Properties

Label 4.4.5125.1-55.1-d4
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

\({y}^2+\left(a+1\right){x}{y}+\left(a^{3}-4a-3\right){y}={x}^{3}+\left(-6a^{3}-a^{2}+25a+16\right){x}-86a^{3}+215a^{2}+239a-646\)
sage: E = EllipticCurve([K([1,1,0,0]),K([0,0,0,0]),K([-3,-4,0,1]),K([16,25,-1,-6]),K([-646,239,215,-86])])
 
gp: E = ellinit([Pol(Vecrev([1,1,0,0])),Pol(Vecrev([0,0,0,0])),Pol(Vecrev([-3,-4,0,1])),Pol(Vecrev([16,25,-1,-6])),Pol(Vecrev([-646,239,215,-86]))], K);
 
magma: E := EllipticCurve([K![1,1,0,0],K![0,0,0,0],K![-3,-4,0,1],K![16,25,-1,-6],K![-646,239,215,-86]]);
 

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: \((15a^3-80a^2-225a+165)\) = \((-a^3+a^2+4a+1)^{6}\cdot(-a)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 27680640625 \) = \(5^{6}\cdot11^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{63264852088520783254}{44289025} a^{3} - \frac{8937579533749840902}{8857805} a^{2} + \frac{258647477139845380633}{44289025} a + \frac{257139780743870750929}{44289025} \)
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: $\left(2 a^{3} - 3 a^{2} - 7 a + 7 : -2 a^{3} - a^{2} + 9 a + 9 : 1\right)$
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: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 1.03839979400277 \)
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\((j)_{-}\)
\((-a^3+a^2+4a+1)\) \(5\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\((-a)\) \(11\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)

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 55.1-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.