Properties

Label 2.2.88.1-144.1-j1
Base field \(\Q(\sqrt{22}) \)
Conductor \((12)\)
Conductor norm \( 144 \)
CM no
Base change yes: 48.a6,23232.do6
Q-curve yes
Torsion order \( 8 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{22}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 22 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-22, 0, 1]))
 
gp: K = nfinit(Pol(Vecrev([-22, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-22, 0, 1]);
 

Weierstrass equation

\(y^2+axy=x^{3}+\left(-64813a+304010\right)x-142529688a+668523501\)
sage: E = EllipticCurve([K([0,1]),K([0,0]),K([0,0]),K([304010,-64813]),K([668523501,-142529688])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([304010,-64813])),Pol(Vecrev([668523501,-142529688]))], K);
 
magma: E := EllipticCurve([K![0,1],K![0,0],K![0,0],K![304010,-64813],K![668523501,-142529688]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((12)\) = \( \left(-3 a - 14\right)^{4} \cdot \left(-a + 5\right) \cdot \left(a + 5\right) \)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 144 \) = \( 2^{4} \cdot 3^{2} \)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((209952)\) = \( \left(-3 a - 14\right)^{10} \cdot \left(-a + 5\right)^{8} \cdot \left(a + 5\right)^{8} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 44079842304 \) = \( 2^{10} \cdot 3^{16} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{207646}{6561} \)
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: \(1\)
Generator $\left(-28 a + \frac{259}{2} : -\frac{16849}{4} a + \frac{39523}{2} : 1\right)$
Height \(1.25108585306857\)
Torsion structure: \(\Z/8\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(35 a - 166 : 3620 a - 16975 : 1\right)$
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: \( 1.25108585306857 \)
Period: \( 5.68350851750159 \)
Tamagawa product: \( 256 \)  =  \(2^{2}\cdot2^{3}\cdot2^{3}\)
Torsion order: \(8\)
Leading coefficient: \(6.06390347137042\)
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\((j)_{-}\)
\( \left(-3 a - 14\right) \) \(2\) \(4\) \(I_{2}^*\) Additive \(-1\) \(4\) \(10\) \(0\)
\( \left(-a + 5\right) \) \(3\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\( \left(a + 5\right) \) \(3\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)

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 144.1-j consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base change of elliptic curves 48.a6, 23232.do6, defined over \(\Q\), so it is also a \(\Q\)-curve.