
\( \left(-2 a + 1\right) \) \(3\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\( \left(-5 a + 3\right) \) \(19\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\( \left(-5 a + 2\right) \) \(19\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\( \left(2\right) \) \(4\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\( \left(5\right) \) \(25\) \(9\) \(I_{9}\) Split multiplicative \(-1\) \(1\) \(9\) \(9\)

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[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 108300.2-i consists of curves linked by isogenies of degrees dividing 6.

Base change

This curve is the base change of elliptic curves 570.k4, 1710.j4, defined over \(\Q\), so it is also a \(\Q\)-curve.