Properties

Label 5.5.70601.1-47.2-b1
Base field 5.5.70601.1
Conductor norm \( 47 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 5.5.70601.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 2, -5, -1, 1]))
 
gp: K = nfinit(Polrev([-1, 3, 2, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 2, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{4}-a^{3}-5a^{2}+3a+3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-2a^{4}+a^{3}+11a^{2}+a-6\right){x}^{2}+\left(-2a^{4}+a^{3}+11a^{2}+a-7\right){x}\)
sage: E = EllipticCurve([K([3,3,-5,-1,1]),K([-6,1,11,1,-2]),K([1,1,0,0,0]),K([-7,1,11,1,-2]),K([0,0,0,0,0])])
 
gp: E = ellinit([Polrev([3,3,-5,-1,1]),Polrev([-6,1,11,1,-2]),Polrev([1,1,0,0,0]),Polrev([-7,1,11,1,-2]),Polrev([0,0,0,0,0])], K);
 
magma: E := EllipticCurve([K![3,3,-5,-1,1],K![-6,1,11,1,-2],K![1,1,0,0,0],K![-7,1,11,1,-2],K![0,0,0,0,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^4+a^3+6a^2-2)\) = \((-a^4+a^3+6a^2-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 47 \) = \(47\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((10a^4-7a^3-47a^2-4a+12)\) = \((-a^4+a^3+6a^2-2)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -103823 \) = \(-47^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{9074112}{103823} a^{4} + \frac{132922005}{103823} a^{3} - \frac{356032536}{103823} a^{2} - \frac{222800132}{103823} a + \frac{342458773}{103823} \)
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: \(1\)
Generator $\left(-1 : 0 : 1\right)$
Height \(0.030642720767374152884458601429825510400\)
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: \(1\)
Regulator: \( 0.030642720767374152884458601429825510400 \)
Period: \( 4853.4030722220209278803249863729934872 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 2.79858346 \)
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\((j)_{-}\)
\((-a^4+a^3+6a^2-2)\) \(47\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

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 47.2-b 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.