Tokens have contract addresses, see https://support.metamask.io/kb/article/24-what-is-a-token-contract-address
But is there some "base" contract that manages the sending of plain vanilla ether?
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No. Ether is the basic currency, it managed by whole Ethereum network.
Almost(see comments) any interaction with contract is actually sending of ether(maybe 0) plus some data.
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Thank you. So does this mean if I start my own test network blockchain then the origin block will not have any code in it? – William Entriken Dec 23 '17 at 16:42
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Yes. Genesis block only contains some configuration data and balances of wallets. – Andrey Putilin Dec 23 '17 at 16:50
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1@AndreyPutilin A small edit to consider for technicality: to interact with
constantmethods (and nothing else) in contracts does not require a transaction and thus does not involve sending any ether (not even a 0 ether transaction). So "Any interaction with a contract that modifies the blockchain is actually ..." – lungj Dec 24 '17 at 17:13
