# Euclidean minima in totally definite quaternion fields

In this table, given a number field $K$ (defined by a polynomial), $F$ is the quaternion field defined by $$F = \left( \dfrac{a,b}{K}\right).$$

$\Lambda$ is an order of $F$, it is described by a quadruple $(q_1,q_2,q_3,q_4)\in F^4$ such that $\Lambda = \oplus_{i=1}^4 q_i \ \mathbb{Z}_K$, where $\mathbb{Z}_K$ is the ring of integers of $K$. In the table below, $q_i = [v_1,v_2,v_3,v_4]$ indicates that $$q_i = v_1 + v_2 i + v_3 j + v_4 k,$$ where $i,j,k \in F$ are such that $i^2 = a$, $j^2 = b$, $k=ij=-ji$.

$\mathcal{C}$ is the set of critical points in $\Phi(F)$ modulo $\mathcal R=\Phi(\Lambda)$, they are grouped by orbits under the action of $\mathbb{Z}_K^\times$. As above, they are written in the basis $\{1,i,j,k\}$.

$T$ is the cardinality of $\mathcal{C}$.

The column $\text{Max}$ indicates if $\Lambda$ is maximal ($\text{Max}=1$) or not ($\text{Max}=0$).

$M(\Lambda)$ is the Euclidean minimum of $\Lambda$.

number field $K$ $a$ $b$ order $\Lambda$ critical points $\mathcal{C}$ $T$ $\text{Max}$ $M(\Lambda)$
x^2-2 -1 -1 [1,0,0,0],[x/2,x/2,0,0],[x/2,0,x/2,0],[1/2,1/2,1/2,1/2] [[[3/4*x + 1/2, 1/4*x + 1/2, 1/4*x, 1/4*x]], [[1/2*x + 1/2, 1/2, 0, 0]], [[1/2*x + 1, 1/2, 1/2, 0]], [[1/4*x + 1, 1/4*x + 1/2, 1/4*x + 1/2, 1/4*x]], [[3/4*x + 1/2, 1/4*x, 1/4*x + 1/2, 1/4*x]], [[1/2*x + 1/2, 0, 1/2, 0]], [[3/4*x + 1, 1/4*x + 1/2, 1/4*x + 1/2, 1/4*x]], [[1/4*x + 1/2, 1/4*x + 1/2, 1/4*x, 1/4*x]], [[1/4*x + 1/2, 1/4*x, 1/4*x + 1/2, 1/4*x]]] 9 1 1/2
x^2-2 -1 -1 [1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1] [[[1/2*x, 1/2*x, 1/2*x, 1/2*x]]] 1 0 4
