Elliptic curve 256.1-CMb1 over number field Q(−11)\Q(\sqrt{-11})

• Label: 2.0.11.1-256.1-CMb1
• Base field: $\Q(\sqrt{-11})$
• Conductor norm: $256$
• CM: yes ($-11$)
• Base change: no
• Q-curve: yes
• Torsion order: $1$
• Rank: $0$

Base field $\Q(\sqrt{-11})$

Generator $a$, with minimal polynomial $x^{2} - x + 3$; class number $1$.

Weierstrass equation

${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a+10\right){x}+12a-1$

This is a global minimal model.

Mordell-Weil group structure

trivial

Invariants

Conductor:	$\frak{N}$	=	$(16)$	=	$(2)^{4}$
Conductor norm:	$N(\frak{N})$	=	$256$	=	$4^{4}$
Discriminant:	$\Delta$	=	$4096$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(4096)$	=	$(2)^{12}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$16777216$	=	$4^{12}$
j-invariant:	$j$	=	$-32768$
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z[(1+\sqrt{-11})/2]$    (complex multiplication)
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z[(1+\sqrt{-11})/2]$
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:	$\Omega(E/K)$	≈	$5.7658126385000336185511624345557420942$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$1$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$1.7384579212176080778970975794514159230$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 1.738457921 \approx L(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 5.765813 \cdot 1 \cdot 1 } { {1^2 \cdot 3.316625} } \approx 1.738457921$

Local data at primes of bad reduction

This elliptic curve is not semistable. There is only one prime $\frak{p}$ of bad reduction.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(2)$	$4$	$1$	$II^{*}$	Additive	$-1$	$4$	$12$	$0$

Galois Representations

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

The image is a Borel subgroup if $p=11$, a split Cartan subgroup if $\left(\frac{ -11 }{p}\right)=+1$ or a nonsplit Cartan subgroup if $\left(\frac{ -11 }{p}\right)=-1$.

Isogenies and isogeny class

This curve has no rational isogenies other than endomorphisms. Its isogeny class 256.1-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.