Properties

Label 2.0.3.1-23808.1-a2
Base field \(\Q(\sqrt{-3}) \)
Conductor \((-176a+112)\)
Conductor norm \( 23808 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 0 \)

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

Weierstrass equation

\({y}^2={x}^{3}+a{x}^{2}+\left(18a+13\right){x}+a-50\)
sage: E = EllipticCurve([K([0,0]),K([0,1]),K([0,0]),K([13,18]),K([-50,1])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,1])),Pol(Vecrev([0,0])),Pol(Vecrev([13,18])),Pol(Vecrev([-50,1]))], K);
 
magma: E := EllipticCurve([K![0,0],K![0,1],K![0,0],K![13,18],K![-50,1]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-176a+112)\) = \((-2a+1)\cdot(2)^{4}\cdot(-6a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 23808 \) = \(3\cdot4^{4}\cdot31\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-1795072a+198656)\) = \((-2a+1)\cdot(2)^{10}\cdot(-6a+1)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2905145868288 \) = \(3\cdot4^{10}\cdot31^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{2499774004}{2770563} a + \frac{1516133908}{2770563} \)
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: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(7 : -4 a - 20 : 1\right)$
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: \( 1.25447673684327 \)
Tamagawa product: \( 16 \)  =  \(1\cdot2^{2}\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 1.44854496341717 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((2)\) \(4\) \(4\) \(I_{2}^{*}\) Additive \(1\) \(4\) \(10\) \(0\)
\((-6a+1)\) \(31\) \(4\) \(I_{4}\) 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 23808.1-a consists of curves linked by isogenies of degrees dividing 4.

Base change

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