# Elimination for coefficients of special characteristic polynomials
# Wilhelm Plesken, Daniel Robertz
# minimal generators for I_(2,4) over the rationals Q:
#
v := [c8 = 8, c7 = 7, c6 = 6, c5 = 5, c4 = 4, c3 = 3, c2 = 2,
c1 = 1]:
min_gen := [-2*c7^3*c1^7-c5*c1^7*c8^2+c4*c1^8*c8^2-c3^3*c1
^3*c8^2-c3^2*c1^6*c8^2-c6^2*c1^8*c8-c5^3*c1^5*c8-c7^2*c4^
2*c2^3+c7^2*c4^3*c1^2+c7^2*c6*c1^8+c7^2*c5^2*c4-c7^3*
c3*c2^2-c7^2*c5^2*c1^4-c6^2*c4*c1^4*c8+c4*c3^2*c1^2*c8^
2-c6*c4^2*c1^6*c8+c5^2*c3^3*c1*c8-c5^3*c2^2*c1*c8-c5^
2*c4*c3^2*c8+c5^2*c4*c2^3*c8-c7*c3^3*c1^4*c8-c7*c5*
c1^8*c8+c5^2*c3^2*c2^2*c8-c7*c3^3*c2^2*c8+c7^2*c4^2
*c2^2*c1^2-c7^2*c4^2*c3*c1^3-c7^2*c6*c4*c1^4+c7^2*c5*c4*
c1^5-c7^2*c5*c3*c2^3-c7^2*c5^2*c3*c1-c8*c7^2*c4*c1^
2+c7^2*c3*c1^3*c8+c5^2*c4*c2*c1^4*c8-c6*c4^2*c2^2*c1^
2*c8+c6*c4*c3^2*c2^2*c8-c7*c4*c3*c2^2*c1^2*c8+c6
*c5*c3*c1^6*c8-c5^2*c3^2*c2*c1^2*c8+c7*c4*c3*c1^6*c8+
c6*c4*c3^2*c1^4*c8-c7^2*c5*c3*c2*c1^4-2*c6*c1^6*c8^
2+2*c3^2*c2^3*c8^2+2*c7^2*c1^6*c8+4*c7^2*c6*c2^4+5*c7^3*
c2^3*c1+2*c7^3*c4*c1^3-12*c7^3*c2^2*c1^3+2*c6*c5^2*c2^2*
c8+2*c5^3*c2*c1^3*c8+2*c6*c5^2*c1^4*c8-8*c6^2*c2^2*c1^
4*c8+4*c5*c2^3*c1*c8^2-6*c3^2*c2^2*c1^2*c8^2+7*c4*c2^2*c1^
4*c8^2-3*c6^2*c3*c1^5*c8-4*c7*c5*c2^4*c8+5*c6^2*c2*c1^
6*c8+3*c7*c6*c1^7*c8-2*c6*c2^2*c1^2*c8^2+4*c6*c2*c1^
4*c8^2-3*c4*c2^3*c1^2*c8^2-9*c5*c2^2*c1^3*c8^2+5*c3^2*c2*
c1^4*c8^2+6*c5*c2*c1^5*c8^2-5*c4*c2*c1^6*c8^2+4*c6^2*
c2^3*c1^2*c8-3*c7^2*c6*c4*c2^2-2*c7^3*c4*c2*c1-12*c7^2
*c6*c2^3*c1^2+13*c7^2*c6*c2^2*c1^4+2*c7^2*c6*c3*c1^5-6*c7^
2*c6*c2*c1^6-2*c7^2*c5*c4^2*c1+3*c7^2*c5^2*c2*c1^2+4
*c7^3*c3*c2*c1^2+7*c8*c7^2*c2^2*c1^2+2*c7^2*c5*c3^2*c1
^3-7*c8*c7^2*c2*c1^4+12*c7*c5*c2^3*c1^2*c8+13*c7*c6*c2
