Database of Local Fields of positive characteristic

The table presents all isomorphism classes of degree 10 extensions of F2((t)) with discriminant exponent < 20.

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^10 + x^6 + x^5 + x^3 + x^2 + x + 1 1 10 0 C10 (10T1) 10 1 [ ] 1 10
x^10 + x^9 + x^8 + x^6 + x^5 + (1 + t)*x^4 + x^2 + (1 + t)*x + 1 + t + t^2 5 2 8 F5 (10T4) 20 5 [ ] 5 4
x^10 + t*x^9 + t*x^8 + t^3*x^6 + t^5*x^5 + (t^4 + t^5)*x^4 + (t^4 + t^5 + t^6 + t^7)*x^2 + (t^5 + t^9)*x + t^5 + t^6 + t^9 + t^10 2 5 10 C2 × (C24 : C5) (10T14) 160 32 [ 2, 2, 2, 2, 2 ] 1 5
x^10 + t*x^8 + t^4*x^6 + (t^4 + t^5)*x^5 + (t^6 + t^7)*x^3 + (t^4 + t^6 + t^7 + t^8)*x^2 + (t^6 + t^7 + t^9)*x + t^5 + t^8 + t^10 2 5 10 C2 × (C24 : C5) (10T14) 160 16 [ 2, 2, 2, 2 ] 1 10
x^10 + t*x^9 + t*x^8 + (t + t^4)*x^6 + (t^2 + t^5)*x^5 + (1 + t + t^2 + t^5)*x^4 + t^4*x^3 + (t^2 + t^3 + t^5)*x^2 + t^5*x + 1 + t^3 + t^5 2 5 10 C10 (10T1) 10 2 [ 2 ] 1 5
x^10 + t*x^8 + t^3*x^7 + t^5*x^5 + t^5*x^3 + t^4*x^2 + t^5 2 5 10 C24 : C5 (10T8) 80 16 [ 2, 2, 2, 2 ] 1 5
x^10 + t*x^9 + (t + t^2)*x^8 + t^3*x^7 + (t^3 + t^4)*x^6 + (t^4 + t^6)*x^4 + (t^5 + t^6 + t^7)*x^3 + (t^4 + t^5 + t^6 + t^7)*x^2 + (t^5 + t^9)*x + t^5 + t^8 + t^10 2 5 10 C2 × (C24 : C5) (10T14) 160 16 [ 2, 2, 2, 2 ] 1 10
x^10 + (t + t^2)*x^8 + t^3*x^6 + t^5*x^5 + t^4*x^4 + (t^4 + t^5)*x^2 + t^5 2 5 10 C24 : C5 (10T8) 80 16 [ 2, 2, 2, 2 ] 1 5
x^10 + t*x^9 + (t + t^2)*x^8 + t^3*x^7 + t^3*x^6 + t^5*x^5 + t^4*x^4 + t^5*x^3 + (t^4 + t^5)*x^2 + t^5*x + t^5 2 5 10 C24 : C5 (10T8) 80 16 [ 2, 2, 2, 2 ] 1 5
x^10 + t*x^9 + t*x^8 + t^3*x^7 + t^4*x^6 + t^5*x^5 + t^5*x^4 + t^5*x^3 + t^4*x^2 + t^5*x + t^5 2 5 10 C2 × (C24 : C5) (10T14) 160 32 [ 2, 2, 2, 2, 2 ] 1 5
x^10 + (t + t^2)*x^8 + t^3*x^7 + t^3*x^6 + t^4*x^5 + t^4*x^4 + (t^5 + t^6 + t^7)*x^3 + (t^4 + t^5 + t^8)*x^2 + (t^6 + t^8 + t^9)*x + t^5 + t^8 + t^10 2 5 10 C2 × (C24 : C5) (10T14) 160 16 [ 2, 2, 2, 2 ] 1 10
x^10 + t*x^9 + (t + t^2)*x^8 + (t + t^4)*x^6 + (t^2 + t^5)*x^5 + (1 + t + t^2 + t^3 + t^5 + t^6)*x^4 + t^4*x^3 + (t^2 + t^3 + t^5 + t^8)*x^2 + (t^5 + t^6 + t^9)*x + 1 + t^3 + t^4 + t^9 + t^10 2 5 10 C10 (10T1) 10 2 [ 2 ] 1 5
x^10 + (t + t^2)*x^8 + t^3*x^7 + (t^3 + t^4)*x^6 + t^5*x^5 + (t^4 + t^5)*x^4 + t^5*x^3 + (t^4 + t^5)*x^2 + t^5 2 5 10 C2 × (C24 : C5) (10T14) 160 32 [ 2, 2, 2, 2, 2 ] 1 5
