Properties

Label 2.2.41.1-361.1-a2
Base field \(\Q(\sqrt{41}) \)
Conductor \((19)\)
Conductor norm \( 361 \)
CM no
Base change yes: 19.a2,31939.e2
Q-curve yes
Torsion order \( 3 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{41}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 10 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-10, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([-10, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10, -1, 1]);
 

Weierstrass equation

\({y}^2+{y}={x}^{3}+{x}^{2}-9{x}-15\)
sage: E = EllipticCurve([K([0,0]),K([1,0]),K([1,0]),K([-9,0]),K([-15,0])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([-9,0])),Pol(Vecrev([-15,0]))], K);
 
magma: E := EllipticCurve([K![0,0],K![1,0],K![1,0],K![-9,0],K![-15,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((19)\) = \((19)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 361 \) = \(361\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-6859)\) = \((19)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 47045881 \) = \(361^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{89915392}{6859} \)
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 $\left(\frac{328071349}{100600900} : \frac{173802949917}{504513513500} a - \frac{678316463417}{1009027027000} : 1\right)$
Height \(20.1099139648163\)
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: $\left(5 : -10 : 1\right)$
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: \( 20.1099139648163 \)
Period: \( 1.84894653281620 \)
Tamagawa product: \( 3 \)
Torsion order: \(3\)
Leading coefficient: \( 3.87125142058114 \)
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\((j)_{-}\)
\((19)\) \(361\) \(3\) \(I_{3}\) 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
\(3\) 3Cs.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 361.1-a consists of curves linked by isogenies of degrees dividing 9.

Base change

This curve is the base change of elliptic curves 19.a2, 31939.e2, defined over \(\Q\), so it is also a \(\Q\)-curve.