Elliptic curve 32400.3-g6 over number field Q(−2)\Q(\sqrt{-2})

• Label: 2.0.8.1-32400.3-g6
• Base field: $\Q(\sqrt{-2})$
• Conductor norm: $32400$
• CM: no
• Base change: yes
• Q-curve: yes
• Torsion order: $4$
• Rank: $0$

Base field $\Q(\sqrt{-2})$

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

Weierstrass equation

${y}^2+a{x}{y}={x}^{3}-{x}^{2}-12006{x}+511434$

This is a global minimal model.

Mordell-Weil group structure

$\Z/{2}\Z \oplus \Z/{2}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(-126 : 63 a : 1\right)$	$0$	$2$
$\left(\frac{123}{2} : -\frac{123}{4} a : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(180)$	=	$(a)^{4}\cdot(-a-1)^{2}\cdot(a-1)^{2}\cdot(5)$
Conductor norm:	$N(\frak{N})$	=	$32400$	=	$2^{4}\cdot3^{2}\cdot3^{2}\cdot25$
Discriminant:	$\Delta$	=	$419904000000$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(419904000000)$	=	$(a)^{24}\cdot(-a-1)^{8}\cdot(a-1)^{8}\cdot(5)^{6}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$176319369216000000000000$	=	$2^{24}\cdot3^{8}\cdot3^{8}\cdot25^{6}$
j-invariant:	$j$	=	$\frac{4102915888729}{9000000}$
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:	$\Omega(E/K)$	≈	$0.215715023391017743943972231469247740380$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$384$  =  $2^{2}\cdot2^{2}\cdot2^{2}\cdot( 2 \cdot 3 )$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$4$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$1.8304026701232403778852285264163093811$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 1.830402670 \approx L(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 0.215715 \cdot 1 \cdot 384 } { {4^2 \cdot 2.828427} } \approx 1.830402670$

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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(a)$	$2$	$4$	$I_{16}^{*}$	Additive	$1$	$4$	$24$	$12$
$(-a-1)$	$3$	$4$	$I_{2}^{*}$	Additive	$-1$	$2$	$8$	$2$
$(a-1)$	$3$	$4$	$I_{2}^{*}$	Additive	$-1$	$2$	$8$	$2$
$(5)$	$25$	$6$	$I_{6}$	Split multiplicative	$-1$	$1$	$6$	$6$

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
$3$	3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 32400.3-g consists of curves linked by isogenies of degrees dividing 12.

Base change

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

Base field	Curve
$\Q$	720.j3
$\Q$	2880.a3