The class group of a number field 
is the Abelian multiplicative
group formed by the equivalence classes
of its nonzero fractional
ideals. Each nonzero fractional
ideal 
of 
belongs to an equivalence
class ![[I]](/images/equations/ClassGroup/Inline4.svg)
consisting of all fractional ideals 
satisfying 
for some nonzero element 
of 
. The number of equivalence
classes of nonzero 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 ![[I][J]=[IJ]](/images/equations/ClassGroup/Inline12.svg)
.
Writing the class group as 
,
its 2-torsion subgroup
is ![Cl(K)[2]={c in Cl(K):c^2=1}](/images/equations/ClassGroup/Inline14.svg)
, where 1
denotes the identity element. This is a vector
space over the finite field 
. Its dimension
![g_2(K)=dim_(F_2)Cl(K)[2]](/images/equations/ClassGroup/NumberedEquation1.svg) |
is called the 2-rank of the class group. In particular, ![|Cl(K)[2]|=2^(g_2(K))](/images/equations/ClassGroup/Inline16.svg)
.
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 from above using the class-group 2-rank together with local reduction data.
Thus the elliptic curve rank computations and
local reduction data give the stated lower bounds
for the class-group 2-ranks; they do not assert that the class-group 2-ranks are
exactly 25 and 26.
and 
Records for Elliptic Curves 
of Ranks 27-30; Record 2-Ranks 25 (Real) and 26 (Complex)
for Class Groups of Cubic Fields." Number Theory List, Aug. 29, 2026.