x^2-2 -3 -x-2 [1,0,0,0],[1/2,1/2,0,0],[0,0,1,0],[0,0,1/2,1/2] [[[1/2*x + 3/4, 1/4, 1/4*x + 1/2, 1/4*x], [1/4*x + 3/4, 1/4*x + 1/4, 3/4*x + 1/2, 1/4*x]], [[1/2*x + 3/4, 1/4, 3/4*x + 1/2, 1/4*x], [1/4*x + 3/4, 1/4*x + 1/4, 1/4*x + 1/2, 1/4*x]], [[1/2*x + 3/4, 1/4, 1/4*x + 3/4, 1/4*x + 1/4], [1/4*x + 3/4, 1/4*x + 1/4, 1/2*x + 3/4, 1/4]], [[1/2*x + 3/4, 1/4, 1/2*x + 1/4, 1/4], [1/4*x + 3/4, 1/4*x + 1/4, 3/4*x + 1/4, 1/4*x + 1/4]], [[1/2*x + 3/4, 1/4, 1/2*x + 3/4, 1/4], [1/4*x + 3/4, 1/4*x + 1/4, 1/4*x + 3/4, 1/4*x + 1/4]], [[1/2*x + 3/4, 1/4, 3/4*x + 1/4, 1/4*x + 1/4], [1/4*x + 3/4, 1/4*x + 1/4, 1/2*x + 1/4, 1/4]], [[3/4*x + 1/4, 1/4*x + 1/4, 1/4*x + 1/2, 1/4*x], [1/2*x + 1/4, 1/4, 3/4*x + 1/2, 1/4*x]], [[3/4*x + 1/4, 1/4*x + 1/4, 1/4*x + 3/4, 1/4*x + 1/4], [1/2*x + 1/4, 1/4, 1/2*x + 3/4, 1/4]], [[3/4*x + 1/2, 1/4*x, 1/4*x + 1/2, 1/4*x], [1/4*x + 1/2, 1/4*x, 3/4*x + 1/2, 1/4*x]], [[3/4*x + 1/4, 1/4*x + 1/4, 1/2*x + 1/4, 1/4], [1/2*x + 1/4, 1/4, 3/4*x + 1/4, 1/4*x + 1/4]], [[3/4*x + 1/2, 1/4*x, 3/4*x + 1/2, 1/4*x], [1/4*x + 1/2, 1/4*x, 1/4*x + 1/2, 1/4*x]], [[3/4*x + 1/4, 1/4*x + 1/4, 1/2*x + 3/4, 1/4], [1/2*x + 1/4, 1/4, 1/4*x + 3/4, 1/4*x + 1/4]], [[3/4*x + 1/4, 1/4*x + 1/4, 3/4*x + 1/4, 1/4*x + 1/4], [1/2*x + 1/4, 1/4, 1/2*x + 1/4, 1/4]], [[3/4*x + 1/2, 1/4*x, 1/4*x + 3/4, 1/4*x + 1/4], [1/4*x + 1/2, 1/4*x, 1/2*x + 3/4, 1/4]], [[3/4*x + 1/4, 1/4*x + 1/4, 3/4*x + 1/2, 1/4*x], [1/2*x + 1/4, 1/4, 1/4*x + 1/2, 1/4*x]], [[1/4*x + 1/2, 1/4*x, 3/4*x + 1/4, 1/4*x + 1/4], [3/4*x + 1/2, 1/4*x, 1/2*x + 1/4, 1/4]], [[1/4*x + 1/2, 1/4*x, 1/4*x + 3/4, 1/4*x + 1/4], [3/4*x + 1/2, 1/4*x, 1/2*x + 3/4, 1/4]], [[3/4*x + 1/2, 1/4*x, 3/4*x + 1/4, 1/4*x + 1/4], [1/4*x + 1/2, 1/4*x, 1/2*x + 1/4, 1/4]]] 36 1 41/16
x^2-2 x-2 -5 [1,0,0,0],[0,1,0,0],[(x+1)/2,x/2,1/2,0],[0,(x+1)/2,0,1/2] [[[3/4*x + 1/2, 3/4*x + 3/2, 1/4*x, 1/4*x], [3/4*x + 1/2, 1/4*x + 3/2, 1/4*x, 1/4*x]], [[1/4*x + 1/2, 3/4*x + 3/2, 1/4*x, 1/4*x], [1/4*x + 1/2, 1/4*x + 3/2, 1/4*x, 1/4*x]]] 4 1 41/8
