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All complex cubic fields with discriminant greater than −500 have class number one, except the fields with discriminants −283, −331 and −491 which have class number 2. The real root of the polynomial for −23 is the reciprocal of the plastic ratio (negated), while that for −31 is the reciprocal of the supergolden ratio .
Equivalently, it is the number of points in a body-centered cubic pattern within a cube that has n + 1 points along each of its edges. The first few centered cube numbers are 1 , 9 , 35 , 91 , 189 , 341, 559, 855, 1241, 1729 , 2331, 3059, 3925, 4941, 6119, 7471, 9009, ...
Besides five cubic groups, there are two more non-crystallographic icosahedral groups (I and I h in Schoenflies notation) and two limit groups (K and K h in Schoenflies notation). The Hermann–Mauguin symbols were not designed for non-crystallographic groups, so their symbols are rather nominal and based on similarity to symbols of the ...
There are 230 space groups in three dimensions, given by a number index, and a full name in Hermann–Mauguin notation, and a short name (international short symbol). The long names are given with spaces for readability. The groups each have a point group of the unit cell.
If, on the other hand, f has a non-real root, then K is called a complex cubic field. A cubic field K is called a cyclic cubic field if it contains all three roots of its generating polynomial f. Equivalently, K is a cyclic cubic field if it is a Galois extension of Q, in which case its Galois group over Q is cyclic of order three.
For each non-linear group, the tables give the most standard notation of the finite group isomorphic to the point group, followed by the order of the group (number of invariant symmetry operations). The finite group notation used is: Z n : cyclic group of order n , D n : dihedral group isomorphic to the symmetry group of an n –sided regular ...
Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x 3 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of the main theorem, which states that if p and q are primary numbers in the ring of Eisenstein integers, both coprime to 3, the congruence x 3 ≡ p (mod q) is solvable if and only if ...
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.