^2*c1^3*c8+2*c6*c5*c4*c2^2*c1*c8-2*c7*c4^2*c3*c1^2*c8+
2*c7*c3^3*c2*c1^2*c8-13*c7*c5*c2^2*c1^4*c8-8*c7*c5*c3*
c2^2*c1*c8+2*c7*c4*c3^2*c1^3*c8-12*c7*c6*c2*c1^5*c8-
4*c7*c5*c3*c1^5*c8-4*c7*c6*c2^3*c1*c8-2*c7*c6*c4
*c1^3*c8+2*c6*c4^2*c2*c1^4*c8+4*c7*c6*c3*c2^2*c8-2*c6*
c4*c3^2*c2*c1^2*c8-4*c6*c5*c4*c2*c1^3*c8-4*c6*c5^2
*c2*c1^2*c8+8*c6^2*c3*c2*c1^3*c8+7*c6*c5*c3*c2^2*c1^
2*c8-4*c6*c5*c3*c2^3*c8-2*c6*c5*c3^2*c1^3*c8-4*c7*c5
*c3^2*c1^2*c8-4*c6*c5*c3*c2*c1^4*c8-2*c7*c4*c3*c2*c1^
4*c8-4*c6^2*c3*c2^2*c1*c8+2*c6*c5*c4*c1^5*c8+12*c7*
c5*c3*c2*c1^3*c8+6*c7*c6*c3*c1^4*c8-2*c5^2*c4*c2^2*c1^
2*c8+2*c6*c5*c4*c3*c1^2*c8-8*c7*c6*c3*c2*c1^2*c8
+6*c7*c5*c2*c1^6*c8+2*c8*c7*c4*c3*c2^3+4*c7^2*c6*c3*
c2^2*c1+4*c7^2*c6*c4*c2*c1^2+9*c7^3*c2*c1^5-2*c7^2*c5^
2*c2^2-2*c8*c7^2*c2^3-3*c7^3*c3*c1^4-6*c7^2*c6*c3*c2*c1^
3+2*c7^2*c5*c4*c3*c2+4*c8*c7*c5*c4*c3*c1-2*c7^2*c5
*c3^2*c2*c1+3*c7^2*c5*c4*c2^2*c1+2*c7^2*c5*c3*c2^2*c1^
2-4*c7^2*c5*c4*c2*c1^3, 5*c7^3*c1^9+c3^3*c1^5*c8^2+c5^
3*c3^2*c1*c8+c4*c3*c1^7*c8^2-c7*c4^2*c1^7*c8+c6^2*c5
*c1^5*c8+c5^3*c3*c2^2*c8+c7*c1^7*c8^2-c5^2*c3^3*c2*c1*
c8-2*c5^3*c3*c2*c1^2*c8+2*c7*c4^2*c2*c1^5*c8+c8*c7
*c5*c3^2*c2^2-c7^2*c5*c4*c3*c2^2-3*c5*c3*c2^2*c1^2*c8^
2+c4*c3*c2^2*c1^3*c8^2-4*c7*c4*c3^2*c2*c1^3*c8+2*c7^
2*c5*c3^2*c2^2*c1-c7^2*c5*c4*c3*c1^4+c7^2*c5*c3*c2^4
+c7^3*c5*c2^2-6*c7^2*c1^8*c8+5*c6*c1^8*c8^2+2*c5*c1^9*c8^
2+2*c3^2*c1^8*c8^2-2*c4*c1^10*c8^2+2*c5^3*c1^7*c8+2*c6^
2*c1^10*c8-3*c7^3*c2^4*c1-5*c7^3*c2^3*c1^3-2*c7^2*c4^3*c1^
4-4*c7^3*c4*c1^5+24*c7^3*c2^2*c1^5-21*c7^3*c2*c1^7-2*c7^
2*c6*c1^10+c7^2*c5^2*c2^3-c7^3*c3*c2^3-3*c7*c5^2*c1^5*c8-
3*c5^3*c2*c1^5*c8-5*c6*c5^2*c1^6*c8+2*c6^2*c4*c1^6*c8+
2*c7*c3^3*c1^6*c8+11*c6^2*c2^2*c1^6*c8+2*c6*c4^2*c1^8
