# Elimination for coefficients of special characteristic polynomials
# Wilhelm Plesken, Daniel Robertz
# minimal generators for I_(2,3) over the rationals Q:
#
v := [c6 = 6, c5 = 5, c4 = 4, c3 = 3, c2 = 2, c1 = 1]:
min_gen := [c5*c4*c2*c1^2*c6-2*c5*c4*c3*c1*c6+3*c4*c3*
c2^2*c1^2*c6-2*c4*c3*c2*c1^4*c6-8*c5*c3*c2*c1^3*c6+c4^
2*c1^5*c6+c5^2*c1^3*c6-5*c4*c3^2*c2*c1*c6+4*c5*c3*
c2^2*c1*c6+c5*c1^8*c6-c5^2*c3*c1^6-c5*c3^4*c1^2-c5^3
*c1^4-2*c3*c1^4*c6^2+3*c2*c1^5*c6^2-2*c2^2*c1^3*c6^2-c5^
2*c4*c2^2*c1-2*c5*c4*c3^2*c2^2+2*c5*c4*c3^3*c1+c5^2*c4*
c3*c1^2-3*c5^2*c3*c2^2*c1^2-3*c5*c4^2*c3*c1^3+c5^2*c4*
c2*c1^3+c5*c4*c3^2*c1^4+3*c5^2*c3*c2*c1^4+c5^3*c4+4
*c4^2*c3*c1^2*c6-c4^3*c3*c2*c1^2+c5^2*c4*c3*c2+c3^3*
c2*c1^2*c6+4*c5*c3*c1^5*c6+5*c5*c2^2*c1^4*c6-c5*c4*c1^
4*c6-2*c4^2*c2*c1^3*c6-c5^2*c2*c1*c6-4*c5*c2*c1^6*
c6+2*c5*c3^2*c1^2*c6-2*c5*c2^3*c1^2*c6+c5*c4*c3^2*c2
*c1^2+c4^4*c1^3-c5*c4^2*c3^2+2*c3*c2*c1^2*c6^2-c1^7*c6
^2+4*c5*c4^2*c3*c2*c1-c5^3*c2^2-c5^2*c4^2*c1-c5^3*c3*c1+
2*c5^3*c2*c1^2+c4^2*c3^3*c2+c5^2*c3*c2^3-c4^3*c3^2*c1-
c5*c4^3*c1^2-c5^2*c3^2*c1^3, -c5^2*c4*c3^2-2*c4*c3*c2*
c1^5*c6-4*c5^2*c3*c1*c6-4*c5^2*c4*c2^2*c1^2+5*c5^2*c4*
c3*c2*c1-2*c5^2*c4^2*c1^2+c5^3*c3*c1^2-2*c5*c4^2*c3^
2*c1+4*c5*c3^2*c1^3*c6+c4^4*c1^4-4*c4^2*c3*c2*c1*c6+
c5*c4*c3^3*c1^2-2*c5*c4*c3^2*c2^2*c1+c5*c4*c3^2*c2*c1^
3+3*c4*c3*c2^2*c1^3*c6-2*c4*c3^2*c2^2*c6+6*c5*c4*c3*
c2*c6+5*c5*c2^2*c1^5*c6-7*c4^2*c2*c1^4*c6+2*c5*c4*c3*
c1^2*c6-5*c5*c3^2*c2*c1*c6+5*c5*c4^2*c3*c2*c1^2-c5^
2*c4^2*c2-c5^3*c3*c2+c5^2*c4*c2^3+2*c5^2*c3^3*c1+2*c5*
c3*c2^3*c6+6*c5^2*c2*c1^2*c6-2*c5*c4^2*c3*c2^2+2*c5*
c4^3*c2*c1+c4^2*c3^3*c2*c1+c5^2*c3*c2^3*c1-4*c5^2*c4
*c3*c1^3-c4^3*c3*c2*c1^3-3*c5^2*c3*c2^2*c1^3-2*c5*c4^2
