This is my first guide and I cant say I really know how to write one but I would like to share the Anivia build I enjoy playing and while you may not want to play this build there is plenty of information that can convert to the classic ability power Anivia. Enjoy.
Why this build?
I started with Anivia like so many others as a glass cannon so I can 100 - 0 in one spell and those times were fun but, a thought crossed my mind of what if my egg could be used strategically rather then a last line of defense and this variation came to be.

Pros:

50% attack speed slow 35% of it aoe

Insane durability

Passive revive

Great base damage

Instant wave clear

Safe mid lane with little counters


Cons:

Mana management

Slow skill shot stun

Low Mobility

Auto attacks take a while to get used to

Quints:
Movement Speed - these help with your bad base movespeed and allow for easier harassing, ganking, and escapes.
Magic Pen - these would raise your burst potential when combined with the other magic pen you can buy.

Marks:
Hybrid Penetration - these serve to help you auto attacks do a little more damage to help with farming as well as your spells.

Seals:
Flat Armor - makes lane minions deal less to you when you harass as well as help deal with the pesky ad mid laners.
Flat HP - good choice if you want specialized pages for ad and ap lane opponents.

Glyphs:
Scaling MR - Ap champs become their strongest from 6 on at which point these almost out-scale their flat MR counter parts
Scaling AP - for that little extra damage when you don't need MR

I currently don't know how to add more mastery pages but the standard 21/0/9 works well too but lets talk about this 6/21/3

Utility
Fleet of Foot: like the Quints these help supplement your low base speeds allowing for you to avoid some harder skill shots, harass, gank ect

Offensive
Everything in the offence tree is taken to help with Anivia's hard early game farming.
Fury: I find this brings Anivias attack animation better in line with other attack animations so your not constantly missing minions by milliseconds.
Butcher: Gives you low base AD the little extra kick to finish that minion.
Feast: honestly this was just an extra point I had but the life and mana gains are a nice bonus

Defensive
Recovery: a little extra hp5
Block & Unyielding: helps with the early trades
Veteran Scars & Juggernaut: extra health to progress deeper into the skill tree
Oppression: With Anivia's kit centered around crowd control this is always active on your target/s
Hardiness & Resistance: these and their upgrades interchange with each other as needed for your lane phase
Perseverance: keeps you healthy as you get more and more HP from your items
Legendary Guardian: With anivia being such a close range mage this helps drastically to get you tons of defenses
Tenacious: Crowd control is your enemy as anivia reducing the druiation so u can keep you ultimate up is a must

Rebirth STATIC COOLDOWN: 240


Upon taking lethal damage, Anivia will transform into an egg with full health. While in egg form, Anivia is unable to make any actions and her armor and magic resistance are modified by −40 / −25 / −10 / +5 / +20. If her egg survives for 6 seconds, Anivia is revived with the same percentage health that her egg had.

Combine this with a jungle gank to bait your opponent to engage into a bad fight. Also makes for a good low level fight that involves death trading. Late game it doubles your effective life for teamfights either soaking excess damage or to clean up after rebirth.


Flash Frost RANGE: 1075 COST: 80 / 90 / 100 / 110 / 120 mana COOLDOWN: 12 / 11 / 10 / 9 / 8


First Cast: Anivia launches an orb of ice in the targeted direction, dealing magic damage to any enemy it passes through. At maximum range, Flash Frost will detonate automatically.

Second Cast: Anivia detonates the orb, dealing magic damage and Stun icon stunning enemies in a 75-radius area for 1 second. Enemies can be damaged by both parts of Flash Frost.

Magic damage: 60 / 90 / 120 / 150 / 180 (+ 50% AP)
Max. Damage: 120 / 180 / 240 / 300 / 360 (+ 100% AP)
Missile Speed: 850
Enemies damaged by Flash Frost are Chilled for 3 seconds, Slow icon slowing their movement speed by 20%.

