Properties

Base field \(\Q(\sqrt{-3}) \)
Label 2.0.3.1-103204.1-a1
Conductor \((370 a - 162)\)
Conductor norm \( 103204 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank \( 2 \)

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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

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((370 a - 162)\) = \( \left(2\right) \cdot \left(185 a - 81\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 103204 \) = \( 4 \cdot 25801 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((370 a - 162)\) = \( \left(2\right) \cdot \left(185 a - 81\right) \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 103204 \) = \( 4 \cdot 25801 \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( \frac{33095871}{51602} a + \frac{119466225}{51602} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \( 2 \)

sage: E.rank()
 
magma: Rank(E);
 

Generators: $\left(a - 1 : -2 a + 1 : 1\right)$, $\left(-a + 1 : -a - 1 : 1\right)$

sage: gens = E.gens(); gens
 
magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 

Heights: 0.640520349132501, 0.474720811241054

sage: [P.height() for P in gens]
 
magma: [Height(P):P in gens];
 

Regulator: 0.140279746294203

sage: E.regulator_of_points(gens)
 
magma: Regulator(gens);
 

Torsion subgroup

Structure: Trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

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\((j)_{-}\)
\( \left(185 a - 81\right) \) \(25801\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\( \left(2\right) \) \(4\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

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 103204.1-a consists of this curve only.

Base change

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