Properties

Label 2.2.57.1-196.1-e3
Base field \(\Q(\sqrt{57}) \)
Conductor \((14)\)
Conductor norm \( 196 \)
CM no
Base change yes: 5054.c6,126.b6
Q-curve yes
Torsion order \( 6 \)
Rank \( 0 \)

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Show commands: Magma / Pari/GP / SageMath

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

\({y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(54108a+177205\right){x}+22267918a+72925587\)
sage: E = EllipticCurve([K([1,0]),K([0,-1]),K([0,1]),K([177205,54108]),K([72925587,22267918])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([0,-1])),Pol(Vecrev([0,1])),Pol(Vecrev([177205,54108])),Pol(Vecrev([72925587,22267918]))], K);
 
magma: E := EllipticCurve([K![1,0],K![0,-1],K![0,1],K![177205,54108],K![72925587,22267918]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((14)\) = \((a-4)\cdot(a+3)\cdot(2a+7)\cdot(-2a+9)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 196 \) = \(2\cdot2\cdot7\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-21952)\) = \((a-4)^{6}\cdot(a+3)^{6}\cdot(2a+7)^{3}\cdot(-2a+9)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 481890304 \) = \(2^{6}\cdot2^{6}\cdot7^{3}\cdot7^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{9938375}{21952} \)
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: $\left(197 a + 644 : 8329 a + 27279 : 1\right)$
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: \( 7.02770810590006 \)
Tamagawa product: \( 324 \)  =  \(( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot3\cdot3\)
Torsion order: \(6\)
Leading coefficient: \( 8.37758410404563 \)
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\((j)_{-}\)
\((a-4)\) \(2\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\((a+3)\) \(2\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\((2a+7)\) \(7\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((-2a+9)\) \(7\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

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
\(3\) 3Cs.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 196.1-e consists of curves linked by isogenies of degrees dividing 18.

Base change

This curve is the base change of 5054.c6, 126.b6, defined over \(\Q\), so it is also a \(\Q\)-curve.