Properties

Label 4.4.5125.1-55.1-c1
Base field 4.4.5125.1
Conductor \((a^3-4a)\)
Conductor norm \( 55 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 4.4.5125.1

Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 6 x^{2} + 7 x + 11 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([11, 7, -6, -2, 1]))
 
gp: K = nfinit(Pol(Vecrev([11, 7, -6, -2, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 7, -6, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-4\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(32a^{3}-33a^{2}-128a-53\right){x}+141a^{3}-137a^{2}-564a-261\)
sage: E = EllipticCurve([K([-4,0,1,0]),K([0,-1,0,0]),K([1,0,0,0]),K([-53,-128,-33,32]),K([-261,-564,-137,141])])
 
gp: E = ellinit([Pol(Vecrev([-4,0,1,0])),Pol(Vecrev([0,-1,0,0])),Pol(Vecrev([1,0,0,0])),Pol(Vecrev([-53,-128,-33,32])),Pol(Vecrev([-261,-564,-137,141]))], K);
 
magma: E := EllipticCurve([K![-4,0,1,0],K![0,-1,0,0],K![1,0,0,0],K![-53,-128,-33,32],K![-261,-564,-137,141]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-4a)\) = \((-a^3+a^2+4a+1)\cdot(-a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 55 \) = \(5\cdot11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-a^2+3a)\) = \((-a^3+a^2+4a+1)\cdot(-a)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 55 \) = \(5\cdot11\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{19204557701238}{55} a^{3} - \frac{2347403181723}{55} a^{2} + \frac{110246499123534}{55} a + \frac{99540191518811}{55} \)
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: $\left(\frac{3}{2} a^{3} - \frac{3}{2} a^{2} - \frac{21}{4} a - \frac{13}{4} : -\frac{3}{8} a^{3} - \frac{5}{8} a^{2} + 3 a + \frac{5}{4} : 1\right)$
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: \( 117.381814862661 \)
Tamagawa product: \( 1 \)  =  \(1\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 1.63966031344996 \)
Analytic order of Ш: \( 4 \) (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\((j)_{-}\)
\((-a^3+a^2+4a+1)\) \(5\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((-a)\) \(11\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

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 and 6.
Its isogeny class 55.1-c consists of curves linked by isogenies of degrees dividing 6.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.