x^2-2 3*x-5 -1 [1,0,0,0],[0,1,0,0],[(x+2)/2,0,(x+2)/2,0],[(x+1)/2,(x+1)/2,1/2,1/2] [[[1/2*x, 1/2*x, 0, 0]], [[1/2*x + 1/2, 1/2*x, 1/2*x + 1/2, 0]]] 2 1 7/2
x^2-2 x-3 -1 [1,0,0,0],[0,1,0,0],[(x+2)/2,0,(x+2)/2,0],[(x+1)/2,(x+1)/2,1/2,1/2] [[[1/2*x, 1/2*x, 0, 0]], [[1/2*x + 1/2, 1/2*x, 1/2*x + 1/2, 0]]] 2 1 7/2
x^2-2 -x-2 2*x-7 [1,0,0,0],[0,1,0,0],[1/2,x/2,1/2,0],[0,1/2,0,1/2] [[[3/4*x + 1/2, 3/4*x + 1/2, 1/4*x, 1/4*x], [1/4*x + 1/2, 3/4*x + 1/2, 1/4*x, 1/4*x]], [[3/4*x + 1/2, 1/4*x + 1/2, 1/4*x, 1/4*x], [1/4*x + 1/2, 1/4*x + 1/2, 1/4*x, 1/4*x]]] 4 1 119/16
x^2-3 -1 -1 [1,0,0,0],[0,1,0,0],[0,x/2,1/2,0],[x/2,0,0,1/2] [[[3/4*x + 5/4, 1/4*x + 3/4, 1/4*x + 1/4, 1/4*x + 1/4]], [[3/4*x + 5/4, 3/4*x + 5/4, 1/4*x + 1/4, 1/4*x + 1/4]], [[3/4*x + 5/4, 1/2*x + 1/2, 0, 1/4*x + 1/4]], [[1/2*x + 1/2, 1/4*x + 3/4, 1/4*x + 1/4, 0]], [[1/2*x + 1/2, 3/4*x + 5/4, 1/4*x + 1/4, 0]], [[1/2*x + 1/2, 1/2*x + 1/2, 0, 0]], [[1/4*x + 3/4, 1/2*x + 1/2, 0, 1/4*x + 1/4]], [[1/4*x + 3/4, 3/4*x + 5/4, 1/4*x + 1/4, 1/4*x + 1/4]], [[1/4*x + 3/4, 1/4*x + 3/4, 1/4*x + 1/4, 1/4*x + 1/4]]] 9 1 1
x^2-3 -1 -1 [1,0,0,0],[(x+1)/2,(x+1)/2,0,0],[(x+1)/2,0,(x+1)/2,0],[1/2,1/2,1/2,1/2] [[[7/4*x + 11/4, 3/4*x + 5/4, 3/4*x + 5/4, 1/4*x + 1/4]], [[5/4*x + 9/4, 3/4*x + 5/4, 3/4*x + 5/4, 1/4*x + 1/4]], [[5/4*x + 7/4, 3/4*x + 5/4, 1/4*x + 1/4, 1/4*x + 1/4]], [[5/4*x + 7/4, 1/4*x + 1/4, 3/4*x + 5/4, 1/4*x + 1/4]], [[3/2*x + 5/2, 1/2*x + 1, 1/2*x + 1, 0]], [[x + 3/2, 0, 1/2*x + 1, 0]], [[x + 3/2, 1/2*x + 1, 0, 0]], [[3/4*x + 5/4, 1/4*x + 1/4, 3/4*x + 5/4, 1/4*x + 1/4]], [[3/4*x + 5/4, 3/4*x + 5/4, 1/4*x + 1/4, 1/4*x + 1/4]]] 9 1 1
x^2-3 -1 -1 [1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1] [[[2/3*x, 1/2*x + 1/2, 1/2*x + 1/2, 1/2*x + 1/2], [1/3*x, 1/2*x + 1/2, 1/2*x + 1/2, 1/2*x + 1/2]]] 2 0 157/36
