Database of Local Fields of positive characteristic

The table presents all isomorphism classes of degree 10 extensions of F5((t)) with discriminant exponent < 16.

The data includes Defining polynomial, ramification index e, inertia degree f, valuation of discriminant c, Galois group G, Inertia Group I, Galois Slope Content.

　Defining Polynomial　 　e　 　f　 　c　 　Galois Group　 　#G　 　#I　 　Galois Slope Content　
x^5 + 4*x + 3 1 5 0 C5 (5T1) 5 1 [ ] 1 5
x^10 + 3*x^6 + x^5 + x^2 + 4*x + 4 + t + 3*t^3 + t^5 2 5 5 C10 (10T1) 10 2 [ ] 2 5
x^10 + 4*t^4*x^2 + 2*t^5 2 5 5 C10 (10T1) 10 2 [ ] 2 5
x^10 + 2*t*x^6 + (4 + t)*x^5 + t^2*x^2 + (4*t + t^2)*x + 2 + t + 2*t^2 5 2 10 F5 × C2 (10T5) 40 20 [ 5/4 ] 4 2
x^10 + t*x^6 + 2*t*x^5 + 2*t^2*x^2 + t^2*x + t^2 5 2 10 C52 : C8 (10T18) 200 100 [ 5/4, 5/4 ] 4 2
x^10 + t*x^6 + (4 + 4*t)*x^5 + 4*t^2*x^2 + (2*t + 2*t^2)*x + 2 + t^2 5 2 10 F5 × C2 (10T5) 40 20 [ 5/4 ] 4 2
x^10 + 2*t*x^6 + 2*t*x^5 + 4*t^2*x^2 + 2*t^2*x + t^2 5 2 10 (C52 : C4) : C2 (10T17) 200 100 [ 5/4, 5/4 ] 4 2
x^10 + 3*t*x^6 + 2*t*x^5 + 3*t^2*x^2 + 3*t^2*x + t^2 5 2 10 C52 : C8 (10T18) 200 100 [ 5/4, 5/4 ] 4 2
x^10 + 2*t*x^5 + 3*t^2*x^2 + t^2 5 2 10 C52 : C8 (10T18) 200 100 [ 5/4, 5/4 ] 4 2
x^10 + t*x^6 + 2*t*x^5 + t^2*x^2 + t^2*x + t^2 5 2 10 (C52 : C4) : C2 (10T17) 200 100 [ 5/4, 5/4 ] 4 2
x^10 + 4*t*x^6 + 4*x^5 + 4*t^2*x^2 + 3*t*x + 2 + 3*t + 2*t^2 5 2 10 F5 × C2 (10T5) 40 20 [ 5/4 ] 4 2
x^10 + 2*t*x^6 + 2*t*x^5 + 3*t^2*x^2 + 2*t^2*x + t^2 5 2 10 C52 : C8 (10T18) 200 100 [ 5/4, 5/4 ] 4 2
x^10 + 4*t*x^6 + 2*t*x^5 + t^2*x^2 + 4*t^2*x + t^2 5 2 10 (C52 : C4) : C2 (10T17) 200 100 [ 5/4, 5/4 ] 4 2
x^10 + 2*t*x^5 + 2*t^2*x^2 + t^2 5 2 10 C52 : C8 (10T18) 200 100 [ 5/4, 5/4 ] 4 2
x^10 + 3*t*x^6 + (4 + 3*t)*x^5 + t^2*x^2 + (t + 2*t^2)*x + 2 + 2*t + 4*t^2 5 2 10 F5 × C2 (10T5) 40 20 [ 5/4 ] 4 2
