Properties

Label 2.2.24.1-10.1-a3
Base field \(\Q(\sqrt{6}) \)
Conductor \((a + 4)\)
Conductor norm \( 10 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{6}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 6 \); class number \(1\).

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 6)
 
gp: K = nfinit(a^2 - 6);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6, 0, 1]);
 

Weierstrass equation

\(y^2+xy+y=x^{3}+\left(9a+22\right)x-133a-326\)
sage: E = EllipticCurve(K, [1, 0, 1, 9*a + 22, -133*a - 326])
 
gp: E = ellinit([1, 0, 1, 9*a + 22, -133*a - 326],K)
 
magma: E := ChangeRing(EllipticCurve([1, 0, 1, 9*a + 22, -133*a - 326]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a + 4)\) = \( \left(-a + 2\right) \cdot \left(-a + 1\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 10 \) = \( 2 \cdot 5 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((-6192 a + 35072)\) = \( \left(-a + 2\right)^{9} \cdot \left(-a + 1\right)^{9} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1000000000 \) = \( 2^{9} \cdot 5^{9} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{21355471243}{62500000} a + \frac{1734442691}{1953125} \)
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: \( 3.96059963476085 \)
Tamagawa product: \( 1 \)  =  \(1\cdot1\)
Torsion order: \(1\)
Leading coefficient: \(0.808454015051459\)
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\((j)_{-}\)
\( \left(-a + 2\right) \) \(2\) \(1\) \(I_{9}\) Non-split multiplicative \(1\) \(1\) \(9\) \(9\)
\( \left(-a + 1\right) \) \(5\) \(1\) \(I_{9}\) Non-split multiplicative \(1\) \(1\) \(9\) \(9\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
Its isogeny class 10.1-a consists of curves linked by isogenies of degrees dividing 9.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.