Properties

Label 2.2.76.1-171.1-d1
Base field \(\Q(\sqrt{19}) \)
Conductor norm \( 171 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{19}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 19 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-19, 0, 1]))
 
gp: K = nfinit(Polrev([-19, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, 0, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}+{y}={x}^{3}+\left(-659572a-2874998\right){x}-609801836a-2658064575\)
sage: E = EllipticCurve([K([0,1]),K([0,0]),K([1,0]),K([-2874998,-659572]),K([-2658064575,-609801836])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([0,0]),Polrev([1,0]),Polrev([-2874998,-659572]),Polrev([-2658064575,-609801836])], K);
 
magma: E := EllipticCurve([K![0,1],K![0,0],K![1,0],K![-2874998,-659572],K![-2658064575,-609801836]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((3a)\) = \((-a-4)\cdot(-a+4)\cdot(a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 171 \) = \(3\cdot3\cdot19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1701a-9234)\) = \((-a-4)^{8}\cdot(-a+4)^{5}\cdot(a)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 30292137 \) = \(3^{8}\cdot3^{5}\cdot19\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{12364905437067631}{124659} a - \frac{2836704000281954}{6561} \)
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: 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: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 1.3164966829745532622875361969558851618 \)
Tamagawa product: \( 40 \)  =  \(2^{3}\cdot5\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 6.0405010532553486831309462364161585526 \)
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\((j)_{-}\)
\((-a-4)\) \(3\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\((-a+4)\) \(3\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)
\((a)\) \(19\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 171.1-d consists of this curve only.

Base change

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

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