Class Group

The class group of a number field is the Abelian multiplicative group formed by the equivalence classes of its fractional ideals. Each fractional ideal of belongs to an equivalence class consisting of all fractional ideals satisfying for some nonzero element of . The number of equivalence classes of fractional ideals of is a finite number, known as the class number of . For , multiplication of equivalence classes of fractional ideals is defined in the obvious way, i.e., by letting .

Writing the class group as , its 2-torsion subgroup is , where 1 denotes the identity element. This is a vector space over the finite field . Its dimension

is called the 2-rank of the class group. In particular, .

Elkies and Klagsbrun (2026) announced a totally real cubic field with class-group 2-rank at least 25 and a complex cubic field with class-group 2-rank at least 26. These were reported as the largest known lower bounds for the two respective types (Elkies 2026; ICARM, curves 404 and 403). The associated elliptic curves have ranks at least 27 and 28, respectively. For each elliptic curve, the cubic field is generated by the -coordinate of a nonidentity 2-torsion point. Brumer and Kramer (1977, Prop. 7.1) bound the elliptic curve rank using the class-group 2-rank together with local reduction data. Thus these computations give the stated lower bounds for the elliptic curve ranks; they do not assert that the class-group 2-ranks are exactly 25 and 26.