x^2-3 -x-3 -1 [1,0,0,0],[0,1,0,0],[x/2,0,1/2,0],[0,x/2,0,1/2] [[[3/4*x + 5/4, 3/4*x + 5/4, 1/4*x + 1/4, 1/4*x + 1/4]], [[3/4*x + 5/4, 1/2*x + 1/2, 1/4*x + 1/4, 0]], [[1/2*x + 1/2, 3/4*x + 5/4, 0, 1/4*x + 1/4]], [[1/2*x + 1/2, 1/4*x + 3/4, 0, 1/4*x + 1/4]], [[1/2*x + 1/2, 1/2*x + 1/2, 0, 0]], [[1/4*x + 3/4, 3/4*x + 5/4, 1/4*x + 1/4, 1/4*x + 1/4]], [[1/4*x + 3/4, 1/4*x + 3/4, 1/4*x + 1/4, 1/4*x + 1/4]], [[1/4*x + 3/4, 1/2*x + 1/2, 1/4*x + 1/4, 0]], [[3/4*x + 5/4, 1/4*x + 3/4, 1/4*x + 1/4, 1/4*x + 1/4]]] 9 1 13/4
x^2-5 -1 -1 [1,0,0,0],[0,1,0,0],[(x+1)/4,(x+3)/4,1/2,0],[1/2,1/2,1/2,1/2] [[[7/20*x + 3/4, 4/5*x + 3/2, 3/10*x + 1/2, 3/20*x + 1/4], [3/20*x + 1/4, 9/20*x + 1/4, 1/5*x, 1/10*x], [1/5*x + 1/2, 7/20*x + 5/4, 1/10*x + 1/2, 1/20*x + 1/4]], [[9/20*x + 3/4, 9/20*x + 3/4, 7/20*x + 1/4, 3/20*x + 1/4], [11/20*x + 3/4, 3/10*x + 1/2, 3/20*x + 1/4, 1/10*x], [13/20*x + 3/4, 2/5*x + 1, 1/5*x + 1/2, 1/20*x + 1/4]], [[2/5*x, 3/5*x, 1/5*x, 1/5*x], [7/20*x + 3/4, 3/20*x + 3/4, 1/20*x + 1/4, 1/20*x + 1/4], [11/20*x + 1/4, 9/20*x + 3/4, 3/20*x + 1/4, 3/20*x + 1/4]], [[3/10*x + 1/2, 13/20*x + 5/4, 1/5*x + 1/2, 3/20*x + 1/4], [1/5*x + 1/2, 3/5*x + 1/2, 3/10*x, 1/10*x], [1/10*x + 1/2, 3/10*x + 1, 3/20*x + 1/4, 1/20*x + 1/4]], [[3/10*x + 1/2, 7/20*x + 3/4, 1/5*x + 1/2, 3/20*x + 1/4], [1/5*x + 1/2, 2/5*x + 1/2, 3/10*x, 1/10*x], [1/10*x + 1/2, 1/5*x + 1/2, 3/20*x + 1/4, 1/20*x + 1/4]], [[1/5*x + 1/2, 13/20*x + 3/4, 1/10*x + 1/2, 1/20*x + 1/4], [3/10*x + 1/2, 3/5*x + 1, 2/5*x, 1/5*x], [3/20*x + 1/4, 11/20*x + 3/4, 1/5*x, 1/10*x]]] 18 1 1/4
x^2-5 (x-5)/2 -2 [1,0,0,0],[(x+1)/4,1/2,0,0],[0,0,-1,0],[0,0,(x+1)/4,1/2] [[[1/4*x + 3/4, 1/20*x + 1/4, -1/4*x - 1/4, 1/10*x], [1/2*x + 1/2, 1/5*x, 1/2, 3/20*x + 1/4], [1/4*x + 1/4, 1/10*x, 0, 1/5*x]], [[1/4*x + 1/4, 1/10*x, -1/4*x + 1/4, 1/20*x + 1/4], [1/2*x + 1, 3/20*x + 1/4, 0, 1/5*x], [1/2*x + 1/2, 1/5*x, -1/4*x - 1/4, 1/10*x]]] 6 1 11/5