x^10 + t*x^9 + (t + t^2)*x^8 + t^3*x^7 + t^4*x^6 + t^5*x^5 + (t^5 + t^6)*x^4 + (t^5 + t^7)*x^3 + (t^4 + t^8)*x^2 + (t^5 + t^9)*x + t^5 + t^6 + t^9 + t^10 2 5 10 C2 × (C24 : C5) (10T14) 160 32 [ 2, 2, 2, 2, 2 ] 1 5
x^10 + t*x^8 + t^3*x^7 + t^3*x^6 + t^5*x^5 + (t^4 + t^5)*x^4 + (t^5 + t^7)*x^3 + (t^4 + t^5 + t^6 + t^7)*x^2 + t^5 + t^6 + t^9 + t^10 2 5 10 C2 × (C24 : C5) (10T14) 160 32 [ 2, 2, 2, 2, 2 ] 1 5
x^10 + t*x^9 + (t + t^2)*x^8 + (t^3 + t^4)*x^6 + t^5*x^5 + (t^4 + t^5)*x^4 + (t^4 + t^5)*x^2 + t^5*x + t^5 2 5 10 C2 × (C24 : C5) (10T14) 160 32 [ 2, 2, 2, 2, 2 ] 1 5
x^10 + t*x + t 10 1 10 (C24 : C5) : C4 (10T24) 320 80 [ 6/5, 6/5, 6/5, 6/5 ] 5 4
x^10 + t*x^2 + t*x + t 10 1 10 (C24 : C5) : C4 (10T25) 320 80 [ 6/5, 6/5, 6/5, 6/5 ] 5 4
x^10 + t*x^6 + t*x^3 + t 10 1 12 (C24 : C5) : C4 (10T24) 320 80 [ 8/5, 8/5, 8/5, 8/5 ] 5 4
x^10 + t*x^6 + t*x^5 + t*x^3 + t 10 1 12 (C24 : C5) : C4 (10T24) 320 80 [ 8/5, 8/5, 8/5, 8/5 ] 5 4
x^10 + t*x^3 + t 10 1 12 (C24 : C5) : C4 (10T25) 320 80 [ 8/5, 8/5, 8/5, 8/5 ] 5 4
x^10 + t*x^5 + t*x^3 + t 10 1 12 (C24 : C5) : C4 (10T25) 320 80 [ 8/5, 8/5, 8/5, 8/5 ] 5 4
x^10 + t*x^8 + t*x^7 + t*x^6 + t*x^5 + t*x^4 + t 10 1 14 F5 × C2 (10T5) 40 10 [ 2 ] 5 4
x^10 + t*x^9 + t*x^5 + t 10 1 14 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 6/5, 6/5, 6/5, 6/5, 2 ] 5 4
x^10 + t*x^9 + t*x^8 + t*x^7 + t*x^5 + t 10 1 14 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 8/5, 8/5, 8/5, 8/5, 2 ] 5 4
x^10 + t*x^8 + t*x^7 + t*x^5 + t 10 1 14 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 8/5, 8/5, 8/5, 8/5, 2 ] 5 4
x^10 + t*x^5 + t + t^2 10 1 14 F5 × C2 (10T5) 40 10 [ 2 ] 5 4
x^10 + t*x^9 + t*x^7 + t*x^5 + t*x^4 + t + t^2 10 1 14 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 6/5, 6/5, 6/5, 6/5, 2 ] 5 4
x^10 + t*x^9 + t*x^8 + t*x^6 + t*x^5 + t*x^4 + t 10 1 14 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 8/5, 8/5, 8/5, 8/5, 2 ] 5 4
x^10 + t*x^5 + t*x^4 + t + t^2 10 1 14 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 8/5, 8/5, 8/5, 8/5, 2 ] 5 4
x^10 + t*x^7 + t^2*x^4 + t^2*x^2 + t 10 1 16 (C24 : C5) : C4 (10T24) 320 80 [ 12/5, 12/5, 12/5, 12/5 ] 5 4
x^10 + t*x^8 + t*x^7 + t*x^6 + t*x^4 + t^2*x^3 + t^2*x^2 + t^2*x + t 10 1 16 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 12/5, 12/5, 12/5, 12/5 ] 5 4
x^10 + t*x^9 + t*x^7 + t^2*x^3 + t + t^2 10 1 16 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 12/5, 12/5, 12/5, 12/5 ] 5 4
x^10 + t*x^8 + t*x^7 + t*x^4 + t^2*x^3 + t^2*x^2 + t 10 1 16 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 12/5, 12/5, 12/5, 12/5 ] 5 4
