You want a link to let u look at the alpha here is one http://paragonwiki.com/wiki/Alpha_Slot_Abilities
There are two ways I tend to look at the Alpha slot, it is either there to strengthen your role, or diversify it. Both ways have their merits. Strengthening your role leads me to think of the design intention behind an Archetype and take an alpha that lets that AT really shine in its role. The other way, to diversify your role, means taking an alpha that lets you spread yourself out and do things that your AT may not be as strong at and make it better at its weak or middling attributes. Agility, Cardiac, and Resilient alphas all help the Tanker strengthen its role. Musculature, Nerve, Spiritual, and Vigor diversify the Tanker's role, and Intuition doesn't add anything.
You were saying that you were having Endurance issues, so I would say take Cardiac alpha before considering others. Cardiac will also help with getting you at or above resistance cap. Musculature is a great alpha for builds that don't have endurance issues, and either deal great damage, or need to boost damage to give them a hand.
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Another vote for Cardiac here. If you're having endurance woes on an Invuln/TW tank, Agility's endmod really won't help you all that much. Cardiac, especially at the higher tiers, should solve your problems there nicely.
As CaptBell mentioned above, there are two general routes to take when choosing an Alpha. For my Tankers, I generally opt to strengthen my tanking role. Also keep in mind that (if you're willing to put in the time) you can craft multiple Alphas and switch between them (although there is a cooldown involved there, it's not very long). So you could have one Alpha for when you're playing main tank to a team or league and another Alpha for soloing or off-tanking. That's something to consider.
If you're not already doing so, I'd strongly recommend downloading Mids Hero Designer since that not only lets you design your IO build very well, but lets you look at the effects various Alphas will have on your build.
Personally, I would go for a Cardiac alpha along the core path as a primary alpha, and possibly a Musculature alpha for a second alpha for soloing purposes if you have the inclination.
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Cardiac is ideal for END issues. I have Cardiac on my stone tank and it serves a dual purpose. When he's NOT using Granite, his massive recharge causes END issues and Cardiac solves that. When he IS using Granite, the extra resistance lets him hit the cap for all damage types (except psi).
On my invuln, he was already built for +regen and didn't have any END issues, so I went with Spiritual. It buffs Dull Pain and Aid Self significantly, plus gives him some extra regen from Health and Physical Perfection. I may end up revisiting him, though, since this was back when we only had 4 Alphas to choose from.
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I have T4 Core Cardiac on my Inv/SS and even with the Rage Crash there is no issue with Endurance. Cardiac seems tailormade for Inv. I would never consider Musculature for a Tanker (maybe FA), our damage is much too low to start with. You get FAR more milage out of Assault Hybrid for damage.. dear Pancake do you....
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Im recently back to the game after a very long break and could use a bit of advice. I finally decided to roll a tank and stick with it till 50. I loved the TW set and decided to go with that and chose invuln after reading a lot of posts that basically stated that it is a pretty solid set and really can't go wrong with it. I am currently at 39 and like it a lot, a lil bit of end problems, but a final build with io sets should help fix that. My question is, what incarnates should i really be going for. I run with a guy in my SG a lot that is a inv/stj tank and i really think he believes he knows a lot more then he really does. He told me he was working on a his build and having trouble with end and when i asked what alpha he was using he told me that he runs musculature. Now i have a stalker at 50 and i run musc. for the damage increase, which works great for me..i am a sty/ea and end isn't a problem whatsoever so the choice was pretty easy. But I've read forums and it seems that tanks are going for agility or cardiac...is this true? it would seem to me that those would be far more beneficial to a tank then must, although i really don't know, but one gives an end reduction the other an end mod, and one has dmg res. bonus the other def. res bonus. To me those just sound like they are tailor made for tanking. Now I'm sure there are certain defensive sets that agility works better for and cardiac work better for. Would really appreciate if someone was willing to explain a bit for me or even put a link down that explains it all. I honestly don't think musculature would be the best choice for a tank, so id rather ask people that know the right answer and could point me in the right direction. Thank you all in advance