x^2-5 -1 x-7 [1,0,0,0],[0,1,0,0],[(x+3)/4,(x+3)/4,1/2,0],[(x+3)/4,(x+3)/4,0,1/2] [[[1/2*x + 1, 1/4*x + 5/4, 1/4, 1/4], [3/4*x + 3/4, 3/4*x + 5/4, 1/8*x + 1/8, 1/8*x + 1/8], [1/4*x + 3/4, 1/2*x, 1/8*x - 1/8, 1/8*x - 1/8]], [[1/4*x + 5/4, 1/2*x + 1, 1/4, 1/4], [3/4*x + 5/4, 3/4*x + 3/4, 1/8*x + 1/8, 1/8*x + 1/8], [1/2*x, 1/4*x + 3/4, 1/8*x - 1/8, 1/8*x - 1/8]]] 6 1 9/4
x^2-5 -1 2*x-8 [1,0,0,0],[0,1,0,0],[1/2,1/2,1/2,0],[1/2,1/2,0,1/2] [[[1/4*x + 3/4, 1/4*x + 1/4, 1/4, 1/4], [1/2*x, 1/4*x + 3/4, 1/8*x + 1/8, 1/8*x + 1/8], [1/4*x + 1/4, 1/2*x, 1/8*x - 1/8, 1/8*x - 1/8]], [[1/4*x + 1/4, 1/4*x + 3/4, 1/4, 1/4], [1/4*x + 3/4, 1/2*x, 1/8*x + 1/8, 1/8*x + 1/8], [1/2*x, 1/4*x + 1/4, 1/8*x - 1/8, 1/8*x - 1/8]]] 6 1 9/4
x^2-13 -1 -1 [1,0,0,0],[0,1,0,0],[(x+3)/4,(x+5)/4,1/2,0],[1/2,1/2,1/2,1/2] [[[1/4*x, 3/4, 1/4, 1/4], [3/8*x + 1/8, 3/8*x - 3/8, 1/8*x - 1/8, 1/8*x - 1/8], [1/8*x + 5/8, 3/8*x + 3/8, 1/8*x + 1/8, 1/8*x + 1/8]], [[3/8*x + 1/8, 1/8*x - 1/8, 1/8*x - 1/8, 1/8*x - 1/8], [1/8*x + 5/8, 1/8*x + 1/8, 1/8*x + 1/8, 1/8*x + 1/8], [1/4*x, 1/4, 1/4, 1/4]], [[3/8*x + 3/8, 1/8*x + 5/8, 1/8*x + 1/8, 1/8*x + 1/8], [3/4, 1/4*x, 1/4, 1/4], [3/8*x - 3/8, 3/8*x + 1/8, 1/8*x - 1/8, 1/8*x - 1/8]], [[3/8*x + 3/8, 3/8*x - 1/8, 1/8*x + 1/8, 1/8*x + 1/8], [3/4, 1/4*x + 1/2, 1/4, 1/4], [3/8*x - 3/8, 1/8*x + 3/8, 1/8*x - 1/8, 1/8*x - 1/8]], [[1/4*x + 1/2, 1/4, 1/4, 1/4], [1/8*x + 3/8, 1/8*x - 1/8, 1/8*x - 1/8, 1/8*x - 1/8], [3/8*x - 1/8, 1/8*x + 1/8, 1/8*x + 1/8, 1/8*x + 1/8]], [[1/8*x + 3/8, 3/8*x - 3/8, 1/8*x - 1/8, 1/8*x - 1/8], [3/8*x - 1/8, 3/8*x + 3/8, 1/8*x + 1/8, 1/8*x + 1/8], [1/4*x + 1/2, 3/4, 1/4, 1/4]], [[1/8*x + 1/8, 1/8*x + 5/8, 1/8*x + 1/8, 1/8*x + 1/8], [1/4, 1/4*x, 1/4, 1/4], [1/8*x - 1/8, 3/8*x + 1/8, 1/8*x - 1/8, 1/8*x - 1/8]], [[1/8*x + 1/8, 3/8*x - 1/8, 1/8*x + 1/8, 1/8*x + 1/8], [1/4, 1/4*x + 1/2, 1/4, 1/4], [1/8*x - 1/8, 1/8*x + 3/8, 1/8*x - 1/8, 1/8*x - 1/8]]] 24 1 3/4