Properties

Label 2.2.12.1-2304.1-e5
Base field \(\Q(\sqrt{3}) \)
Conductor norm \( 2304 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Base field \(\Q(\sqrt{3}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 3 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, 0, 1]))
 
gp: K = nfinit(Polrev([-3, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);
 

Weierstrass equation

\({y}^2={x}^{3}+a{x}^{2}+\left(44a-84\right){x}+160a-272\)
sage: E = EllipticCurve([K([0,0]),K([0,1]),K([0,0]),K([-84,44]),K([-272,160])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([0,1]),Polrev([0,0]),Polrev([-84,44]),Polrev([-272,160])], K);
 
magma: E := EllipticCurve([K![0,0],K![0,1],K![0,0],K![-84,44],K![-272,160]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((48)\) = \((a+1)^{8}\cdot(a)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 2304 \) = \(2^{8}\cdot3^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-124416)\) = \((a+1)^{18}\cdot(a)^{10}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 15479341056 \) = \(2^{18}\cdot3^{10}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1122088}{9} a + \frac{1989808}{9} \)
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: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-4 : 0 : 1\right)$ $\left(-4 a + 8 : 0 : 1\right)$
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: \( 7.7830915032296410870543107897663836461 \)
Tamagawa product: \( 16 \)  =  \(2^{2}\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 2.2467849872585611634644197290321034328 \)
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\((j)_{-}\)
\((a+1)\) \(2\) \(4\) \(I_{6}^{*}\) Additive \(-1\) \(8\) \(18\) \(0\)
\((a)\) \(3\) \(4\) \(I_{4}^{*}\) Additive \(-1\) \(2\) \(10\) \(4\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 2304.1-e consists of curves linked by isogenies of degrees dividing 8.

Base change

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

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