Before level 6 this is the only way to trigger your critical hits of your Frostbite. Get one point at level one and by 7 decide if you want to try to snowball your lane maxing it second or gear towards team fighting and max it last. The latter will help keep the mana costs lower for a already terrible mana sustain lane. Flash frost is a good spell to save in lane to escape ganks or follow up on your own ganks.


Crystallize RANGE: 1000 COST: 70 mana COOLDOWN: 25


Active: Anivia creates an impassable wall of ice at the targeted location for 5 seconds, perpendicular to which way she's facing. The wall grants sight in a large area around it.
Wall length: 400 / 500 / 600 / 700 / 800

Anivia's walls separate the good players from the bad players, effective use of this spell makes a hard 5v5 into an easy 2v5 keeping the carrys away from their tanks allowing you to safely kill the party in front or make them waste precious summoner spells.


Frostbite RANGE: 650 COST: 50 / 60 / 70 / 80 / 90 mana COOLDOWN: 5


Active: Anivia sends a shard of ice at the targeted enemy, dealing magic damage.
Magic damage: 55 / 85 / 115 / 145 / 175 (+ 50% AP)
Frostbite deals 200% damage to Chilled targets.
Chilled Damage: 110 / 170 / 230 / 290 / 350 (+ 100% AP)

This is you main burst damage throughout the game always max this first. when trading blows with an enemy try sticking with the E+R combo using R as the missile is in flight. Q+E is a good follow up once they take and are slowed by the E+R.


Glacial Storm RANGE: 625 COST: 75 mana + 40 / 50 / 60 per second COOLDOWN: 6


Active: Anivia creates a blizzard at the targeted location. Enemies within the blizzard take magic damage per second, and are Chilled for 1 second, Slow icon slowing their movement speed and attack speed by 20%.
Magic damage per second: 80 / 120 / 160 (+ 25% AP)

Anivia can re-activate Glacial Storm at any time to disable its effects, and it will deactivate automatically if she gets too far away or runs out of mana. Additionally, Glacial Storm will be deactivated by all forms of interrupting crowd control, excluding fear, taunt and Charm icon charming

This ultimate serves several tasks:
1. wave clear-
when your team is forced to defend there is no one better then anivia at defending turrets with her large low CD ranged aoe

2. Zone control-
Deciding placement in a team fight is a crucial skill for an anivia player to learn a well placed ultimate will separate the back lane damage from their front line and keeps the front line slowed in movement speed and attack speed.

3. Tons of Damage-
That's right TONS OF DAMAGE even with our meager looking ability power total our ultimate will still tick for 200+ damage and apply the lindrys debuff for us quickly melting whatever stands in it for to long.

Core Items

Rod of Ages: this item has no wasted stats increasing our small mana pool giving a lot of health and some decent ability power and if all that wasn't enough it also gives extra sustain for every level we gain making it a strong first buy.

Liandrys Torment: with all of anivias abilitys focusing on crowd control there is never a time you cant use this item to its full potential.

Merc Treds VS Ninja Tabi VS Sorc Shoes
Each of these boots are great for different reasons and there is no wrong choice when deciding it just comes down to:
1. do they have the ability to lock me down?? if so get merc treds a bird under cc cant defend her team zone the enemy or fight back
2. are you getting bullied by an ad champion mid lane?? if so ninja tabis will reduce the damage you take conserderably
3. if u want to snowball your lane or make some devastating ganks Sorc shoes are the way to go. with them most mid lanes and all adc champions will end up with 0 MR when you combine the penetration of the Haunting Guise, Sorc Shoes, and the penetration runes you use.

Frozen Heart: this item has great synergy with anivias kit making her 20% aoe attack speed slow on her ultimate add up to 35% along with an increased mana pool and some CDR do make our short CDs even shorter.

Omen: this when combined with a frozen heart and you ultimate will make all ad champs really have to decide weather or not to commit on you or try and get past your defenses to get to the prize that is your adc.

Abyssal Scepter: with anivia scaling so well on magic pen and her close quarters play style this item is a great choice when you need that little extra magic resist.

