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Properties

Label 4.4.2225.1-1.1-a8
Base field 4.4.2225.1
Conductor norm \( 1 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 4.4.2225.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} + 2 x + 4 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([4, 2, -5, -1, 1]))
 
gp: K = nfinit(Polrev([4, 2, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 2, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a\right){x}{y}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-2\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}+\frac{1}{2}a-1\right){x}^{2}+\left(a^{3}-72a^{2}-205a-123\right){x}-14a^{3}-662a^{2}-1638a-908\)
sage: E = EllipticCurve([K([0,-1/2,-1/2,1/2]),K([-1,1/2,1/2,-1/2]),K([-2,-5/2,1/2,1/2]),K([-123,-205,-72,1]),K([-908,-1638,-662,-14])])
 
gp: E = ellinit([Polrev([0,-1/2,-1/2,1/2]),Polrev([-1,1/2,1/2,-1/2]),Polrev([-2,-5/2,1/2,1/2]),Polrev([-123,-205,-72,1]),Polrev([-908,-1638,-662,-14])], K);
 
magma: E := EllipticCurve([K![0,-1/2,-1/2,1/2],K![-1,1/2,1/2,-1/2],K![-2,-5/2,1/2,1/2],K![-123,-205,-72,1],K![-908,-1638,-662,-14]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1)\) = \((1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1 \) = 1
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1)\) = \((1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -1 \) = -1
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{3464946905534287422448807585}{2} a^{3} - \frac{6307067008351224343169529695}{2} a^{2} - \frac{12151367540253075049485032893}{2} a + 8448521965185185103366369798 \)
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: $\left(-\frac{11}{8} a^{3} + \frac{21}{8} a^{2} + \frac{109}{8} a + \frac{31}{4} : -\frac{33}{8} a^{3} - \frac{41}{8} a^{2} + \frac{79}{8} a + \frac{73}{8} : 1\right)$
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: \( 7.6913614842102498395918828051015952505 \)
Tamagawa product: \( 1 \)
Torsion order: \(2\)
Leading coefficient: \( 0.366877209043145 \)
Analytic order of Ш: \( 9 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
No primes of bad reduction.

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, 4, 6, 8, 12 and 24.
Its isogeny class 1.1-a 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.