Properties

Label 2.2.57.1-4.1-b1
Base field \(\Q(\sqrt{57}) \)
Conductor norm \( 4 \)
CM no
Base change no
Q-curve yes
Torsion order \( 6 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{57}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 14 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, -1, 1]))
 
gp: K = nfinit(Polrev([-14, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(91a-386\right){x}-817a+3486\)
sage: E = EllipticCurve([K([1,0]),K([0,-1]),K([0,1]),K([-386,91]),K([3486,-817])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([0,-1]),Polrev([0,1]),Polrev([-386,91]),Polrev([3486,-817])], K);
 
magma: E := EllipticCurve([K![1,0],K![0,-1],K![0,1],K![-386,91],K![3486,-817]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2)\) = \((a-4)\cdot(a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 4 \) = \(2\cdot2\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-8a+24)\) = \((a-4)^{3}\cdot(a+3)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -512 \) = \(-2^{3}\cdot2^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{20297286875}{64} a + \frac{43383068421}{32} \)
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: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-a + 6 : 4 a - 20 : 1\right)$
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: \( 37.293973247474262896767627306329999867 \)
Tamagawa product: \( 18 \)  =  \(3\cdot( 2 \cdot 3 )\)
Torsion order: \(6\)
Leading coefficient: \( 2.4698537144898885856119803717851474036 \)
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\((j)_{-}\)
\((a-4)\) \(2\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((a+3)\) \(2\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)

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
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 4.1-b consists of curves linked by isogenies of degrees dividing 6.

Base change

This elliptic curve is a \(\Q\)-curve.

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