Abacus is a 6th-level spellcaster. His spellcasting ability is Intelligence.
Spell save DC 14, +6 to hit with spell attacks (+1 from Arcane Grimoire) He has has the following wizard spells prepared:
Cantrips (at will): Ray of Frost, Light, Minor Illusion, Prestidigitation
1st level (4 slots): Mage armor, Shield, Detect magic (R)
2nd level (3 slots): Icingdeath's Frost, Earthbind, Vortex Warp, Misty step
3rd level (3 slots): Erupting Earth, Slow     Abacus casts one of the following spells, using Intelligence as the spellcasting ability (spell save DC 14, +6 to hit with spell attacks): At will: Mage Hand, Light, Minor Illusion, Prestidigitation 1/day each: Disguise Self, Ice Knife, Mage Armor (included in AC), Thunderwave

Ritual caster. Abacus can cast specific spells as rituals, these spells do not expend a spell slot but the casting takes an extra 10 minutes.
Arcane Grimoire. A spellcasting focus for wizard spells, and Abacus gains a +1 bonus to spell attack rolls and to the saving throw DCs of your wizard spells. He can use this book as a spellbook.