Properties

Label 2.2.37.1-9.1-a2
Base field \(\Q(\sqrt{37}) \)
Conductor norm \( 9 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{37}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 9 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-9, -1, 1]))
 
gp: K = nfinit(Polrev([-9, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-345a-876\right){x}-4855a-12339\)
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([1,1]),K([-876,-345]),K([-12339,-4855])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([1,0]),Polrev([1,1]),Polrev([-876,-345]),Polrev([-12339,-4855])], K);
 
magma: E := EllipticCurve([K![1,0],K![1,0],K![1,1],K![-876,-345],K![-12339,-4855]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((3)\) = \((a-3)\cdot(a+2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 9 \) = \(3\cdot3\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1404a+1269)\) = \((a-3)^{3}\cdot(a+2)^{12}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 14348907 \) = \(3^{3}\cdot3^{12}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{21580736655842}{531441} a + \frac{8491991437009}{59049} \)
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: $\left(-6 a - \frac{61}{4} : \frac{5}{2} a + \frac{57}{8} : 1\right)$
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: \( 12.433164740040620502028966364814808649 \)
Tamagawa product: \( 2 \)  =  \(1\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 1.0219998461316769133749775600079428676 \)
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\((j)_{-}\)
\((a-3)\) \(3\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\((a+2)\) \(3\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)

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
\(3\) 3Ns

Isogenies and isogeny class

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

Base change

This elliptic curve is a \(\Q\)-curve.

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