Properties

Label 2.2.8.1-4050.1-a3
Base field \(\Q(\sqrt{2}) \)
Conductor \((45 a)\)
Conductor norm \( 4050 \)
CM no
Base change yes: 90.c2,2880.a2
Q-curve yes
Torsion order \( 6 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{2}) \)

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

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

Weierstrass equation

\(y^2+xy+y=x^{3}-x^{2}-4082x+14681\)
sage: E = EllipticCurve(K, [1, -1, 1, -4082, 14681])
 
gp: E = ellinit([1, -1, 1, -4082, 14681],K)
 
magma: E := ChangeRing(EllipticCurve([1, -1, 1, -4082, 14681]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((45 a)\) = \( \left(a\right) \cdot \left(3\right)^{2} \cdot \left(5\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 4050 \) = \( 2 \cdot 9^{2} \cdot 25 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((4271484375000)\) = \( \left(a\right)^{6} \cdot \left(3\right)^{7} \cdot \left(5\right)^{12} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 18245578765869140625000000 \) = \( 2^{6} \cdot 9^{7} \cdot 25^{12} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{10316097499609}{5859375000} \)
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: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-9 : 229 : 1\right)$
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: \( 0.447290761184899 \)
Tamagawa product: \( 288 \)  =  \(( 2 \cdot 3 )\cdot2^{2}\cdot( 2^{2} \cdot 3 )\)
Torsion order: \(6\)
Leading coefficient: \(1.26512932158374\)
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\((j)_{-}\)
\( \left(a\right) \) \(2\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\( \left(3\right) \) \(9\) \(4\) \(I_{1}^*\) Additive \(1\) \(2\) \(7\) \(1\)
\( \left(5\right) \) \(25\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)

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
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 4050.1-a consists of curves linked by isogenies of degrees dividing 12.

Base change

This curve is the base change of elliptic curves 90.c2, 2880.a2, defined over \(\Q\), so it is also a \(\Q\)-curve.