x^10 + 4*t*x^6 + 2*t*x^5 + 2*t^2*x^2 + 4*t^2*x + t^2 5 2 10 C52 : C8 (10T18) 200 100 [ 5/4, 5/4 ] 4 2
x^10 + 3*t*x^6 + 2*t*x^5 + 4*t^2*x^2 + 3*t^2*x + t^2 5 2 10 (C52 : C4) : C2 (10T17) 200 100 [ 5/4, 5/4 ] 4 2
x^10 + t*x + t 10 1 10 (C52 : C8) : C2 (10T28) 400 200 [ 9/8, 9/8 ] 8 2
x^10 + 2*t*x + t 10 1 10 (C52 : C8) : C2 (10T28) 400 200 [ 9/8, 9/8 ] 8 2
x^10 + t*x + 2*t 10 1 10 (C52 : C8) : C2 (10T28) 400 200 [ 9/8, 9/8 ] 8 2
x^10 + 2*t*x + 2*t 10 1 10 (C52 : C8) : C2 (10T28) 400 200 [ 9/8, 9/8 ] 8 2
x^10 + 4*t*x^2 + t 10 1 11 F5 (10T4) 20 20 [ 5/4 ] 4 1
x^10 + t*x^2 + t 10 1 11 F5 (10T4) 20 20 [ 5/4 ] 4 1
x^10 + 2*t*x^2 + t 10 1 11 F5 × C2 (10T5) 40 20 [ 5/4 ] 4 2
x^10 + 3*t*x^2 + t 10 1 11 F5 × C2 (10T5) 40 20 [ 5/4 ] 4 2
x^10 + 3*t*x^2 + 2*t 10 1 11 F5 (10T4) 20 20 [ 5/4 ] 4 1
x^10 + 2*t*x^2 + 2*t 10 1 11 F5 (10T4) 20 20 [ 5/4 ] 4 1
x^10 + t*x^2 + 2*t 10 1 11 F5 × C2 (10T5) 40 20 [ 5/4 ] 4 2
x^10 + 4*t*x^2 + 2*t 10 1 11 F5 × C2 (10T5) 40 20 [ 5/4 ] 4 2
x^10 + 2*t*x^7 + 3*t*x^6 + (4 + 4*t)*x^5 + t^2*x^4 + 3*t^2*x^3 + (4*t + 2*t^2)*x^2 + (4*t + 4*t^2)*x + 2 + 4*t + 2*t^2 5 2 12 F5 (10T4) 20 10 [ 3/2 ] 2 2
x^10 + t*x^7 + 2*t*x^5 + 2*t^2*x^4 + t^2*x^2 + t^2 5 2 12 C52 : C8 (10T18) 200 50 [ 3/2, 3/2 ] 2 4
x^10 + t*x^7 + 4*t*x^6 + (4 + 3*t)*x^5 + 4*t^2*x^4 + 2*t^2*x^3 + (2*t + t^2)*x^2 + (2*t + 3*t^2)*x + 2 + 4*t + 3*t^2 5 2 12 D10 (10T3) 20 10 [ 3/2 ] 2 2
x^10 + 2*t*x^7 + 2*t*x^5 + 4*t^2*x^4 + 2*t^2*x^2 + t^2 5 2 12 D52 (10T9) 100 50 [ 3/2, 3/2 ] 2 2
x^10 + 3*t*x^7 + 2*t*x^5 + 3*t^2*x^4 + 3*t^2*x^2 + t^2 5 2 12 C52 : C8 (10T18) 200 50 [ 3/2, 3/2 ] 2 4
x^10 + 2*t*x^5 + 3*t^2*x^4 + t^2 5 2 12 C52 : C8 (10T18) 200 50 [ 3/2, 3/2 ] 2 4
x^10 + t*x^7 + 2*t*x^5 + t^2*x^4 + t^2*x^2 + t^2 5 2 12 C52 : C4 (10T10) 100 50 [ 3/2, 3/2 ] 2 2
x^10 + 4*t*x^7 + t*x^6 + (4 + t)*x^5 + 4*t^2*x^4 + 2*t^2*x^3 + (3*t + 4*t^2)*x^2 + 3*t*x + 2 + 4*t + t^2 5 2 12 D10 (10T3) 20 10 [ 3/2 ] 2 2
