Properties

Label 4.4.2777.1-704.3-q11
Base field 4.4.2777.1
Conductor norm \( 704 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 4.4.2777.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 4 x^{2} + x + 2 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 1, -4, -1, 1]))
 
gp: K = nfinit(Polrev([2, 1, -4, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 1, -4, -1, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(27a^{3}+a^{2}-122a-77\right){x}-103a^{3}-32a^{2}+512a+327\)
sage: E = EllipticCurve([K([0,1,0,0]),K([-2,-1,1,0]),K([-2,-1,1,0]),K([-77,-122,1,27]),K([327,512,-32,-103])])
 
gp: E = ellinit([Polrev([0,1,0,0]),Polrev([-2,-1,1,0]),Polrev([-2,-1,1,0]),Polrev([-77,-122,1,27]),Polrev([327,512,-32,-103])], K);
 
magma: E := EllipticCurve([K![0,1,0,0],K![-2,-1,1,0],K![-2,-1,1,0],K![-77,-122,1,27],K![327,512,-32,-103]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^3+5a^2-a-6)\) = \((-a)^{6}\cdot(-a^3+2a^2+2a-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 704 \) = \(2^{6}\cdot11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((871a^3-565a^2-2604a-44)\) = \((-a)^{34}\cdot(-a^3+2a^2+2a-1)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2078764171264 \) = \(2^{34}\cdot11^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{22702786953003143813}{7929856} a^{3} + \frac{30928910991628706537}{7929856} a^{2} - \frac{8873274696408279559}{3964928} a - \frac{19220593607745698479}{7929856} \)
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: $\left(-2 a^{3} + \frac{3}{4} a^{2} + 7 a + 6 : \frac{5}{8} a^{3} - \frac{7}{2} a - 1 : 1\right)$
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: \( 53.856972788828019140987035063703612587 \)
Tamagawa product: \( 8 \)  =  \(2^{2}\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 2.04401460481171 \)
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\((j)_{-}\)
\((-a)\) \(2\) \(4\) \(I_{24}^{*}\) Additive \(-1\) \(6\) \(34\) \(16\)
\((-a^3+2a^2+2a-1)\) \(11\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6, 8, 12 and 24.
Its isogeny class 704.3-q consists of curves linked by isogenies of degrees dividing 24.

Base change

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

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