Elliptic curve 48384.2-g3 over number field Q(−3)\Q(\sqrt{-3})

• Label: 2.0.3.1-48384.2-g3
• Base field: $\Q(\sqrt{-3})$
• Conductor norm: $48384$
• CM: no
• Base change: no
• Q-curve: no
• Torsion order: $1$
• Rank: $1$

Base field $\Q(\sqrt{-3})$

Generator $a$, with minimal polynomial $x^{2} - x + 1$; class number $1$.

Weierstrass equation

${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-22a-25\right){x}+73a-9$

This is a global minimal model.

Mordell-Weil group structure

$\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(12 a - 1 : 48 a - 32 : 1\right)$	$0.12539665589001946873738694357532586991$	$\infty$

Invariants

Conductor:	$\frak{N}$	=	$(48a+192)$	=	$(-2a+1)^{3}\cdot(2)^{4}\cdot(3a-2)$
Conductor norm:	$N(\frak{N})$	=	$48384$	=	$3^{3}\cdot4^{4}\cdot7$
Discriminant:	$\Delta$	=	$1966080a+1671168$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(1966080a+1671168)$	=	$(-2a+1)^{3}\cdot(2)^{15}\cdot(3a-2)^{3}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$9943923032064$	=	$3^{3}\cdot4^{15}\cdot7^{3}$
j-invariant:	$j$	=	$\frac{1598955}{686} a + \frac{947913}{2744}$
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z$    (no potential complex multiplication)
Sato-Tate group:	$\mathrm{ST}(E)$	=	$\mathrm{SU}(2)$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$1$
Mordell-Weil rank:	$r$	=	$1$
Regulator:	$\mathrm{Reg}(E/K)$	≈	$0.12539665589001946873738694357532586991$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$0.250793311780038937474773887150651739820$
Global period:	$\Omega(E/K)$	≈	$2.2120949694505331200974455779075360722$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$12$  =  $1\cdot2^{2}\cdot3$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$1$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$3.8436190504537616227339422193491281347$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 3.843619050 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 2.212095 \cdot 0.250793 \cdot 12 } { {1^2 \cdot 1.732051} } \approx 3.843619050$

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 3 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(-2a+1)$	$3$	$1$	$II$	Additive	$1$	$3$	$3$	$0$
$(2)$	$4$	$4$	$I_{7}^{*}$	Additive	$-1$	$4$	$15$	$3$
$(3a-2)$	$7$	$3$	$I_{3}$	Split multiplicative	$-1$	$1$	$3$	$3$

Galois Representations

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

prime	Image of Galois Representation
$3$	3Cs[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 48384.2-g consists of curves linked by isogenies of degrees dividing 9.

Base change

This elliptic curve is not a $\Q$-curve.

It is not the base change of an elliptic curve defined over any subfield.