Other MR items

Banshees Veil: great for facing the nasty ap burst champs like Annie and LeBlanc and also good for countering high cc initiation such as Amumu Sejuani Leona and Malphite

Athene's Unholy Grail: this item can make your lane phase trivial and sustain for the longer team fights.

If MR is unneeded get a void staff, the void staff is the most effective item at boosting anivias damage late game. with Sorc shoes Void Staff and Liandrys Torment any champion still at 30 MR ends up with a whopping -16 MR actually amplifying your damage on all spells and for the targets with 150 after calculations they are at about 62 magic resist.

Items not to get

Spirit Visage: with no innate heals and no spell vamp in our items this item is very wasted on anivia

Zonyahs Hourglass: while this is normally a staple mage item it defeats the purpose of this build that actually wants you to get attacked. while good on a glass cannon anivia this item just does not fit in our build

Rabadons Deathcap: magic pennetration generally gives anivia more damage then a lot of ability power and with our kit core only netting around 130 ap its just not that beneficial compared to abyssal scepter and void staff.

Levels 1-2 Melee
Highly recommended to take a Dorans Shield then we can freely auto attack between farm sending our Q for a Q+E combo when they go for last hits. During this time is when your put yourself forward as the dominant champion of the lane.

Levels 3-5 Melee
Right when we hit level 3 stop harassing with the Q+E but still harass with autos and wait for them to decide they can take you on at which point we instantly stun them reaffirming our status as the dominant one. Only use your Q if they waste a spell on minions farming. The more your Q is off CD the harder it is to gank you and you are less likely to need flash if you are.

Levels 1-5 Ranged
Depending on how comfortable with the lane choose Dorans Ring or Dorans Shield. Ranged are generally a safer lane then the melee with their lack of all in abilitys and multiple skill shots that aren't that hard to dodge with all our extra % movespeed. During levels 1-5 use your combo when you dodge one of their spells or they are obviously walking in for a minion.

Level 6+
Now is the time Anivia shines, at this point we are hoping we have a kind jungle that has given us a blue buff to harass and farm with. After level 6 we never want to open with our stun while harassing, instead the swap to our E+R combo so we never miss an E crit. once the mid lane is at about half health we will go in for our E+R combo and while they are slowed by the ultimate send your Q out forcing a flash or Netting ourselves a kill with our second E and ignite.

Jungle Ganks Our Team
when our jungle comes in for a gank send out a wall for the mid lane to path around for a very predictable Stun from our Q. Alternatively if we have a jungle with very strong assassination qualities such as lee sin or Kha'zix another good strategy it to bait out the fight with your egg putting the enemy in an all in situation they cant escape.

Jungle Ganks Their Team
Send out your wall and calmly walk towards your turret or if possible quickly finish the enemy mid lane at which point we get to safety flashing only when needed and if entering egg form try to separate yourself from them as you die with a wall to buy a little extra time to complete your escape.

Ganking Bottom and Top
When gankning lanes Start by using our E+R combo placing your ultimate in the path of their escape and follow up with you Q+E, When they enter a bush use your walls to grant you vision and possibly pin them to the wall scoring an easy kill the longer your walls the more likely this will occur.

Mid Game
In the mid game is when our build shines dealing devastating damage to the opposing teams back line. At this point we want to have both our core items and be working on our frozen heart starting with a wardens mail for the 35% slow so the consistent damage from an ad wont kill you. Our positioning should be in the front lines nuking the high priority targets using your walls to either lock down the enemy carries or separate the enemy front line from our ad and support.

Late Game
At this point we should be virtually unkillable with our egg up doubling our effective health. Our positioning and this point is between our front line and back line so we can respond to kill the squishy backline or defend our squishy backline as needed if all goes as planned they will focus you while the adc is safe shredding away their health

While you may decide this build is not for you, everything stated in this guide about laning and abilities translates over to the ability stacked version. This is a different spin on an old champion and I hope you enjoy this version.

If I missed anything or need to clarify anything about this build or even its ap counterpart ill reply to the best of my ability.

Thanks for Reading