*c3*c1^4+4*c5^2*c4*c2*c1^4+c5*c4*c3^2*c1^5+3*c5^2*c3*c2*
c1^5-c4*c3^2*c2*c1^2*c6+c5*c3*c2^2*c1^2*c6+c4^3*c3^
2*c2-c4^3*c3^2*c1^2-c4^4*c2*c1^2-c5*c3^4*c1^3-c5^2*c3^
2*c1^4-c5^2*c4*c1^6-c5^2*c3*c1^7-7*c5*c3*c2*c1^4*c6
-3*c5^2*c1^4*c6-3*c5^2*c2^2*c6+2*c4^2*c1^6*c6+c5*c1^9*c6+
4*c5*c3*c1^6*c6-4*c5*c2*c1^7*c6+c5*c1^3*c6^2-4*c2^3
*c1^2*c6^2+6*c2^2*c1^4*c6^2+c4*c1^4*c6^2+5*c4^2*c2^2*c1^
2*c6+c5^4-4*c5*c4*c2^2*c1*c6-3*c4*c2*c1^2*c6^2-5*c3*c2*
c1^3*c6^2+6*c3*c2^2*c1*c6^2-2*c2*c1^6*c6^2+4*c4^2*c3*
c1^3*c6+3*c5*c4*c2*c1^3*c6+c3^3*c2*c1^3*c6-2*c5*c2^3
*c1^3*c6-c4*c3^2*c1^4*c6, 4*c5*c2^2*c1^6*c6+2*c5*c2^4*
c1^2*c6+2*c5^2*c2*c1^3*c6-c3^3*c2^2*c1^2*c6-5*c5*c3^2*
c2*c1^2*c6+5*c5*c4*c2*c1^4*c6-4*c5*c4*c2^2*c1^2*c6+2
*c4*c3*c2^2*c1^4*c6-4*c4^2*c3*c2*c1^2*c6+c5*c3^2*c1^
4*c6-5*c5*c2^3*c1^4*c6-2*c5^2*c3*c1^2*c6-3*c4*c3*c2^3*
c1^2*c6+5*c4*c3^2*c2^2*c1*c6-c5^2*c4^2*c2*c1-c5^3*c3
*c2*c1+c5^2*c4*c2^3*c1+c5^2*c3^2*c2*c1^3-c5^2*c4*c2^
2*c1^3-2*c5*c4*c3^3*c2*c1+c5^2*c3^3*c1^2+2*c5^3*c3*c1^
3-2*c5^3*c2*c1^4+c5^3*c2^2*c1^2+6*c5*c4*c3*c2*c1*c6+3*
c5*c4^2*c3*c2*c1^3+c5^3*c1^6+c5^2*c4*c3*c2^2-c5*c2*c1^
8*c6-c4^2*c2*c1^5*c6+c5*c1^4*c6^2+c4*c1^5*c6^2-3*c2^
2*c1^5*c6^2+c2*c1^7*c6^2+2*c2^3*c1^3*c6^2-c5*c4*c3^2*c2^
2*c1^2+2*c5*c4*c3^2*c2^3-c5*c4*c3^2*c2*c1^4-3*c4*c2*
c1^3*c6^2+c3*c2*c1^4*c6^2-2*c5^2*c1^5*c6+8*c5*c3*c2^
2*c1^3*c6-4*c5*c4^2*c3*c2^2*c1-4*c5*c3*c2^3*c1*c6-4*c5*
c3*c2*c1^5*c6+c4^3*c3^2*c2*c1+2*c5*c4^3*c2*c1^2+c5*
c3^4*c2*c1^2+c4^3*c3*c2^2*c1^2+3*c5^2*c3*c2^3*c1^2-c5^
2*c4*c3*c1^4-3*c5^2*c3*c2^2*c1^4+c5^2*c3*c2*c1^6-c4^
2*c3^3*c2^2-c5^2*c3*c2^4-c4^4*c2*c1^3-2*c5^2*c4*c3*c2*
c1^2+2*c4^2*c2^2*c1^3*c6-c5*c4*c1^6*c6, -3*c3*c1^6*c6^
