Properties

Label 4.4.2225.1-1.1-a4
Base field 4.4.2225.1
Conductor norm \( 1 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
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

\({y}^2+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{3}{2}a-3\right){x}{y}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-2\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{3}{2}a+4\right){x}^{2}+\left(14a^{3}+9a^{2}-71a-88\right){x}+\frac{245}{2}a^{3}+\frac{93}{2}a^{2}-\frac{1209}{2}a-580\)
sage: E = EllipticCurve([K([-3,-3/2,1/2,1/2]),K([4,3/2,-1/2,-1/2]),K([-2,-5/2,1/2,1/2]),K([-88,-71,9,14]),K([-580,-1209/2,93/2,245/2])])
 
gp: E = ellinit([Polrev([-3,-3/2,1/2,1/2]),Polrev([4,3/2,-1/2,-1/2]),Polrev([-2,-5/2,1/2,1/2]),Polrev([-88,-71,9,14]),Polrev([-580,-1209/2,93/2,245/2])], K);
 
magma: E := EllipticCurve([K![-3,-3/2,1/2,1/2],K![4,3/2,-1/2,-1/2],K![-2,-5/2,1/2,1/2],K![-88,-71,9,14],K![-580,-1209/2,93/2,245/2]]);
 

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: \( \frac{34051050961357}{2} a^{3} - \frac{61982369024223}{2} a^{2} - \frac{119414392544609}{2} a + 83028186306741 \)
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: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(\frac{5}{2} a^{3} - \frac{5}{2} a^{2} - \frac{19}{2} a + 1 : 3 a^{3} - 2 a^{2} - 12 a - 2 : 1\right)$ $\left(\frac{1}{4} a^{3} + \frac{5}{2} a^{2} - \frac{7}{2} a - 9 : \frac{7}{8} a^{3} + \frac{7}{4} a^{2} - \frac{13}{2} a - 10 : 1\right)$
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: \( 30.765445936840999358367531220406381002 \)
Tamagawa product: \( 1 \)
Torsion order: \(4\)
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
\(2\) 2Cs
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
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.