x^10 + 2*t*x^7 + 2*t*x^5 + 3*t^2*x^4 + 2*t^2*x^2 + t^2 5 2 12 C52 : C8 (10T18) 200 50 [ 3/2, 3/2 ] 2 4
x^10 + 4*t*x^7 + 2*t*x^5 + t^2*x^4 + 4*t^2*x^2 + t^2 5 2 12 C52 : C4 (10T10) 100 50 [ 3/2, 3/2 ] 2 2
x^10 + 2*t*x^5 + 2*t^2*x^4 + t^2 5 2 12 C52 : C8 (10T18) 200 50 [ 3/2, 3/2 ] 2 4
x^10 + 3*t*x^7 + 2*t*x^6 + 4*x^5 + t^2*x^4 + 3*t^2*x^3 + (t + 3*t^2)*x^2 + (t + 3*t^2)*x + 2 + 4*t + 3*t^2 5 2 12 F5 (10T4) 20 10 [ 3/2 ] 2 2
x^10 + 4*t*x^7 + 2*t*x^5 + 2*t^2*x^4 + 4*t^2*x^2 + t^2 5 2 12 D52 (10T9) 200 50 [ 3/2, 3/2 ] 2 4
x^10 + 3*t*x^7 + 2*t*x^5 + 4*t^2*x^4 + 3*t^2*x^2 + t^2 5 2 12 C52 : C8 (10T18) 200 50 [ 3/2, 3/2 ] 2 4
x^10 + t*x^3 + t 10 1 12 (C52 : C8) : C2 (10T28) 400 200 [ 11/8, 11/8 ] 8 2
x^10 + 2*t*x^3 + 2*t 10 1 12 (C52 : C8) : C2 (10T28) 400 200 [ 11/8, 11/8 ] 8 2
x^10 + 2*t*x^3 + t 10 1 12 (C52 : C8) : C2 (10T28) 400 200 [ 11/8, 11/8 ] 8 2
x^10 + t*x^3 + 2*t 10 1 12 (C52 : C8) : C2 (10T28) 400 200 [ 11/8, 11/8 ] 8 2
x^10 + 3*t*x^4 + t 10 1 13 D5 (10T2) 10 5 [ 3/2 ] 2 1
x^10 + 2*t*x^4 + 2*t 10 1 13 F5 × C2 (10T5) 40 10 [ 3/2 ] 2 4
x^10 + 2*t*x^5 + t*x^4 + 2*t 10 1 13 D5 × C5 (10T6) 50 10 [ 3/2 ] 2 5
x^10 + t*x^5 + t*x^4 + 2*t 10 1 13 D5 × C5 (10T6) 50 10 [ 3/2 ] 2 5
x^10 + 2*t*x^4 + t 10 1 13 D10 (10T3) 20 10 [ 3/2 ] 2 2
x^10 + 4*t*x^4 + t 10 1 13 F5 × C2 (10T5) 40 10 [ 3/2 ] 2 4
x^10 + t*x^4 + t 10 1 13 F5 × C2 (10T5) 40 10 [ 3/2 ] 2 4
x^10 + t*x^5 + 3*t*x^4 + t 10 1 13 D5 × C5 (10T6) 50 10 [ 3/2 ] 2 5
x^10 + 2*t*x^5 + 3*t*x^4 + t 10 1 13 D5 × C5 (10T6) 50 10 [ 3/2 ] 2 5
x^10 + t*x^4 + 2*t 10 1 13 D5 (10T2) 10 5 [ 3/2 ] 2 1
x^10 + 4*t*x^4 + 2*t 10 1 13 D10 (10T3) 20 10 [ 3/2 ] 2 2
x^10 + 3*t*x^4 + 2*t 10 1 13 F5 × C2 (10T5) 40 10 [ 3/2 ] 2 4
x^10 + 2*t*x^8 + 2*t*x^7 + (t + t^2)*x^6 + (4 + 2*t + 2*t^2)*x^5 + 4*t^2*x^4 + (4*t + 4*t^2)*x^3 + t*x^2 + 4*t^2*x + 2 + 4*t^2 5 2 14 F5 × C2 (10T5) 40 20 [ 7/4 ] 4 2
