Properties

Label 4.4.2225.1-19.2-a2
Base field 4.4.2225.1
Conductor norm \( 19 \)
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

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1/2a^3-1/2a^2-1/2a+2)\) = \((1/2a^3-1/2a^2-1/2a+2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 19 \) = \(19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((3/2a^3-7/2a^2-5/2a+8)\) = \((1/2a^3-1/2a^2-1/2a+2)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 361 \) = \(19^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{1170199359725}{722} a^{3} - \frac{162147898405}{722} a^{2} + \frac{5666384793219}{722} a + \frac{2055574156969}{361} \)
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: $\left(-a^{2} + 2 a + 2 : -a^{2} + a + 2 : 1\right)$ $\left(\frac{3}{8} a^{3} - \frac{7}{8} a^{2} - \frac{7}{8} a + \frac{7}{4} : -\frac{1}{8} a^{2} - \frac{1}{8} a - \frac{1}{8} : 1\right)$
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: \( 434.85492064667811289629217138118454955 \)
Tamagawa product: \( 2 \)
Torsion order: \(4\)
Leading coefficient: \( 1.15236323498953 \)
Analytic order of Ш: \( 1 \) (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\((j)_{-}\)
\((1/2a^3-1/2a^2-1/2a+2)\) \(19\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 19.2-a consists of curves linked by isogenies of degrees dividing 8.

Base change

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

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