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Properties

Label 4.4.19821.1-49.1-b1
Base field 4.4.19821.1
Conductor norm \( 49 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank not available

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Base field 4.4.19821.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 8 x^{2} + 6 x + 3 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, 6, -8, -1, 1]))
 
gp: K = nfinit(Polrev([3, 6, -8, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 6, -8, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a-1\right){x}{y}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-2\right){y}={x}^{3}+\left(-\frac{2}{3}a^{3}+\frac{1}{3}a^{2}+4a-1\right){x}^{2}+\left(\frac{46}{3}a^{3}+\frac{7}{3}a^{2}-125a-54\right){x}+117a^{3}+7a^{2}-929a-296\)
sage: E = EllipticCurve([K([-1,-2,1/3,1/3]),K([-1,4,1/3,-2/3]),K([-2,3,2/3,-1/3]),K([-54,-125,7/3,46/3]),K([-296,-929,7,117])])
 
gp: E = ellinit([Polrev([-1,-2,1/3,1/3]),Polrev([-1,4,1/3,-2/3]),Polrev([-2,3,2/3,-1/3]),Polrev([-54,-125,7/3,46/3]),Polrev([-296,-929,7,117])], K);
 
magma: E := EllipticCurve([K![-1,-2,1/3,1/3],K![-1,4,1/3,-2/3],K![-2,3,2/3,-1/3],K![-54,-125,7/3,46/3],K![-296,-929,7,117]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2/3a^3+1/3a^2+6a-3)\) = \((-1/3a^3-1/3a^2+2a)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 49 \) = \(7^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-30a^3+52a^2+105a-95)\) = \((-1/3a^3-1/3a^2+2a)^{10}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 282475249 \) = \(7^{10}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 6360 a^{3} - 378485 a^{2} + 78352 a + 2833106 \)
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: \(0 \le r \le 1\)
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: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0 \le r \le 1\)
Regulator: not available
Period: \( 51.312042001624570617238531632223118536 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 6.31142597792801 \)
Analytic order of Ш: not available

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\((j)_{-}\)
\((-1/3a^3-1/3a^2+2a)\) \(7\) \(1\) \(II^{*}\) Additive \(-1\) \(2\) \(10\) \(0\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 49.1-b consists of this curve only.

Base change

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

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