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Properties

Label 3.3.1620.1-1.1-a1
Base field 3.3.1620.1
Conductor norm \( 1 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 3 \)
Rank \( 1 \)

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Base field 3.3.1620.1

Generator \(a\), with minimal polynomial \( x^{3} - 12 x - 14 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, -12, 0, 1]))
 
gp: K = nfinit(Polrev([-14, -12, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, -12, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-a-8\right){x}{y}+\left(a^{2}-2a-7\right){y}={x}^{3}+\left(-a^{2}+3a+7\right){x}^{2}+\left(3a^{2}-5a-19\right){x}-2a^{2}+6a+11\)
sage: E = EllipticCurve([K([-8,-1,1]),K([7,3,-1]),K([-7,-2,1]),K([-19,-5,3]),K([11,6,-2])])
 
gp: E = ellinit([Polrev([-8,-1,1]),Polrev([7,3,-1]),Polrev([-7,-2,1]),Polrev([-19,-5,3]),Polrev([11,6,-2])], K);
 
magma: E := EllipticCurve([K![-8,-1,1],K![7,3,-1],K![-7,-2,1],K![-19,-5,3],K![11,6,-2]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1)\) = \((1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1 \) = 1
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1)\) = \((1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -1 \) = -1
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -316368 \)
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 $\left(a^{2} - 2 a - 9 : 2 a^{2} - 2 a - 22 : 1\right)$
Height \(0.19424341991560637818547261867367932874\)
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: $\left(\frac{1}{3} a^{2} - a - \frac{7}{3} : -\frac{1}{3} a^{2} + \frac{4}{3} a + 2 : 1\right)$
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: \( 0.19424341991560637818547261867367932874 \)
Period: \( 685.20009975084139466811148321442734169 \)
Tamagawa product: \( 1 \)
Torsion order: \(3\)
Leading coefficient: \( 1.1022623445806833334489159663292698170 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
No primes of bad reduction.

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
\(5\) 5Ns

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 1.1-a consists of curves linked by isogenies of degree 3.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following elliptic curve:

Base field Curve
\(\Q\) 324.b1