Properties

Label 2.2.76.1-171.1-i4
Base field \(\Q(\sqrt{19}) \)
Conductor \((3 a)\)
Conductor norm \( 171 \)
CM no
Base change yes: 57.c2,17328.u2
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

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

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

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 19)
 
gp: K = nfinit(a^2 - 19);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, 0, 1]);
 

Weierstrass equation

\(y^2+xy+y=x^{3}-7x+5\)
sage: E = EllipticCurve(K, [1, 0, 1, -7, 5])
 
gp: E = ellinit([1, 0, 1, -7, 5],K)
 
magma: E := ChangeRing(EllipticCurve([1, 0, 1, -7, 5]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((3 a)\) = \( \left(-a - 4\right) \cdot \left(-a + 4\right) \cdot \left(a\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 171 \) = \( 3^{2} \cdot 19 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((3249)\) = \( \left(-a - 4\right)^{2} \cdot \left(-a + 4\right)^{2} \cdot \left(a\right)^{4} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 10556001 \) = \( 3^{4} \cdot 19^{4} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{30664297}{3249} \)
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: \(\Z/2\Z\times\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-3 : 1 : 1\right)$ $\left(1 : -1 : 1\right)$
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: \( 18.8460963550300 \)
Tamagawa product: \( 8 \)  =  \(2\cdot2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \(1.08089775647113\)
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\((j)_{-}\)
\( \left(-a - 4\right) \) \(3\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\( \left(-a + 4\right) \) \(3\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\( \left(a\right) \) \(19\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 171.1-i consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base change of elliptic curves 57.c2, 17328.u2, defined over \(\Q\), so it is also a \(\Q\)-curve.