Define $$K=\int_0^1 \frac{dx}{\sqrt{1-x^4}}$$ and the Jacobi elliptic function $\operatorname{cd}$ with modulus $i$ by $$\int_{\operatorname{cd}z}^1\dfrac{dx}{\sqrt{1-x^4}}=z$$ on $z\in [0,2K]$ and by $$\operatorname{cd}(z+2K)=-\operatorname{cd}z$$ on the rest of the real line.
Then this function is $4K$-periodic and is an analog of the cosine function which is $2\pi$-periodic. Let $\operatorname{dmp}$ denote the ''degree of the minimal polynomial'' of an algebraic number over $\mathbb{Q}$. Then it is well-known that $$\operatorname{dmp}\cos\frac{2\pi}{n}\ge \dfrac{\sqrt{2n}}{4}$$ for all $n\in\mathbb{N}^\times$. (See also https://en.wikipedia.org/wiki/Minimal_polynomial_of_2cos(2pi/n)).
This led me to the following conjecture:
$$\operatorname{dmp}\operatorname{cd}\frac{4K}{n}\ge \frac{\sqrt{2n}}{4}$$
for all $n\in\mathbb{N}^\times$. Is this conjecture true or false? If this conjecture is true, then it has an application in an algorithm for evaluating elliptic integrals which I'm developing. This should provide an alternative to a known algorithm that is using elliptic curves. If the conjecture is false, can we come up with a simple non-trivial lower bound for $\operatorname{dmp}\operatorname{cd}\frac{4K}{n}$?
Example: $$\operatorname{dmp}\operatorname{cd}\frac{4K}{5}=8$$ with the corresponding minimal polynomial $$X^8-4X^7+4X^5-6X^4+4X^3-4X+1,$$ whereas $$\frac{\sqrt{2\cdot 5}}{4}\approx 0.79$$ to two decimal places.
Some thoughts...
We have $cdz=sn(K-z).$ Using the addition formula of the Jacobi's elliptic function $sn$ and some $+K$ tricks I obtained the half-angle formula $$cd(2z)=\frac{cd^4z+2cd^2z-1}{-cd^4z+2cd^2z+1}.$$ From this, I roughly observed that $$\text{dmp}\, cd\left(\frac{4K}{2^n}\right)\stackrel{?}{=}2^{2n-4}$$ for $n\geq2.$ Whereas, in trigonometric case, we have $\text{dmp}\cos\left(\frac{2\pi}{2^n}\right)=2^{n-2}.$ Therefore I would conjecture that $$\text{dmp}\, cd\left(\frac{4K}{n}\right)\geq=\left(\frac{\sqrt{2n}}{4}\right)^2=\frac n8.$$