Properties

Label 4.4.5125.1-55.2-d3
Base field 4.4.5125.1
Conductor \((-3a^2+4a+10)\)
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

\({y}^2+a{x}{y}+\left(a^{3}-a^{2}-3a\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(189a^{3}-180a^{2}-756a-392\right){x}+2259a^{3}-2017a^{2}-9352a-4703\)
sage: E = EllipticCurve([K([0,1,0,0]),K([1,-1,0,0]),K([0,-3,-1,1]),K([-392,-756,-180,189]),K([-4703,-9352,-2017,2259])])
 
gp: E = ellinit([Pol(Vecrev([0,1,0,0])),Pol(Vecrev([1,-1,0,0])),Pol(Vecrev([0,-3,-1,1])),Pol(Vecrev([-392,-756,-180,189])),Pol(Vecrev([-4703,-9352,-2017,2259]))], K);
 
magma: E := EllipticCurve([K![0,1,0,0],K![1,-1,0,0],K![0,-3,-1,1],K![-392,-756,-180,189],K![-4703,-9352,-2017,2259]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-3a^2+4a+10)\) = \((-a^3+a^2+4a+1)\cdot(a-1)\)
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: \((-3a^3+12a^2+14a-34)\) = \((-a^3+a^2+4a+1)^{3}\cdot(a-1)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 166375 \) = \(5^{3}\cdot11^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{133990069789633150033955762628}{6655} a^{3} + \frac{496615724716951900546923293583}{6655} a^{2} - \frac{8693172568676612045861200086}{1331} a - \frac{78523816804016589414106934741}{605} \)
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: $\left(4 a^{3} - \frac{25}{4} a^{2} - 13 a + 1 : -\frac{11}{8} a^{3} - 5 a^{2} + 15 a + 22 : 1\right)$
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: \( 8.25979024593038 \)
Tamagawa product: \( 1 \)  =  \(1\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 1.03839979400277 \)
Analytic order of Ш: \( 36 \) (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\((j)_{-}\)
\((-a^3+a^2+4a+1)\) \(5\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\((a-1)\) \(11\) \(1\) \(I_{3}\) Non-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
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 55.2-d 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.