*c8-2*c5^2*c3^3*c1^3*c8-9*c6^2*c2*c1^8*c8+c7*c2^2*c1^3
*c8^2-2*c7*c2*c1^5*c8^2-c3^3*c2^2*c1*c8^2-2*c4*c3^2*c1
^4*c8^2+c6*c2^2*c1^4*c8^2+7*c3^2*c2^2*c1^4*c8^2+12*c5*c2
^2*c1^5*c8^2-3*c5*c3*c1^6*c8^2-8*c6*c2*c1^6*c8^2-9*c3^2*
c2*c1^6*c8^2-9*c4*c2^2*c1^6*c8^2-11*c5*c2*c1^7*c8^2+9*c4*
c2*c1^8*c8^2-c5*c3^2*c1^3*c8^2+2*c3^2*c2^3*c1^2*c8^2+
c5*c2^3*c1^3*c8^2-c4*c2^3*c1^4*c8^2+2*c6*c2^3*c1^2*c8^
2+6*c6^2*c3*c1^7*c8+2*c7*c5*c1^10*c8-7*c7*c6*c1^9*c8+c7^
2*c4^2*c2^3*c1^2+4*c7^2*c6*c2^4*c1^2+4*c7^3*c4*c2*c1
^3-2*c7^2*c4^2*c2^2*c1^4+11*c7^2*c6*c2^3*c1^4+2*c7^2*c4^2*
c3*c1^5+c7^2*c6*c4*c1^6-20*c7^2*c6*c2^2*c1^6-4*c7^2*c6
*c3*c1^7+11*c7^2*c6*c2*c1^8+3*c7^2*c6*c4*c2^3+3*c7^3*c3*
c2^2*c1^2+c8*c7^2*c5*c1^3+3*c7^2*c5*c4^2*c1^3+2*c8*c7^
2*c4*c1^4-8*c7^2*c5^2*c2*c1^4-9*c7^3*c3*c2*c1^4-15*c8*
c7^2*c2^2*c1^4-4*c7^2*c5*c3^2*c1^5+19*c8*c7^2*c2*c1^6-2*
c7^2*c5*c4*c1^7-3*c7^2*c3*c1^5*c8-4*c6*c5*c4*c3*c1^4
*c8+4*c6*c5*c3^2*c1^5*c8-2*c6*c5^2*c3*c1^3*c8-13*c6^2*
c3*c2*c1^5*c8-11*c7*c5*c2^3*c1^4*c8-19*c7*c6*c2^2*c1
^5*c8+3*c6*c4*c3^2*c2*c1^4*c8+3*c5^2*c4*c2^2*c1^4*c8+
10*c7*c6*c3*c2*c1^4*c8-3*c7*c3^3*c2*c1^4*c8-24*c7*c5
*c3*c2*c1^5*c8-2*c5^2*c4*c2*c1^6*c8-3*c6*c4^2*c2*c1^
6*c8+6*c6*c5*c4*c2*c1^5*c8+9*c7*c5*c3*c2^2*c1^3*c8+c6*
c5*c3*c2^3*c1^2*c8-2*c6*c4*c3^2*c1^6*c8+6*c7*c5*c3*
c2^3*c1*c8-4*c7*c5*c2^4*c1^2*c8-2*c6*c5*c3*c1^8*c8-
2*c7*c4*c3*c1^8*c8+2*c5^2*c4*c3^2*c1^2*c8-10*c6*c5*c3*
c2^2*c1^4*c8+c7*c4*c3^2*c2^2*c1*c8+7*c6*c5*c3*c2
*c1^6*c8+3*c7*c4*c3*c2*c1^6*c8+2*c5^2*c3^2*c2*c1^4*c8-
c5^2*c3^2*c2^2*c1^2*c8-c6*c5^2*c2^2*c1^2*c8-3*c7*c4*
c3*c2^3*c1^2*c8+4*c7*c6*c4*c1^5*c8+9*c7*c5*c3^2*c1^4
*c8+20*c7*c5*c2^2*c1^6*c8+25*c7*c6*c2*c1^7*c8+3*c7*c5*
c4*c1^6*c8-3*c7*c4*c3^2*c1^5*c8+2*c6*c5*c3^2*c2*c1
