
Samsung Galaxy Core
The latest firmware for Samsung Galaxy Core (GT-I8260E) is I8260EUHAMK3 with Jelly(Android Bean), released on December 31, 2013 for 1 region (CSC). Every official firmware build for GT-I8260E, newest first: one row per build with all regions (CSC) that received it.
Each firmware build has its own page at https://samfirm.shop/build/{PDA}, where {PDA} is the build number from the list above. Example: https://samfirm.shop/build/I8260EUHAMK3

GT-I9308

GT-I9268

GT-I9228

GT-I9210

GT-I9108

GT-I9103

GT-I9100T

GT-I9050
You can find the firmware on the model: GT-I8260E page.
Yes, all firmware files for GT-I8260E are 100% official, original, and obtained directly from Samsung servers.
To flash GT-I8260E, you need to download the appropriate firmware, put your device into Download Mode, and use the Odin tool to load the BL, AP, CP, and CSC files.
Downgrading is only possible if the bootloader version (binary bit) of the old firmware matches the current one. Always check the binary version before flashing GT-I8260E.