10. For the curve $C: x^{4}+y^{2}=1$, consider the following statements:
(1) The curve $C$ is symmetric with respect to the origin;
(2) The curve $C$ is symmetric with respect to the line $y=x$;
(3) The area enclosed by the curve $C$ is less than $\pi$;
(4) The area enclosed by the curve $C$ is greater than $\pi$.

The true statements are $\qquad$ (write down the numbers of all true statements).
10. (1), (4).

Notice that, by substituting $-x$ for $x$ and $-y$ for $y$, the equation of curve $C$ remains unchanged, so statement (1) is correct.

By substituting $x$ for $y$ and $y$ for $x$, the equation of curve $C$ changes, so statement (2) is incorrect.

Since $1=x^{4}+y^{2}<x^{2}+y^{2}(0<|x|<1)$, the points on curve $C$ lie outside the circle $x^{2}+y^{2}=1$, thus the area enclosed by curve $C$ is greater than $\pi$. Therefore, statement (4) is correct.