^3*c8+8*c6*c5^2*c2*c1^4*c8-3*c7*c6*c2^3*c1^3*c8+4*c7*c4*
c3*c2^2*c1^4*c8-11*c7*c5*c2*c1^8*c8+6*c5*c3*c2*c1^
4*c8^2-12*c7*c6*c3*c1^6*c8-4*c6*c5*c4*c1^7*c8+9*c7*c5*
c3*c1^7*c8+2*c7*c6*c5*c1^4*c8+4*c7*c4^2*c3*c1^4*c8-
c7^2*c6*c4*c2^2*c1^2-2*c7^2*c6*c5*c2*c1^3-3*c7^2*c6
*c4*c2*c1^4+10*c7^2*c6*c3*c2*c1^5+2*c7^2*c6*c5*c2^2*c1-
4*c7^2*c6*c3*c2^3*c1+2*c8*c7^2*c2^4-c7^3*c5*c1^4+3*c7^
2*c5^2*c1^6+7*c7^3*c3*c1^6-2*c3^2*c2^4*c8^2+c7^2*c4^2*c2^
4-4*c7^2*c6*c2^5-c7^2*c5^3*c1-c7^2*c5^2*c3*c2*c1-3*c8*
c7*c5^2*c2^2*c1-c8*c7^2*c3*c2^2*c1-4*c8*c7*c5^2*c3*
c1^2-4*c7^2*c5*c4*c3*c2*c1^2+2*c8*c7*c5*c3^2*c2*c1^
2+3*c8*c7*c5*c4*c2^2*c1^2-6*c8*c7*c5*c4*c3*c1^3+6*c8*
c7*c5^2*c2*c1^3+c8*c7^2*c3*c2*c1^3+2*c7^2*c5*c3^2*c2*
c1^3-2*c7^2*c5*c4*c2^2*c1^3-6*c8*c7*c5*c4*c2*c1^4-3*
c7^2*c5*c3*c2^2*c1^4+7*c7^2*c5*c4*c2*c1^5+2*c7^2*c5*
c3*c2*c1^6-c5^2*c3^2*c2^3*c8+c7*c3^3*c2^3*c8+c5^3*c3
*c1^4*c8-4*c6^2*c2^4*c1^2*c8+3*c4*c2^4*c1^2*c8^2+3*c3^3*
c2*c1^3*c8^2-4*c5*c2^4*c1*c8^2-c5^2*c4*c2^4*c8-c6*c4
*c3^2*c2^3*c8+4*c6*c5*c3*c2^4*c8+c6*c4^2*c2^3*c1^2*c8-
2*c6*c5*c4*c2^3*c1*c8+4*c6^2*c3*c2^3*c1*c8+4*c7*c6
*c2^4*c1*c8-2*c6*c5^2*c2^3*c8+c5^3*c2^3*c1*c8+4*c8*c7*
c5*c2^5+4*c7^2*c5^2*c2^2*c1^2+4*c7^2*c5^2*c3*c1^3-2*c4
*c3*c2*c1^5*c8^2-c7*c4^2*c2^2*c1^3*c8+c7^2*c4^2*c3*c2*
c1^3-2*c7^2*c6*c3*c2^2*c1^3-2*c8*c7*c4*c3*c2^4-4*c8*
c7*c6*c3*c2^3-3*c7^2*c5*c4*c2^3*c1, -3*c1^6*c8^3+3*c7^
3*c1^9-3*c5^4*c1^2*c8+18*c5^2*c1^4*c8^2-2*c3^3*c1^5*c8^2-c7*
c6^2*c1^11+c6^3*c5*c1^7+c6^2*c4^3*c1^6-c6^2*c5^2*c1^
8+c6^3*c4*c1^8+c7^2*c3^4*c1^4-c5^4*c3^2*c2^2-c5^5*c3*c1^
2-c7*c5^3*c1^8-c5^4*c3^3*c1+c7*c5^3*c2^4+c5^5*c2*c1^
3-4*c7^2*c3^2*c1^2*c8+c5^3*c3^2*c1*c8-12*c5*c4*c1^5*c8
^2+3*c4*c3*c1^7*c8^2-c7*c4^2*c1^7*c8+6*c6^2*c5*c1^5*c8
