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Properties

Label 4.4.17600.1-4.1-a2
Base field 4.4.17600.1
Conductor norm \( 4 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 4.4.17600.1

Generator \(a\), with minimal polynomial \( x^{4} - 14 x^{2} + 44 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([44, 0, -14, 0, 1]))
 
gp: K = nfinit(Polrev([44, 0, -14, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![44, 0, -14, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(\frac{1}{2}a^{3}-4a+1\right){x}{y}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-3a-4\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+3a+1\right){x}^{2}+\left(-2a^{3}+3a^{2}+16a-27\right){x}-\frac{461}{2}a^{3}+\frac{1003}{2}a^{2}+2128a-4637\)
sage: E = EllipticCurve([K([1,-4,0,1/2]),K([1,3,0,-1/2]),K([-4,-3,1/2,1/2]),K([-27,16,3,-2]),K([-4637,2128,1003/2,-461/2])])
 
gp: E = ellinit([Polrev([1,-4,0,1/2]),Polrev([1,3,0,-1/2]),Polrev([-4,-3,1/2,1/2]),Polrev([-27,16,3,-2]),Polrev([-4637,2128,1003/2,-461/2])], K);
 
magma: E := EllipticCurve([K![1,-4,0,1/2],K![1,3,0,-1/2],K![-4,-3,1/2,1/2],K![-27,16,3,-2],K![-4637,2128,1003/2,-461/2]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-1/2a^3+4a+2)\) = \((-1/2a^3+4a+2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 4 \) = \(4\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-8a^3-16a^2+64a+128)\) = \((-1/2a^3+4a+2)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -262144 \) = \(-4^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{2103593}{64} a^{3} - \frac{1259703}{16} a^{2} + \frac{2476531}{8} a + \frac{11343627}{16} \)
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: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 216.93117694562527560287970649173869762 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.63518027104281 \)
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\((j)_{-}\)
\((-1/2a^3+4a+2)\) \(4\) \(1\) \(I_{9}\) Non-split multiplicative \(1\) \(1\) \(9\) \(9\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 4.1-a consists of curves linked by isogenies of degree 3.

Base change

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

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