Properties

Label 3.3.49.1-56.1-a1
Base field \(\Q(\zeta_{7})^+\)
Conductor \((-2a^2-2a+4)\)
Conductor norm \( 56 \)
CM no
Base change yes: 14.a1
Q-curve yes
Torsion order \( 2 \)
Rank \( 0 \)

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Base field \(\Q(\zeta_{7})^+\)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -2, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([1, -2, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}-2731{x}-55146\)
sage: E = EllipticCurve([K([1,0,0]),K([0,0,0]),K([1,0,0]),K([-2731,0,0]),K([-55146,0,0])])
 
gp: E = ellinit([Pol(Vecrev([1,0,0])),Pol(Vecrev([0,0,0])),Pol(Vecrev([1,0,0])),Pol(Vecrev([-2731,0,0])),Pol(Vecrev([-55146,0,0]))], K);
 
magma: E := EllipticCurve([K![1,0,0],K![0,0,0],K![1,0,0],K![-2731,0,0],K![-55146,0,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^2-2a+4)\) = \((-a^2-a+2)\cdot(2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 56 \) = \(7\cdot8\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((25088)\) = \((-a^2-a+2)^{6}\cdot(2)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 15790581481472 \) = \(7^{6}\cdot8^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{2251439055699625}{25088} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{121}{4} : \frac{117}{8} : 1\right)$
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: \( 0.288080952338379 \)
Tamagawa product: \( 6 \)  =  \(( 2 \cdot 3 )\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 0.555584693795445 \)
Analytic order of Ш: \( 9 \) (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\((j)_{-}\)
\((-a^2-a+2)\) \(7\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\((2)\) \(8\) \(1\) \(I_{9}\) Non-split multiplicative \(1\) \(1\) \(9\) \(9\)

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.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 6, 9 and 18.
Its isogeny class 56.1-a consists of curves linked by isogenies of degrees dividing 18.

Base change

This curve is the base change of elliptic curves 14.a1, defined over \(\Q\), so it is also a \(\Q\)-curve.