This show is about a young man who has to pilot a mecha that was created by an elder family member, and this mecha is the only defense against a new mysterious threat from out of nowhere.
He is initially pressed into piloting it against his will by said family member, with zero experience, and while in its first sortie it goes berserk and destroys both the monster and a lot of property.
This mecha is nigh invincible, and is made of the most expensive and rare materials in the world. It takes an entire massive base (which is pyramid-shaped and has lots of built in defense) to support it.
If any of the monsters breaches the base and gets to what's within, the world could possibly end due to the combination of what's in the base and the nature of the monsters. The monsters seem to have the explicit goal of either breaching the base or destroying the mecha.
The name of the mecha?
No, it doesn't start with an "E", it is: