Диоптрии и астигматизм указаны в рецепте на очки.

Because the subject is compound: диоптрии и астигматизм = the diopters and the astigmatism.

In Russian, when two things are joined by и (and), the verb or short-form passive participle normally goes in the plural:

• Диоптрии и астигматизм указаны... = The diopters and the astigmatism are indicated...

If there were only one subject, it would be singular:

• Астигматизм указан...
• Диоптрии указаны... — still plural here, because диоптрии itself is plural in this sentence.

Указаны is the short-form passive participle, plural, from указать (to indicate / specify / state).

So the sentence literally has the sense of:

• Diopters and astigmatism are specified in the glasses prescription.

This is a very common formal written style in Russian, especially in documents, instructions, and medical or technical contexts.

Compare:

• указаны = are indicated / are specified
• написаны = are written
• отмечены = are marked
• перечислены = are listed

Because in the context of eyewear prescriptions, диоптрии is commonly treated as a plural quantity or value.

A glasses prescription usually includes lens values for one or both eyes, so Russian often uses the plural noun:

• диоптрии = diopters

Even when English might say the diopter value or just the prescription strength, Russian often naturally uses the plural form.

You may also see:

• оптическая сила указана в диоптриях = the optical power is given in diopters

So here диоптрии is simply the normal plural noun, not a special grammatical trick.

They are in the nominative case, because they form the subject of the sentence.

You can test that by asking what is indicated?

• Что указано / указаны?
• Диоптрии и астигматизм.

So:

• диоптрии — nominative plural
• астигматизм — nominative singular

Together they create a compound subject, which is why указаны is plural.

Because Russian uses в with many documents to mean in a document / in a form / in a record.

So:

• в рецепте = in the prescription
• в паспорте = in the passport
• в договоре = in the contract
• в анкете = in the form/questionnaire

Using на рецепте would sound more like on the physical surface of the prescription, which is not the usual meaning here.

So в рецепте means as part of the information contained in the prescription.

Рецепт на очки is a standard Russian expression meaning a prescription for glasses.

This pattern is common with рецепт на + accusative when talking about what the prescription is for:

• рецепт на очки = prescription for glasses
• рецепт на линзы = prescription for contact lenses
• рецепт на лекарство = prescription for medicine

About the alternatives:

• рецепт очков is not the normal way to say this
• рецепт для очков sounds unnatural in this context

So the idiomatic phrase to learn is:

• рецепт на очки

No. Although the words are related historically, in modern Russian рецепт usually means:

• prescription in a medical/pharmacy sense

Depending on context, it can also mean recipe, but that is a different usage.

Here, because of на очки, it clearly means:

• prescription for glasses

Examples:

• врач выписал рецепт = the doctor wrote a prescription
• рецепт на очки = glasses prescription
• рецепт торта = cake recipe

So learners should be careful not to assume рецепт always means recipe.

Because и simply connects two nouns in the same syntactic role. Each noun stays in the case required by the sentence.

Here both nouns are subjects, so both stay in the nominative:

• диоптрии — nominative plural
• астигматизм — nominative singular

Russian does not force both nouns to look grammatically alike in number. One can be plural and the other singular, and together they still form a plural subject:

• Мама и дети пришли.
• Цена и качество указаны.
• Диоптрии и астигматизм указаны.

So nothing special is happening to астигматизм—it is just in the nominative singular.

Yes. Russian word order is flexible, though each version may sound a little different in emphasis.

The given sentence:

• Диоптрии и астигматизм указаны в рецепте на очки.

This is neutral and natural.

Other possible orders:

• В рецепте на очки указаны диоптрии и астигматизм.
  This puts the location first: In the glasses prescription, the diopters and astigmatism are indicated.

• Указаны в рецепте на очки диоптрии и астигматизм.
  Grammatically possible, but less neutral and more stylistically marked.

For learners, the two most useful versions are:

• Диоптрии и астигматизм указаны в рецепте на очки.
• В рецепте на очки указаны диоптрии и астигматизм.

Yes, especially in written, formal, medical, or explanatory contexts.

It sounds like something you might see:

• in a medical explanation
• on a website about eye care
• in instructions about how to read a glasses prescription
• in administrative or clinical language

In casual everyday speech, someone might say something a little more conversational, for example:

• В рецепте на очки указаны диоптрии и астигматизм.
• В рецепте написаны диоптрии и астигматизм.

So the original sentence is natural, but somewhat formal/neutral rather than chatty.

Both can work, but they are not exactly the same.

• указаны = indicated / specified / stated
• написаны = written

So:

• Диоптрии и астигматизм указаны в рецепте на очки.
  Focuses on the fact that this information is officially included or specified.

• Диоптрии и астигматизм написаны в рецепте на очки.
  Focuses more literally on the fact that they are written there.

In formal or document-related language, указаны is often the more natural choice.

In the present tense, Russian usually omits the verb to be.

So English:

• Diopters and astigmatism are indicated in the prescription

becomes Russian:

• Диоптрии и астигматизм указаны в рецепте на очки.

There is no separate present-tense word for are here. That is completely normal in Russian.

You can think of it as:

• [subject] + [short-form participle/adjective]

This pattern is extremely common:

• Дверь закрыта. = The door is closed.
• Все документы готовы. = All the documents are ready.
• Данные указаны в анкете. = The data are indicated in the form.