2-3*c3^2*c1^3*c6^2+9*c5*c2^2*c1^6*c6+2*c5*c2^4*c1^2*c6+4
*c3^2*c2*c1*c6^2-2*c3*c2^2*c1^2*c6^2-c5*c2*c1^2*c6^2+c5^
2*c4^2*c3+4*c5^2*c2*c1^3*c6-c3^3*c2^2*c1^2*c6+c5*c1^
10*c6-c4^3*c1^3*c6-2*c4*c3*c2*c1^6*c6+c5^3*c3^2-10*c5*
c3^2*c2*c1^2*c6-c1^9*c6^2-4*c5*c4*c3^2*c6-c5^2*c2^2*
c1*c6+c3^3*c2*c1^4*c6+c5^2*c4*c1*c6+c5*c3^2*c2^2*c6+
7*c5*c4*c2*c1^4*c6-5*c5*c4*c2^2*c1^2*c6+5*c4*c3*c2
^2*c1^4*c6-11*c4^2*c3*c2*c1^2*c6+6*c5*c3^2*c1^4*c6-7*c5*
c2^3*c1^4*c6-3*c5^2*c3*c1^2*c6-3*c4*c3*c2^3*c1^2*c6+
4*c4*c3^2*c2^2*c1*c6+2*c5*c4^2*c3^2*c2-c5^2*c4*c3^2*c1-
2*c5^2*c4^2*c2*c1-2*c5^3*c3*c2*c1+2*c5^2*c4*c2^3*c1
-3*c5*c4^2*c3^2*c1^2+c5*c4*c3^3*c1^3+c5^2*c3^2*c2*c1^3
-5*c5^2*c4*c2^2*c1^3-2*c5*c4*c3^3*c2*c1+3*c5^2*c3^3*c1
^2+2*c5^3*c3*c1^3-2*c5^3*c2*c1^4+c5^3*c2^2*c1^2-c5*c4*c3*
c1^3*c6+10*c5*c4*c3*c2*c1*c6+8*c5*c4^2*c3*c2*c1^3
+c5^3*c1^6-c3^4*c2*c1*c6+5*c5*c3*c1^7*c6-2*c4*c3^3*c1^
2*c6+7*c4^2*c3*c1^4*c6-2*c4*c3^2*c1^5*c6+5*c4^2*c3^2*c1*
c6-c4*c3^2*c2*c1^3*c6-5*c5*c2*c1^8*c6-7*c4^2*c2*c1^
5*c6+c5*c4*c3^2*c1^6+2*c5*c1^4*c6^2+2*c4*c1^5*c6^2-5*
c2^2*c1^5*c6^2+4*c2*c1^7*c6^2+2*c2^3*c1^3*c6^2-3*c5*c4*c3^
2*c2^2*c1^2+2*c5*c4*c3^2*c2^3+c4^4*c1^5+c3^4*c1^3*c6
-5*c4*c2*c1^3*c6^2+6*c3*c2*c1^4*c6^2+c5*c4*c3^4-3*c5^2
*c1^5*c6+14*c5*c3*c2^2*c1^3*c6-6*c5*c4^2*c3*c2^2*c1-c5^
2*c4*c1^7-c5^2*c3*c1^8-c4^3*c3^3-4*c5*c3*c2^3*c1*c6
-15*c5*c3*c2*c1^5*c6+2*c4^2*c1^7*c6-2*c5*c4^3*c3*c1+2*
c4^3*c3^2*c2*c1+4*c5*c4^3*c2*c1^2+c4^2*c3^3*c2*c1^2+c5*
c3^4*c2*c1^2+c4^3*c3*c2^2*c1^2+4*c5^2*c3*c2^3*c1^2-4*
c5^2*c4*c3*c1^4-c4^3*c3*c2*c1^4-6*c5^2*c3*c2^2*c1^4-2*
c5*c4^2*c3*c1^5+4*c5^2*c4*c2*c1^5+4*c5^2*c3*c2*c1^6-c4^
2*c3^3*c2^2-c5^2*c3*c2^4+c4^4*c3*c1^2-c4^3*c3^2*c1^3-2*
c4^4*c2*c1^3-c5*c4^3*c1^4-c5*c3^4*c1^4-c5^2*c3^2*c1^
