Properties

Label 4.4.9909.1-13.1-a1
Base field 4.4.9909.1
Conductor norm \( 13 \)
CM no
Base change no
Q-curve no
Torsion order \( 3 \)
Rank \( 1 \)

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Base field 4.4.9909.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -3, -6, 0, 1]))
 
gp: K = nfinit(Polrev([3, -3, -6, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -3, -6, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}-5a-1\right){x}{y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-17a^{3}-3a^{2}+34a-30\right){x}+45a^{3}+106a^{2}+25a-23\)
sage: E = EllipticCurve([K([-1,-5,0,1]),K([0,-5,-1,1]),K([0,0,0,0]),K([-30,34,-3,-17]),K([-23,25,106,45])])
 
gp: E = ellinit([Polrev([-1,-5,0,1]),Polrev([0,-5,-1,1]),Polrev([0,0,0,0]),Polrev([-30,34,-3,-17]),Polrev([-23,25,106,45])], K);
 
magma: E := EllipticCurve([K![-1,-5,0,1],K![0,-5,-1,1],K![0,0,0,0],K![-30,34,-3,-17],K![-23,25,106,45]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^2+2a+2)\) = \((-a^2+2a+2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 13 \) = \(13\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a^3-a^2-4a-2)\) = \((-a^2+2a+2)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 13 \) = \(13\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{7837506146388204859}{13} a^{3} - \frac{20303559893752613332}{13} a^{2} - \frac{5572637861243472815}{13} a + \frac{9076222771033149887}{13} \)
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 $\left(\frac{2}{3} a^{2} + 2 a + 1 : -\frac{8}{3} a^{3} - \frac{1}{3} a^{2} + \frac{25}{3} a : 1\right)$
Height \(1.0792081230232533500294488918118566163\)
Torsion structure: \(\Z/3\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(\frac{4}{3} a^{3} - a^{2} - \frac{14}{3} a + \frac{14}{3} : \frac{7}{9} a^{3} - \frac{34}{9} a^{2} - 4 a + \frac{20}{3} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 1.0792081230232533500294488918118566163 \)
Period: \( 441.35516528255297854490464229289901069 \)
Tamagawa product: \( 1 \)
Torsion order: \(3\)
Leading coefficient: \( 2.12664983546438 \)
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\((j)_{-}\)
\((-a^2+2a+2)\) \(13\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

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.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
Its isogeny class 13.1-a consists of curves linked by isogenies of degrees dividing 9.

Base change

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

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