Properties

Label 2.2.24.1-48.1-b4
Base field \(\Q(\sqrt{6}) \)
Conductor norm \( 48 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 8 \)
Rank \( 0 \)

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

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

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

Weierstrass equation

\({y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(120a-297\right){x}-891a+2181\)
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([0,1]),K([-297,120]),K([2181,-891])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([1,-1]),Polrev([0,1]),Polrev([-297,120]),Polrev([2181,-891])], K);
 
magma: E := EllipticCurve([K![0,1],K![1,-1],K![0,1],K![-297,120],K![2181,-891]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

\(\Z/{2}\Z \oplus \Z/{4}\Z\)

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(-2 a + 5 : -3 a + 6 : 1\right)$$0$$2$
$\left(0 : -14 a + 33 : 1\right)$$0$$4$

Invariants

Conductor: $\frak{N}$ = \((4a+12)\) = \((-a+2)^{4}\cdot(a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 48 \) = \(2^{4}\cdot3\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: $\Delta$ = $1296$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((1296)\) = \((-a+2)^{8}\cdot(a+3)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 1679616 \) = \(2^{8}\cdot3^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: $j$ = \( \frac{1556068}{81} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: $\mathrm{End}(E)$ = \(\Z\)   
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$ = \(\Z\)    (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{ST}(E)$ = $\mathrm{SU}(2)$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$= \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: $r$ = \(0\)
Regulator: $\mathrm{Reg}(E/K)$ = \( 1 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ = \( 1 \)
Global period: $\Omega(E/K)$ \( 22.734034070006347604148590511773734758 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 16 \)  =  \(2\cdot2^{3}\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(8\)
Special value: $L^{(r)}(E/K,1)/r!$ \( 1.1601413180534136645530850446401178325 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$\displaystyle 1.160141318 \approx L(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 22.734034 \cdot 1 \cdot 16 } { {8^2 \cdot 4.898979} } \approx 1.160141318$

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 

This elliptic curve is not semistable. There are 2 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$ $N(\mathfrak{p})$ Tamagawa number Kodaira symbol Reduction type Root number \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((-a+2)\) \(2\) \(2\) \(I_0^{*}\) Additive \(-1\) \(4\) \(8\) \(0\)
\((a+3)\) \(3\) \(8\) \(I_{8}\) 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
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 48.1-b consists of curves linked by isogenies of degrees dividing 8.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 48.a3
\(\Q\) 576.d3