I have hyperelliptic curve of genus 2 over $\mathbb Q$. The questions are:
- Is it true that I can extract Tamagawa numbers ($c_v$ in definition) from Sage function
genus2reduction (see below) output and calculate Tamagawa product (called $c$ in definition) from them? Is it ok that group of connected components is over an algebraic closure of $\mathbb F_p$?
- Is it true that I can use Magma's
RegularModel (see below) and get tamagawa numbers $c_p$ ($=c_v$) as orders of Magma's ComponentGroup for corresponding $p$? I'm confused by group of components of the Neron model of the Jacobian of C over the **completion** at P. There is no mentioned of completition in Sage function. I'm not sure that I understand all things right.
- Is it true that in second case if I want to get Tamagawa product I should just calculate product of $c_p$ for all $p$ that divide discriminant of my curve?
Sage genus2reduction function ducumentation says following:
I have hyperelliptic curve of genus 2 over $\mathbb Q$. The questions are:
- Is it true that I can extract Tamagawa numbers ($c_v$ in definition) from Sage function
genus2reduction output and calculate Tamagawa product (called $c$ in definition) from them? Is it ok that group of connected components is over an algebraic closure of $\mathbb F_p$?
- Is it true that I can use
RegularModel and get tamagawa numbers $c_p$ ($=c_v$) as orders of Magma's ComponentGroup for corresponding $p$? I'm confused by group of components of the Neron model of the Jacobian of C over the **completion** at P. There is no mentioned of completition in Sage function. I'm not sure that I understand all things right.
- Is it true that in second case if I want to get Tamagawa product I should just calculate product of $c_p$ for all $p$ that divide discriminant of my curve?
