Elliptic curve 32400.1-CMb1 over number field Q(−1)\Q(\sqrt{-1})

• Label: 2.0.4.1-32400.1-CMb1
• Base field: $\Q(\sqrt{-1})$
• Conductor norm: $32400$
• CM: yes ($-4$)
• Base change: no
• Q-curve: yes
• Torsion order: $2$
• Rank: $0$

Base field $\Q(\sqrt{-1})$

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

Weierstrass equation

${y}^2={x}^{3}+\left(-18i+99\right){x}$

This is a global minimal model.

Mordell-Weil group structure

$\Z/{2}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(0 : 0 : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(144i+108)$	=	$(i+1)^{4}\cdot(-i-2)^{2}\cdot(3)^{2}$
Conductor norm:	$N(\frak{N})$	=	$32400$	=	$2^{4}\cdot5^{2}\cdot9^{2}$
Discriminant:	$\Delta$	=	$33499008i-55940544$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(33499008i-55940544)$	=	$(i+1)^{12}\cdot(-i-2)^{9}\cdot(3)^{6}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$4251528000000000$	=	$2^{12}\cdot5^{9}\cdot9^{6}$
j-invariant:	$j$	=	$1728$
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z[\sqrt{-1}]$    (complex multiplication)
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z[\sqrt{-1}]$
Sato-Tate group:	$\mathrm{ST}(E)$	=	$\mathrm{U}(1)$

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:	$\Omega(E/K)$	≈	$1.37077343115282462308804099624222274768$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$4$  =  $1\cdot2\cdot2$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$0.68538671557641231154402049812111137384$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 0.685386716 \approx L(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 1.370773 \cdot 1 \cdot 4 } { {2^2 \cdot 2.000000} } \approx 0.685386716$

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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(i+1)$	$2$	$1$	$II^{*}$	Additive	$1$	$4$	$12$	$0$
$(-i-2)$	$5$	$2$	$III^{*}$	Additive	$-1$	$2$	$9$	$0$
$(3)$	$9$	$2$	$I_0^{*}$	Additive	$1$	$2$	$6$	$0$

Galois Representations

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

prime	Image of Galois Representation
$5$	5Cs.4.1

For all other primes $p$, the image is a Borel subgroup if $p=2$, a split Cartan subgroup if $\left(\frac{ -1 }{p}\right)=+1$ or a nonsplit Cartan subgroup if $\left(\frac{ -1 }{p}\right)=-1$.

Isogenies and isogeny class

This curve has no rational isogenies other than endomorphisms. Its isogeny class 32400.1-CMb consists of this curve only.

Base change

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

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