Elliptic curve 54684.2-a5 over number field Q(−3)\Q(\sqrt{-3})

• Label: 2.0.3.1-54684.2-a5
• Base field: $\Q(\sqrt{-3})$
• Conductor norm: $54684$
• CM: no
• Base change: no
• Q-curve: no
• Torsion order: $1$
• Rank: $0$

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

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

Weierstrass equation

${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-32887a+47339\right){x}+1679851a+2156108$

This is a global minimal model.

Mordell-Weil group structure

trivial

Invariants

Conductor:	$\frak{N}$	=	$(-270a+132)$	=	$(-2a+1)^{2}\cdot(2)\cdot(-3a+1)^{2}\cdot(6a-5)$
Conductor norm:	$N(\frak{N})$	=	$54684$	=	$3^{2}\cdot4\cdot7^{2}\cdot31$
Discriminant:	$\Delta$	=	$125715456a-212184576$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(125715456a-212184576)$	=	$(-2a+1)^{6}\cdot(2)^{9}\cdot(-3a+1)^{8}\cdot(6a-5)$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$34151789441581056$	=	$3^{6}\cdot4^{9}\cdot7^{8}\cdot31$
j-invariant:	$j$	=	$-\frac{19926242340409933}{388864} a + \frac{18769373204677155}{777728}$
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:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/K)$	=	$1$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	=	$1$
Global period:	$\Omega(E/K)$	≈	$0.216929058318908098458925836395784211340$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$2$  =  $1\cdot1\cdot2\cdot1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$1$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$2.2543929038785251052712157553570879198$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$9$ (rounded)

BSD formula

$\begin{aligned}2.254392904 \approx L(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 9 \cdot 0.216929 \cdot 1 \cdot 2 } { {1^2 \cdot 1.732051} } \\ & \approx 2.254392904 \end{aligned}$

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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(-2a+1)$	$3$	$1$	$I_0^{*}$	Additive	$-1$	$2$	$6$	$0$
$(2)$	$4$	$1$	$I_{9}$	Non-split multiplicative	$1$	$1$	$9$	$9$
$(-3a+1)$	$7$	$2$	$I_{2}^{*}$	Additive	$-1$	$2$	$8$	$2$
$(6a-5)$	$31$	$1$	$I_{1}$	Split multiplicative	$-1$	$1$	$1$	$1$

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[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3 and 9.
Its isogeny class 54684.2-a 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.