Properties

Label 3.3.148.1-20.1-a11
Base field 3.3.148.1
Conductor norm \( 20 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 3.3.148.1

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

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

Weierstrass equation

\({y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-40a^{2}+102a-30\right){x}-285a^{2}+712a-203\)
sage: E = EllipticCurve([K([-1,0,1]),K([-2,-1,1]),K([-2,-1,1]),K([-30,102,-40]),K([-203,712,-285])])
 
gp: E = ellinit([Polrev([-1,0,1]),Polrev([-2,-1,1]),Polrev([-2,-1,1]),Polrev([-30,102,-40]),Polrev([-203,712,-285])], K);
 
magma: E := EllipticCurve([K![-1,0,1],K![-2,-1,1],K![-2,-1,1],K![-30,102,-40],K![-203,712,-285]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^2+2a+3)\) = \((a^2-a-2)^{2}\cdot(a^2-a-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 20 \) = \(2^{2}\cdot5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-8a^2+6a+14)\) = \((a^2-a-2)^{4}\cdot(a^2-a-1)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2000 \) = \(2^{4}\cdot5^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{3238786394956992}{125} a^{2} - \frac{8036058359040256}{125} a + \frac{2186604679552576}{125} \)
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: \(0\)
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: $\left(-\frac{5}{2} a^{2} + \frac{9}{2} a + 2 : -\frac{3}{2} a + 3 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 4.1661379972939733536126697996404754190 \)
Tamagawa product: \( 1 \)  =  \(1\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 0.77052247619568588510982188104015429810 \)
Analytic order of Ш: \( 9 \) (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\((j)_{-}\)
\((a^2-a-2)\) \(2\) \(1\) \(IV\) Additive \(-1\) \(2\) \(4\) \(0\)
\((a^2-a-1)\) \(5\) \(1\) \(I_{3}\) 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
\(2\) 2B
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6, 8, 12 and 24.
Its isogeny class 20.1-a consists of curves linked by isogenies of degrees dividing 24.

Base change

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

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