Properties

Label 2.2.76.1-72.1-j2
Base field \(\Q(\sqrt{19}) \)
Conductor \((-18 a + 78)\)
Conductor norm \( 72 \)
CM no
Base change yes: 48.a5,8664.j5
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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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(Pol(Vecrev([-19, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, 0, 1]);
 

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-18 a + 78)\) = \( \left(-3 a + 13\right)^{3} \cdot \left(-a - 4\right) \cdot \left(-a + 4\right) \)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 72 \) = \( 2^{3} \cdot 3^{2} \)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((48)\) = \( \left(-3 a + 13\right)^{8} \cdot \left(-a - 4\right) \cdot \left(-a + 4\right) \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2304 \) = \( 2^{8} \cdot 3^{2} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{2048}{3} \)
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 $\left(\frac{25}{9} : \frac{35}{27} a : 1\right)$
Height \(3.12519016772384\)
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: $\left(0 : 0 : 1\right)$
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: \( 3.12519016772384 \)
Period: \( 11.3670170350032 \)
Tamagawa product: \( 2 \)  =  \(2\cdot1\cdot1\)
Torsion order: \(2\)
Leading coefficient: \(4.07489257427996\)
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\((j)_{-}\)
\( \left(-3 a + 13\right) \) \(2\) \(2\) \(I_{1}^*\) Additive \(-1\) \(3\) \(8\) \(0\)
\( \left(-a - 4\right) \) \(3\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\( \left(-a + 4\right) \) \(3\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

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, 4 and 8.
Its isogeny class 72.1-j consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base change of elliptic curves 48.a5, 8664.j5, defined over \(\Q\), so it is also a \(\Q\)-curve.