Establishing an Order: What School Child Should Go to

My question involves child support in the State of: CA

My ex-wife and I are divorced and have a daughter. My ex makes twice what I do and pays court ordered child support to me. We're currently battling over what school our daughter should attend for her intermediate education. There is a great (newer) public HS that has won awards in our neighborhood. My ex refuses to send our daughter there. She wants to send our daughter to a private school, regardless of the fact that it is in bad part of town and does not have as good of an academic reputation. She just wants to send our daughter there so she can tell her friends "her" daughter goes to a private school.

The school costs 18k a year. I cannot afford that; morever, I want her to go to the better public school. It's close by and I feel she would get a better education. My ex makes 6 figures but her finances are in complete disarray, she often cannot pay her rent or utilities and has begged me to loan her money. I know how this scenario will play out, a few weeks/months into our daughter going to the private schol the ex will approach me to pay a portion of the tuition/costs/fees b/c she can't and then she'll expect me to drive our daughter to this school on the other side of town. Eventually she won't be able to pay and my daughter will end up having to switch to a public schhol and the "good" one might be full by then. Changing schools in HS would be very disruptive to her.

Talking to my ex is fruitless, she's like a tantrum throwing child.

My questions are these:

1) Can the court order me to send my daughter to a private school against my wishes just because my ex wants me to send her there?
2) Can the private school enroll her if they do not have the permission of both parents?

Thank you.