5+3*c5^2*c4*c3*c2*c1^2-c5^2*c3^3*c2+6*c4^2*c2^2*c1^3*c6-
2*c5*c4*c1^6*c6, c5^3*c1^7-4*c5^2*c4*c3*c1^5+c5*c4^2
*c3*c1^6+c5^3*c3^2*c1+3*c5^2*c3^3*c1^3+c5^3*c2^2*c1^3+2*
c5^3*c3*c1^4-2*c5^3*c2*c1^5-2*c5*c4^2*c3^2*c2*c1-c5*c4*
c3^3*c2*c1^2+3*c5^2*c3^2*c2^2*c1^2+c5^2*c4*c2^3*c1^2+
c5*c4^2*c3^2*c1^3-2*c5*c4*c3^3*c1^4-3*c5^2*c3^2*c2*c1^
4-3*c5^2*c4*c2^2*c1^4+3*c5^2*c4*c3*c2*c1^3+c5*c4^2*c3*
c2*c1^4-4*c5*c3^2*c2^2*c1*c6-2*c5*c4^2*c3*c2^2*c1^2-c4^
4*c2*c1^4+c5^2*c3^2*c1^6+c4*c1^6*c6^2+c5*c1^5*c6^2+4*
c4^2*c2^2*c1^4*c6+4*c5*c4*c3*c2*c1^2*c6-2*c5^2*c1^6*c6+
c3^4*c1^4*c6+c4^2*c1^8*c6+6*c4*c3^2*c2*c1^4*c6-3*c5
*c3*c2^2*c1^4*c6+2*c5*c4*c2*c1^5*c6+2*c4^2*c3*c1^5*c6-
c5*c3^2*c1^5*c6-2*c4*c3^2*c1^6*c6-4*c4^2*c2*c1^6*c6-2*
c5*c3^3*c1^2*c6-2*c5^2*c3*c1^3*c6-2*c4*c3^3*c1^3*c6+
2*c5^2*c2*c1^4*c6-c5^2*c4*c1^8+2*c3*c2^2*c1^3*c6^2-3*c4*
c2*c1^4*c6^2-c3*c2*c1^5*c6^2-4*c4*c3^2*c2^2*c1^2*c6+
c5*c3*c2*c1^6*c6-2*c3^4*c2*c1^2*c6+2*c5*c4*c3*c1^4*c6+
2*c5*c3*c2^3*c1^2*c6-6*c4^2*c3*c2*c1^3*c6+3*c5*c3^
2*c2*c1^3*c6+6*c4*c3^3*c2*c1*c6-2*c5*c4*c2^2*c1^3*c6
-c4^2*c3^4*c2+c5*c3^5*c1^2+c5^2*c4*c3^2*c2+2*c4^3*c3^2
*c2*c1^2+2*c5*c4^3*c2*c1^3+3*c5^2*c4*c2*c1^6-c5^2*c3^2
*c2^3+2*c5*c4*c3^3*c2^2-c5*c4*c3^4*c1-c5^2*c3^3*c2*c1-
2*c5^2*c4*c3^2*c1^2-c5^2*c4^2*c2*c1^2-2*c5^3*c3*c2*c1^
2, 24*c5*c4*c3^2*c2^2*c1^3-4*c5^3*c1^7-c5^2*c4*c3*c1^5
+17*c5*c4^2*c3*c1^6+4*c5^3*c3^2*c1-2*c5^3*c2^3*c1+2*c5^2
*c4^2*c1^4-8*c5^3*c3*c1^4+6*c5^3*c2*c1^5-26*c5*c4^2*c3^2
*c2*c1+20*c5*c4*c3^3*c2*c1^2+21*c5^2*c3^2*c2^2*c1^2-15*c5^
2*c4*c2^3*c1^2+21*c5*c4^2*c3^2*c1^3-19*c5*c4*c3^3*c1
^4-35*c5^2*c3^2*c2*c1^4+16*c5^2*c4*c2^2*c1^4+16*c5^2*c4*c3*
c2*c1^3-c1^4*c6^3-59*c5*c4^2*c3*c2*c1^4+9*c5*c4*c3^