x^10 + t*x^8 + 2*t^2*x^6 + 2*t*x^5 + t^2*x^3 + t^2 5 2 14 C52 : C8 (10T18) 200 100 [ 7/4, 7/4 ] 4 2
x^10 + t*x^8 + t*x^7 + (3*t + 4*t^2)*x^6 + (4 + 2*t + 3*t^2)*x^5 + t^2*x^4 + (2*t + 4*t^2)*x^3 + (3*t + 3*t^2)*x^2 + 2 + 2*t + 3*t^2 5 2 14 F5 × C2 (10T5) 40 20 [ 7/4 ] 4 2
x^10 + 2*t*x^8 + 4*t^2*x^6 + 2*t*x^5 + 2*t^2*x^3 + t^2 5 2 14 (C52 : C4) : C2 (10T17) 200 100 [ 7/4, 7/4 ] 4 2
x^10 + 3*t*x^8 + 3*t^2*x^6 + 2*t*x^5 + 3*t^2*x^3 + t^2 5 2 14 C52 : C8 (10T18) 200 100 [ 7/4, 7/4 ] 4 2
x^10 + 3*t^2*x^6 + 2*t*x^5 + t^2 5 2 14 C52 : C8 (10T18) 200 100 [ 7/4, 7/4 ] 4 2
x^10 + t*x^8 + t^2*x^6 + 2*t*x^5 + t^2*x^3 + t^2 5 2 14 (C52 : C4) : C2 (10T17) 200 100 [ 7/4, 7/4 ] 4 2
x^10 + 4*t*x^8 + 4*t*x^7 + (2*t + 4*t^2)*x^6 + (4 + 2*t + 3*t^2)*x^5 + t^2*x^4 + (3*t + 2*t^2)*x^3 + (2*t + t^2)*x^2 + 4*t^2*x + 2 + t + 3*t^2 5 2 14 F5 × C2 (10T5) 40 20 [ 7/4 ] 4 2
x^10 + 2*t*x^8 + 3*t^2*x^6 + 2*t*x^5 + 2*t^2*x^3 + t^2 5 2 14 C52 : C8 (10T18) 200 100 [ 7/4, 7/4 ] 4 2
x^10 + 4*t*x^8 + t^2*x^6 + 2*t*x^5 + 4*t^2*x^3 + t^2 5 2 14 (C52 : C4) : C2 (10T17) 200 100 [ 7/4, 7/4 ] 4 2
x^10 + 2*t^2*x^6 + 2*t*x^5 + t^2 5 2 14 C52 : C8 (10T18) 200 100 [ 7/4, 7/4 ] 4 2
x^10 + 3*t*x^8 + 3*t*x^7 + (4*t + t^2)*x^6 + (4 + 2*t + 2*t^2)*x^5 + 4*t^2*x^4 + t*x^3 + (4*t + t^2)*x^2 + 2*t^2*x + 2 + 3*t + 4*t^2 5 2 14 F5 × C2 (10T5) 40 20 [ 7/4 ] 4 2
x^10 + 4*t*x^8 + 2*t^2*x^6 + 2*t*x^5 + 4*t^2*x^3 + t^2 5 2 14 C52 : C8 (10T18) 200 100 [ 7/4, 7/4 ] 4 2
x^10 + 3*t*x^8 + 4*t^2*x^6 + 2*t*x^5 + 3*t^2*x^3 + t^2 5 2 14 (C52 : C4) : C2 (10T17) 200 100 [ 7/4, 7/4 ] 4 2
x^10 + 3*t*x^6 + t 10 1 15 F5 (10T4) 20 20 [ 7/4 ] 4 1
x^10 + 2*t*x^8 + 4*t*x^7 + t*x^6 + t*x^5 + t 10 1 15 (C52 : C4) : C2 (10T17) 200 100 [ 5/4, 7/4 ] 4 2
x^10 + 3*t*x^8 + 2*t*x^7 + t*x^6 + 3*t*x^5 + t 10 1 15 (C52 : C4) : C2 (10T17) 200 100 [ 5/4, 7/4 ] 4 2
