Properties

Label 2.0.3.1-50700.2-d2
Base field \(\Q(\sqrt{-3}) \)
Conductor \((-260a+130)\)
Conductor norm \( 50700 \)
CM no
Base change yes: 390.e1,1170.e1
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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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

\({y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+45a{x}-127\)
sage: E = EllipticCurve([K([0,1]),K([-1,1]),K([1,0]),K([0,45]),K([-127,0])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([-1,1])),Pol(Vecrev([1,0])),Pol(Vecrev([0,45])),Pol(Vecrev([-127,0]))], K);
 
magma: E := EllipticCurve([K![0,1],K![-1,1],K![1,0],K![0,45],K![-127,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-260a+130)\) = \((-2a+1)\cdot(2)\cdot(-4a+1)\cdot(4a-3)\cdot(5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 50700 \) = \(3\cdot4\cdot13\cdot13\cdot25\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1232010)\) = \((-2a+1)^{12}\cdot(2)\cdot(-4a+1)^{2}\cdot(4a-3)^{2}\cdot(5)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1517848640100 \) = \(3^{12}\cdot4\cdot13^{2}\cdot13^{2}\cdot25\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{10779215329}{1232010} \)
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: \(1\)
Generator $\left(-10 a + 10 : -27 a + 8 : 1\right)$
Height \(0.683588771940206\)
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: $\left(-\frac{13}{4} a + \frac{13}{4} : -\frac{17}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.683588771940206 \)
Period: \( 1.21906926899628 \)
Tamagawa product: \( 8 \)  =  \(2\cdot1\cdot2\cdot2\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 3.84904212214376 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)
\((2)\) \(4\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((-4a+1)\) \(13\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((4a-3)\) \(13\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((5)\) \(25\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

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.
Its isogeny class 50700.2-d consists of curves linked by isogenies of degree 2.

Base change

This curve is the base change of 390.e1, 1170.e1, defined over \(\Q\), so it is also a \(\Q\)-curve.