Local data of an elliptic curve (reviewed)

The local data of an elliptic curve \(E\) defined over a number field \(K\) at a prime \(\mathfrak{p}\) of \(K\) consists of

• the Tamagawa number \(c_{\mathfrak{p}}\)
• the Kodaira symbol
• the reduction type
• the local root number
• the conductor valuation \(\text{ord}_{\mathfrak{p}}(\mathfrak{N})\)
• the discriminant_valuation \(\text{ord}_{\mathfrak{p}}(\mathfrak{D})\)
• the j-invariant denominator valuation \(\text{ord}_{\mathfrak{p}}(j)_{-}\)