x^10 + 2*t*x^8 + 3*t*x^7 + 4*t*x^6 + 3*t*x^5 + t 10 1 15 (C52 : C4) : C2 (10T17) 200 100 [ 5/4, 7/4 ] 4 2
x^10 + t*x^6 + 2*t 10 1 15 F5 (10T4) 20 20 [ 7/4 ] 4 1
x^10 + 4*t*x^6 + 2*t 10 1 15 F5 (10T4) 20 20 [ 7/4 ] 4 1
x^10 + 2*t*x^6 + 2*t 10 1 15 F5 × C2 (10T5) 40 20 [ 7/4 ] 4 2
x^10 + 3*t*x^6 + 2*t 10 1 15 F5 × C2 (10T5) 40 20 [ 7/4 ] 4 2
x^10 + 3*t*x^8 + t*x^7 + 4*t*x^6 + t*x^5 + 2*t 10 1 15 C52 : C4 (10T10) 100 100 [ 5/4, 7/4 ] 4 1
x^10 + 2*t*x^8 + 4*t*x^7 + t*x^6 + t*x^5 + 2*t 10 1 15 C52 : C4 (10T10) 100 100 [ 5/4, 7/4 ] 4 1
x^10 + 3*t*x^8 + 2*t*x^7 + t*x^6 + t*x^5 + 2*t 10 1 15 C52 : C4 (10T10) 100 100 [ 5/4, 7/4 ] 4 1
x^10 + 2*t*x^6 + t 10 1 15 F5 (10T4) 20 20 [ 7/4 ] 4 1
x^10 + 2*t*x^8 + 3*t*x^7 + 4*t*x^6 + t*x^5 + 2*t 10 1 15 C52 : C4 (10T10) 100 100 [ 5/4, 7/4 ] 4 1
x^10 + t*x^8 + 4*t*x^7 + 2*t*x^6 + t*x^5 + 2*t 10 1 15 (C52 : C4) : C2 (10T17) 200 100 [ 5/4, 7/4 ] 4 2
x^10 + 4*t*x^8 + t*x^7 + 3*t*x^6 + t*x^5 + 2*t 10 1 15 (C52 : C4) : C2 (10T17) 200 100 [ 5/4, 7/4 ] 4 2
x^10 + 4*t*x^8 + 3*t*x^7 + 2*t*x^6 + t*x^5 + 2*t 10 1 15 (C52 : C4) : C2 (10T17) 200 100 [ 5/4, 7/4 ] 4 2
x^10 + t*x^8 + 2*t*x^7 + 3*t*x^6 + t*x^5 + 2*t 10 1 15 (C52 : C4) : C2 (10T17) 200 100 [ 5/4, 7/4 ] 4 2
x^10 + t*x^6 + t 10 1 15 F5 × C2 (10T5) 40 20 [ 7/4 ] 4 2
x^10 + 4*t*x^6 + t 10 1 15 F5 × C2 (10T5) 40 20 [ 7/4 ] 4 2
x^10 + 4*t*x^8 + 3*t*x^7 + 2*t*x^6 + t*x^5 + t 10 1 15 C52 : C4 (10T10) 100 100 [ 5/4, 7/4 ] 4 1
x^10 + t*x^8 + 4*t*x^7 + 2*t*x^6 + 3*t*x^5 + t 10 1 15 C52 : C4 (10T10) 100 100 [ 5/4, 7/4 ] 4 1
x^10 + t*x^8 + 2*t*x^7 + 3*t*x^6 + t*x^5 + t 10 1 15 C52 : C4 (10T10) 100 100 [ 5/4, 7/4 ] 4 1
x^10 + 4*t*x^8 + t*x^7 + 3*t*x^6 + 3*t*x^5 + t 10 1 15 C52 : C4 (10T10) 100 100 [ 5/4, 7/4 ] 4 1
x^10 + 3*t*x^8 + t*x^7 + 4*t*x^6 + t*x^5 + t 10 1 15 (C52 : C4) : C2 (10T17) 200 100 [ 5/4, 7/4 ] 4 2
inserted by FC2 system