2*c2*c1^5-5*c5*c4*c3^2*c1^7-29*c5^2*c3*c2*c1^7+4*c5*c4
*c3*c2^2*c6+15*c5*c4*c3^2*c1*c6+4*c5*c4^2*c2*c1*c6-34*
c5*c3^2*c2^2*c1*c6-28*c5*c4*c3^2*c2^3*c1+53*c5*c4^2
*c3*c2^2*c1^2+12*c4^4*c2*c1^4+4*c5*c4^3*c1^5+5*c5*c3^4*c1^
5+12*c5^2*c3^2*c1^6+5*c5^2*c3*c1^9-9*c4*c1^6*c6^2-6*c5
*c1^5*c6^2-26*c4^2*c2^2*c1^4*c6-5*c4^4*c1^6-2*c5*c4^4*c1
+4*c4^3*c3^3*c1-4*c4^4*c3*c1^3+c5^2*c3*c2*c1*c6-4*c5*c4*
c2^3*c1*c6-64*c5*c4*c3*c2*c1^2*c6+13*c5^2*c1^6*c6+
3*c3^4*c1^4*c6-3*c4^2*c1^8*c6+46*c4*c3^2*c2*c1^4*c6-157*
c5*c3*c2^2*c1^4*c6-58*c5*c4*c2*c1^5*c6+c4^5*c1^2-3*c5^
2*c2^3*c6-c5^3*c1*c6-5*c5*c1^11*c6+2*c5*c4^3*c3*c2-2
*c5*c4^2*c3*c2^3+2*c5*c4^3*c2^2*c1-19*c4^2*c3*c1^5*c6-42*
c5*c3^2*c1^5*c6-5*c4*c3^2*c1^6*c6+18*c4^2*c2*c1^6*c6-
16*c5*c3^3*c1^2*c6+12*c5^2*c3*c1^3*c6-5*c4*c3^3*c1^3*c6-
17*c5^2*c2*c1^4*c6+c4^3*c3^2*c2^2-c4^4*c2^2*c1^2+5*c4^
3*c3^2*c1^4-2*c5^2*c4*c1^8+9*c3*c2^2*c1^3*c6^2+23*c4*c2*
c1^4*c6^2-32*c3*c2*c1^5*c6^2+5*c4^3*c1^4*c6-88*c4*c3^2
*c2^2*c1^2*c6+114*c5*c3*c2*c1^6*c6-10*c3^4*c2*c1^2*c6+
16*c5*c4*c1^7*c6-81*c5*c2^2*c1^7*c6-24*c5*c3*c1^8*c6+
34*c5*c2*c1^9*c6-6*c5*c4^2*c3*c6+3*c5^2*c4*c2*c6-5*c5^
2*c4*c1^2*c6-2*c4*c3^2*c2^3*c6+2*c5*c3*c2^4*c6-c4^4*
c3^2+4*c1^10*c6^2+19*c5*c4*c3*c1^4*c6-20*c4^2*c3^2*c1^2*
c6-5*c4^3*c2*c1^2*c6+7*c5^2*c2^2*c1^2*c6+5*c4^2*c2^3*c1^
2*c6-3*c5*c4^2*c1^3*c6+14*c3^3*c2^2*c1^3*c6-28*c5*c2^4
*c1^3*c6-5*c3^3*c2*c1^5*c6+80*c5*c2^3*c1^5*c6+65*c5*c3
*c2^3*c1^2*c6+49*c4^2*c3*c2*c1^3*c6+108*c5*c3^2*c2*c1^
3*c6+39*c4*c3^3*c2*c1*c6+46*c5*c4*c2^2*c1^3*c6+10*c4*c3*
c2*c1^7*c6+42*c4*c3*c2^3*c1^3*c6-43*c4*c3*c2^2*c1^
5*c6-7*c4^2*c3^4*c2+7*c5*c3^5*c1^2-6*c4*c3*c2*c1*c6^2-4*
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