Properties

Label 2.0.3.1-43776.2-a3
Base field \(\Q(\sqrt{-3}) \)
Conductor norm \( 43776 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 2 \)

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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

\({y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(114a+63\right){x}-87a+867\)
sage: E = EllipticCurve([K([0,0]),K([1,1]),K([0,0]),K([63,114]),K([867,-87])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([1,1]),Polrev([0,0]),Polrev([63,114]),Polrev([867,-87])], K);
 
magma: E := EllipticCurve([K![0,0],K![1,1],K![0,0],K![63,114],K![867,-87]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-240a+96)\) = \((-2a+1)^{2}\cdot(2)^{4}\cdot(-5a+2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 43776 \) = \(3^{2}\cdot4^{4}\cdot19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-114960384a-92067840)\) = \((-2a+1)^{10}\cdot(2)^{11}\cdot(-5a+2)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 32276531292143616 \) = \(3^{10}\cdot4^{11}\cdot19^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1018942694}{390963} a - \frac{1263770528}{1172889} \)
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: \(2\)
Generators $\left(-34 a + 11 : -72 a - 108 : 1\right)$ $\left(22 a + 19 : 192 a + 60 : 1\right)$
Heights \(0.19906882987648826376042885719837972233\) \(0.51502249883034584281497657311192930321\)
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: $\left(11 a - 7 : 0 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 0.095213010062462069327286253671938133857 \)
Period: \( 0.56293924313301726002595399226717480047 \)
Tamagawa product: \( 64 \)  =  \(2^{2}\cdot2^{2}\cdot2^{2}\)
Torsion order: \(2\)
Leading coefficient: \( 3.9610211589087325671723427724835135904 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(4\) \(I_{4}^{*}\) Additive \(-1\) \(2\) \(10\) \(4\)
\((2)\) \(4\) \(4\) \(I_{3}^{*}\) Additive \(-1\) \(4\) \(11\) \(0\)
\((-5a+2)\) \(19\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 43776.2-a consists of curves linked by isogenies of degrees dividing 4.

Base change

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

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