Elliptic curve 138384.2-b2 over number field Q(−3)\Q(\sqrt{-3})

• Label: 2.0.3.1-138384.2-b2
• Base field: $\Q(\sqrt{-3})$
• Conductor norm: $138384$
• CM: no
• Base change: no
• Q-curve: yes
• Torsion order: $3$
• Rank: $2$

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

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

Weierstrass equation

${y}^2={x}^{3}+\left(-36a+3\right){x}+84a-47$

This is a global minimal model.

Mordell-Weil group structure

$\Z \oplus \Z \oplus \Z/{3}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(4 a + 1 : 6 a - 5 : 1\right)$	$1.1868364784358381697913212721369349892$	$\infty$
$\left(-a + 1 : -9 a : 1\right)$	$1.2510842975811834387540865657497474871$	$\infty$
$\left(3 : 6 a - 5 : 1\right)$	$0$	$3$

Invariants

Conductor:	$\frak{N}$	=	$(372)$	=	$(-2a+1)^{2}\cdot(2)^{2}\cdot(-6a+1)\cdot(6a-5)$
Conductor norm:	$N(\frak{N})$	=	$138384$	=	$3^{2}\cdot4^{2}\cdot31\cdot31$
Discriminant:	$\Delta$	=	$-321408a-147312$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-321408a-147312)$	=	$(-2a+1)^{6}\cdot(2)^{4}\cdot(-6a+1)\cdot(6a-5)^{3}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$172351183104$	=	$3^{6}\cdot4^{4}\cdot31\cdot31^{3}$
j-invariant:	$j$	=	$-\frac{185048064}{29791} a + \frac{379461888}{29791}$
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:	$r_{\mathrm{an}}$	=	$2$
Mordell-Weil rank:	$r$	=	$2$
Regulator:	$\mathrm{Reg}(E/K)$	≈	$0.46383067915812151697601387049061572289$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$1.85532271663248606790405548196246289156$
Global period:	$\Omega(E/K)$	≈	$2.8798055469335009862035990878990204366$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$18$  =  $2\cdot3\cdot1\cdot3$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$3$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$6.1695287775182601686499565680088645650$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 6.169528778 \approx L^{(2)}(E/K,1)/2! \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 2.879806 \cdot 1.855323 \cdot 18 } { {3^2 \cdot 1.732051} } \approx 6.169528778$

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 4 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(-2a+1)$	$3$	$2$	$I_0^{*}$	Additive	$-1$	$2$	$6$	$0$
$(2)$	$4$	$3$	$IV$	Additive	$1$	$2$	$4$	$0$
$(-6a+1)$	$31$	$1$	$I_{1}$	Split multiplicative	$-1$	$1$	$1$	$1$
$(6a-5)$	$31$	$3$	$I_{3}$	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
$3$	3B.1.1[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 138384.2-b consists of curves linked by isogenies of degree 3.

Base change

This elliptic curve is a $\Q$-curve.

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