Properties

Label 2.0.3.1-106875.1-a3
Base field \(\Q(\sqrt{-3}) \)
Conductor \((-375a+225)\)
Conductor norm \( 106875 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Show commands: Magma / Pari/GP / SageMath

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(Pol(Vecrev([1, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(12a+8\right){x}-19a+10\)
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([0,1]),K([8,12]),K([10,-19])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,1])),Pol(Vecrev([8,12])),Pol(Vecrev([10,-19]))], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,1],K![8,12],K![10,-19]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-375a+225)\) = \((-2a+1)^{2}\cdot(-5a+3)\cdot(5)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 106875 \) = \(3^{2}\cdot19\cdot25^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-328125a+375000)\) = \((-2a+1)^{3}\cdot(-5a+3)\cdot(5)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 125244140625 \) = \(3^{3}\cdot19\cdot25^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{9153}{19} a + \frac{36801}{19} \)
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: $\left(-5 a + 4 : 2 a - 2 : 1\right)$
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: \( 1.62788251674446 \)
Tamagawa product: \( 4 \)  =  \(2\cdot1\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 1.87971681850300 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(2\) \(III\) Additive \(1\) \(2\) \(3\) \(0\)
\((-5a+3)\) \(19\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((5)\) \(25\) \(2\) \(I_0^{*}\) Additive \(1\) \(2\) \(6\) \(0\)

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[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 106875.1-a consists of curves linked by isogenies of degrees dividing 6.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.