Properties

Label 3.3.361.1-152.1-b1
Base field 3.3.361.1
Conductor norm \( 152 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 1 \)
Rank \( 2 \)

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Base field 3.3.361.1

Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 6 x + 7 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([7, -6, -1, 1]))
 
gp: K = nfinit(Polrev([7, -6, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, -6, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}-86{x}-2456\)
sage: E = EllipticCurve([K([1,0,0]),K([0,0,0]),K([1,0,0]),K([-86,0,0]),K([-2456,0,0])])
 
gp: E = ellinit([Polrev([1,0,0]),Polrev([0,0,0]),Polrev([1,0,0]),Polrev([-86,0,0]),Polrev([-2456,0,0])], K);
 
magma: E := EllipticCurve([K![1,0,0],K![0,0,0],K![1,0,0],K![-86,0,0],K![-2456,0,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a^2-2a-8)\) = \((2)\cdot(a^2-a-4)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 152 \) = \(8\cdot19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2550136832)\) = \((2)^{27}\cdot(a^2-a-4)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -16584044393473483018619322368 \) = \(-8^{27}\cdot19^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{69173457625}{2550136832} \)
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: \(2\)
Generators $\left(56 a^{2} + 70 a - 152 : -1064 a^{2} - 1330 a + 3239 : 1\right)$ $\left(14 a^{2} - 56 a + 72 : 252 a^{2} - 1008 a + 944 : 1\right)$
Heights \(3.5310205292547226350389848862382583360\) \(3.5310205292547226350389848862382583360\)
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: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 9.3510794835137261610841082385429325727 \)
Period: \( 0.25029801601354496512378004733500609506 \)
Tamagawa product: \( 3 \)  =  \(1\cdot3\)
Torsion order: \(1\)
Leading coefficient: \( 3.3260541759120084807228687381942275689 \)
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\((j)_{-}\)
\((2)\) \(8\) \(1\) \(I_{27}\) Non-split multiplicative \(1\) \(1\) \(27\) \(27\)
\((a^2-a-4)\) \(19\) \(3\) \(I_{3}\) 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.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
Its isogeny class 152.1-b consists of curves linked by isogenies of degrees dividing 9.

Base change

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

Base field Curve
\(\Q\) 38.a1