+2*c5^3*c3*c2^2*c8+c7^2*c5*c3^3*c2+4*c7^3*c5*c3*c1+7*c7^
2*c5*c3^3*c1^2-c5^2*c3^2*c1^6*c8+c6*c3^3*c1^7*c8-c5*
c3^5*c2*c8-c6*c5^3*c2^2*c1^5+c6^3*c5*c2^2*c1^3-c4^2*
c3^4*c1^2*c8+c5*c3^5*c1^2*c8+c3*c2*c1^9*c8^2+7*c7*c1
^7*c8^2+c6*c4^2*c3*c1^5*c8+8*c7*c3*c1^4*c8^2+4*c6*c5*c3^
3*c2*c8-3*c6*c5*c3^3*c1^2*c8-4*c5^2*c3^3*c2*c1*c8+
c5^2*c4*c3*c2^2*c1*c8+c5^3*c3*c2*c1^2*c8+c6*c4*c3^
3*c1^3*c8+4*c8*c7*c5*c3^2*c2^2+c7^2*c5*c4*c3*c2^2-10*c5*
c3*c2^2*c1^2*c8^2+8*c5*c4*c2*c1^3*c8^2+9*c4*c3*c2^
2*c1^3*c8^2+2*c6*c4*c3^3*c2*c1*c8-11*c5^2*c4*c3*c2*c1^
3*c8-13*c7*c4*c3^2*c2*c1^3*c8-4*c6*c4^2*c3*c2*c1^3*c8+
2*c7^2*c4^2*c3*c2^2*c1+c7^2*c5*c3^2*c2^2*c1-4*c7^2*c5*
c4*c3*c1^4+c5*c4^2*c2*c1^7*c8-c5^3*c4^2*c3*c2*c1
^2+c5^2*c3*c2*c1^7*c8+c5*c4*c3^2*c2*c1^5*c8-c5*c4^2*
c2^3*c1^3*c8+c5*c4*c3^2*c2^3*c1*c8-c5*c3^3*c2^3*c1^2
*c8-c5^2*c4*c3*c1^5*c8+c7*c6*c5*c3*c1^9-c4^3*c3*c2
*c1^5*c8+c6^2*c5*c4*c2*c1^7-c7*c3^2*c2^4*c1*c8-c5*c4^
2*c3*c2^2*c1^2*c8+c4^2*c3^3*c2^2*c1*c8-c6*c4*c3*c1^
9*c8-c5^2*c3*c2^2*c1^5*c8-c6^2*c4^2*c3*c2*c1^5-c6^2*
c4*c3^3*c2^2*c1+c7*c6*c3^4*c2^2*c1-c6*c5*c4^2*c3^3*c1^
2+c5*c3^3*c2^2*c1^4*c8+c7*c3^2*c2*c1^7*c8-c5*c4^2*
c3*c1^6*c8+c6^2*c4*c3^3*c2*c1^3+c5*c4*c3^3*c2^2*c8-c4^
2*c3^2*c2^3*c1^2*c8+c5*c4*c3*c2^3*c1^4*c8-c7*c4*c2^
4*c1^3*c8+c6^2*c4^2*c3*c2^2*c1^3-c6^2*c5*c3^2*c2^2*c1^
3+c4^3*c2^3*c1^4*c8-c6*c5^3*c2^4*c1+c6*c5^2*c3^4*c1
^2-c6^2*c4^2*c3^2*c1^4+c7*c6*c3^4*c1^5-c6*c5^2*c4^2*c1
^6-c7*c6^2*c4*c1^7-c6^3*c3*c2*c1^7+c4^3*c3^2*c1^4*c8-c7^
2*c5*c2^4*c1^3-c7*c6*c4^3*c1^5-c7^2*c5*c2^3*c1^5+c7^
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*c1^4-364544*c6*c5*c4^2*c3*c1^4*c8+7458111*c5^2*c3^2*c2*c1^
8*c8-479488*c6^2*c5*c4*c3*c1^10+2