Properties

Label 2.2.57.1-256.1-a1
Base field \(\Q(\sqrt{57}) \)
Conductor norm \( 256 \)
CM no
Base change no
Q-curve yes
Torsion order \( 1 \)
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(Pol(Vecrev([-14, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, -1, 1]);
 

Weierstrass equation

\({y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-10282063a-33672998\right){x}+34886965210a+114251923320\)
sage: E = EllipticCurve([K([0,0]),K([-1,-1]),K([0,0]),K([-33672998,-10282063]),K([114251923320,34886965210])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([-1,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([-33672998,-10282063])),Pol(Vecrev([114251923320,34886965210]))], K);
 
magma: E := EllipticCurve([K![0,0],K![-1,-1],K![0,0],K![-33672998,-10282063],K![114251923320,34886965210]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((16)\) = \((a-4)^{4}\cdot(a+3)^{4}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 256 \) = \(2^{4}\cdot2^{4}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-24576a-81920)\) = \((a-4)^{15}\cdot(a+3)^{13}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 268435456 \) = \(2^{15}\cdot2^{13}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{293180476215589246298781}{8} a - \frac{960141789432972248487021}{8} \)
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: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 1.91553793752032 \)
Tamagawa product: \( 16 \)  =  \(2^{2}\cdot2^{2}\)
Torsion order: \(1\)
Leading coefficient: \( 4.05950716709009 \)
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\((j)_{-}\)
\((a-4)\) \(2\) \(4\) \(I_{7}^{*}\) Additive \(-1\) \(4\) \(15\) \(3\)
\((a+3)\) \(2\) \(4\) \(I_{5}^{*}\) Additive \(-1\) \(4\) \(13\) \(1\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B
\(7\) 7B.6.3

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 7 and 21.
Its isogeny class 256.1-a consists of curves linked by isogenies of degrees dividing 21.

Base change

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

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