Properties

Label 4.4.725.1-89.1-a4
Base field 4.4.725.1
Conductor norm \( 89 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 4.4.725.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 1, -3, -1, 1]))
 
gp: K = nfinit(Polrev([1, 1, -3, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -3, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}-3a\right){x}{y}+a^{2}{y}={x}^{3}+\left(a^{3}-3a+1\right){x}^{2}+\left(70a^{3}-189a^{2}+25a+30\right){x}+801a^{3}-2028a^{2}+111a+647\)
sage: E = EllipticCurve([K([0,-3,0,1]),K([1,-3,0,1]),K([0,0,1,0]),K([30,25,-189,70]),K([647,111,-2028,801])])
 
gp: E = ellinit([Polrev([0,-3,0,1]),Polrev([1,-3,0,1]),Polrev([0,0,1,0]),Polrev([30,25,-189,70]),Polrev([647,111,-2028,801])], K);
 
magma: E := EllipticCurve([K![0,-3,0,1],K![1,-3,0,1],K![0,0,1,0],K![30,25,-189,70],K![647,111,-2028,801]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^3+5a)\) = \((-2a^3+5a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 89 \) = \(89\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-519a^3+945a^2+542a-1235)\) = \((-2a^3+5a)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -496981290961 \) = \(-89^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{101672510628391532828120426}{496981290961} a^{3} + \frac{239507317407110136025004820}{496981290961} a^{2} - \frac{19676384011560149188326968}{496981290961} a - \frac{74997740941951418606146221}{496981290961} \)
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: $\left(5 a^{3} - \frac{25}{4} a^{2} - \frac{21}{4} a + \frac{5}{2} : -3 a^{3} + \frac{13}{4} a^{2} - \frac{1}{8} a - \frac{13}{4} : 1\right)$
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: \( 4.8214262022697171776448708264838895454 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 0.805784732213557 \)
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\((j)_{-}\)
\((-2a^3+5a)\) \(89\) \(2\) \(I_{6}\) 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 89.1-a consists of curves linked by isogenies of degrees dividing 6.

Base change

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

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