Properties

Label 4.4.15952.1-9.1-a3
Base field 4.4.15952.1
Conductor norm \( 9 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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Base field 4.4.15952.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -2, -6, 0, 1]))
 
gp: K = nfinit(Polrev([1, -2, -6, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -6, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-3\right){x}{y}+{y}={x}^{3}+\left(a^{3}-7a-1\right){x}^{2}+\left(-38a^{3}+25a^{2}+209a-57\right){x}-264a^{3}+170a^{2}+1472a-413\)
sage: E = EllipticCurve([K([-3,0,1,0]),K([-1,-7,0,1]),K([1,0,0,0]),K([-57,209,25,-38]),K([-413,1472,170,-264])])
 
gp: E = ellinit([Polrev([-3,0,1,0]),Polrev([-1,-7,0,1]),Polrev([1,0,0,0]),Polrev([-57,209,25,-38]),Polrev([-413,1472,170,-264])], K);
 
magma: E := EllipticCurve([K![-3,0,1,0],K![-1,-7,0,1],K![1,0,0,0],K![-57,209,25,-38],K![-413,1472,170,-264]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^3+6a+4)\) = \((a^2+2a)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 9 \) = \(3^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2a^3+2a^2+12a+3)\) = \((a^2+2a)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -2187 \) = \(-3^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{11187968}{3} a^{3} - \frac{7136320}{3} a^{2} - \frac{62574976}{3} a + \frac{17544448}{3} \)
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 $\left(-2 a^{3} + 2 a^{2} + 12 a - 3 : -2 a^{3} + a^{2} + 15 a - 4 : 1\right)$
Height \(0.32363630688438263996662183817381777955\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-2 a^{3} + a^{2} + \frac{23}{2} a - 3 : -\frac{11}{4} a^{3} + 2 a^{2} + \frac{61}{4} a - \frac{9}{2} : 1\right)$
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: \( 0.32363630688438263996662183817381777955 \)
Period: \( 451.08632628849794254771318930750723524 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 2.31174179264793 \)
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\((j)_{-}\)
\((a^2+2a)\) \(3\) \(2\) \(I_{1}^{*}\) Additive \(-1\) \(2\) \(7\) \(1\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 9.1-a consists of curves linked by isogenies of degrees dividing 12.

Base change

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

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