Properties

Label 3.3.316.1-37.1-a1
Base field 3.3.316.1
Conductor norm \( 37 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 3.3.316.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -4, -1, 1]))
 
gp: K = nfinit(Polrev([2, -4, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -4, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-8602a^{2}+4553a+36552\right){x}+7282823a^{2}-3854918a-30945737\)
sage: E = EllipticCurve([K([1,0,0]),K([0,1,0]),K([1,1,0]),K([36552,4553,-8602]),K([-30945737,-3854918,7282823])])
 
gp: E = ellinit([Polrev([1,0,0]),Polrev([0,1,0]),Polrev([1,1,0]),Polrev([36552,4553,-8602]),Polrev([-30945737,-3854918,7282823])], K);
 
magma: E := EllipticCurve([K![1,0,0],K![0,1,0],K![1,1,0],K![36552,4553,-8602],K![-30945737,-3854918,7282823]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^2+2a+5)\) = \((-2a^2+2a+5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 37 \) = \(37\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((5a^2-3a-9)\) = \((-2a^2+2a+5)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1369 \) = \(37^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{231143481}{1369} a^{2} - \frac{650356337}{1369} a + \frac{254921595}{1369} \)
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: $\left(-\frac{259}{4} a^{2} + \frac{67}{2} a + \frac{1097}{4} : \frac{259}{8} a^{2} - \frac{69}{4} a - \frac{1101}{8} : 1\right)$
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: \( 61.225121792654115375211395713487934645 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 1.7220910940409345116243036898995417107 \)
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\((j)_{-}\)
\((-2a^2+2a+5)\) \(37\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 37.1-a consists of curves linked by isogenies of degree 2.

Base change

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

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