x^10 + t*x^8 + t*x^7 + t*x^6 + (t + t^2)*x^4 + t^2*x^3 + t^2*x + t + t^2 10 1 16 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 12/5, 12/5, 12/5, 12/5 ] 5 4
x^10 + t*x^9 + t*x^8 + t*x^7 + t^2*x^4 + t^2*x^3 + t^2*x + t 10 1 16 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 12/5, 12/5, 12/5, 12/5 ] 5 4
x^10 + t*x^8 + t*x^7 + t*x^6 + t*x^4 + t^2*x^2 + t 10 1 16 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 12/5, 12/5, 12/5, 12/5 ] 5 4
x^10 + t*x^9 + t*x^8 + t*x^7 + t^2*x^4 + t^2*x^2 + t 10 1 16 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 12/5, 12/5, 12/5, 12/5 ] 5 4
x^10 + t*x^8 + t*x^7 + t^2*x^3 + t^2*x^2 + t + t^2 10 1 16 (C24 : C5) : C4 (10T24) 320 80 [ 12/5, 12/5, 12/5, 12/5 ] 5 4
x^10 + t*x^7 + t^2*x^4 + t^2*x^2 + t + t^2 10 1 16 (C24 : C5) : C4 (10T24) 320 80 [ 12/5, 12/5, 12/5, 12/5 ] 5 4
x^10 + t*x^9 + t*x^8 + t*x^7 + t*x^4 + t^2*x^2 + t^2*x + t 10 1 16 (C24 : C5) : C4 (10T24) 320 80 [ 12/5, 12/5, 12/5, 12/5 ] 5 4
x^10 + t*x^9 + t*x^8 + t*x^7 + t*x^6 + t*x^4 + t^2*x^3 + t^2*x^2 + t 10 1 16 (C24 : C5) : C4 (10T25) 320 80 [ 12/5, 12/5, 12/5, 12/5 ] 5 4
x^10 + t*x^8 + t*x^7 + t^2*x^4 + t^2*x + t 10 1 16 (C24 : C5) : C4 (10T25) 320 80 [ 12/5, 12/5, 12/5, 12/5 ] 5 4
x^10 + t*x^9 + t*x^8 + t*x^7 + t*x^6 + t*x^4 + t^2*x^3 + t^2*x + t + t^2 10 1 16 (C24 : C5) : C4 (10T25) 320 80 [ 12/5, 12/5, 12/5, 12/5 ] 5 4
x^10 + t*x^9 + t*x^8 + t*x^7 + t*x^4 + t^2*x^2 + t^2*x + t + t^2 10 1 16 (C24 : C5) : C4 (10T25) 320 80 [ 12/5, 12/5, 12/5, 12/5 ] 5 4
x^10 + t*x^8 + t*x^7 + t*x^6 + (t + t^2)*x^4 + t + t^2 10 1 16 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 12/5, 12/5, 12/5, 12/5 ] 5 4
x^10 + t*x^9 + (t + t^2)*x^8 + t^2*x^7 + t^2*x^6 + t^2*x + t 10 1 18 (C24 : C5) : C4 (10T24) 320 80 [ 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t*x^8 + t^2*x^6 + t^2*x + t 10 1 18 (C24 : C5) : C4 (10T25) 320 80 [ 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t^2*x^8 + t^2*x^7 + t^2*x^6 + t^2*x^4 + t^2*x^3 + t 10 1 18 (C24 : C5) : C4 (10T25) 320 80 [ 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t^2*x^8 + t^2*x^7 + t^2*x^6 + t^2*x^5 + t^2*x^4 + t^2*x^3 + t 10 1 18 (C24 : C5) : C4 (10T25) 320 80 [ 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t*x^8 + t^2*x^7 + t^2*x^6 + t^2*x + t 10 1 18 (C24 : C5) : C4 (10T25) 320 80 [ 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t*x^8 + t^2*x^7 + t^2*x^6 + t^2*x^5 + t^2*x + t 10 1 18 (C24 : C5) : C4 (10T25) 320 80 [ 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t^2*x^8 + t^2*x^6 + t^2*x^5 + t^2*x^4 + t^2*x^3 + t 10 1 18 (C24 : C5) : C4 (10T25) 320 80 [ 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t*x^8 + t^2*x^6 + t^2*x^5 + t^2*x + t 10 1 18 (C24 : C5) : C4 (10T25) 320 80 [ 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t*x^8 + t^2*x^7 + t^2*x^6 + t^2*x^3 + t^2*x^2 + t^2*x + t 10 1 18 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t^2*x^8 + t^2*x^4 + t 10 1 18 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t*x^8 + t^2*x^7 + t^2*x^6 + t^2*x^5 + t^2*x^3 + t^2*x^2 + t^2*x + t 10 1 18 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + (t + t^2)*x^8 + t^2*x^7 + t^2*x^6 + t^2*x^5 + t^2*x + t 10 1 18 (C24 : C5) : C4 (10T24) 320 80 [ 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t*x^8 + t^2*x^6 + t^2*x^3 + t^2*x^2 + t^2*x + t 10 1 18 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t^2*x^8 + t^2*x^5 + t^2*x^4 + t 10 1 18 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t^2*x^8 + t^2*x^7 + t^2*x^4 + t 10 1 18 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t^2*x^8 + t^2*x^7 + t^2*x^5 + t^2*x^4 + t 10 1 18 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t*x^8 + t^2*x^6 + t^2*x^5 + t^2*x^3 + t^2*x^2 + t^2*x + t 10 1 18 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + (t + t^2)*x^8 + t^2*x^7 + t^2*x^6 + t^2*x^5 + t^2*x^3 + t^2*x^2 + t^2*x + t 10 1 18 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t^2*x^7 + t^2*x^5 + t^2*x^4 + t 10 1 18 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + (t + t^2)*x^8 + t^2*x^6 + t^2*x^5 + t^2*x^3 + t^2*x^2 + t^2*x + t 10 1 18 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t^2*x^5 + t^2*x^4 + t 10 1 18 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t^2*x^7 + t^2*x^4 + t 10 1 18 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t^2*x^7 + t^2*x^6 + t^2*x^5 + t^2*x^4 + t^2*x^3 + t 10 1 18 (C24 : C5) : C4 (10T24) 320 80 [ 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + (t + t^2)*x^8 + t^2*x^7 + t^2*x^6 + t^2*x^3 + t^2*x^2 + t^2*x + t 10 1 18 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + (t + t^2)*x^8 + t^2*x^6 + t^2*x^3 + t^2*x^2 + t^2*x + t 10 1 18 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t^2*x^4 + t 10 1 18 ((C24 : C5) : C4) × C2 (10T29) 640 160 [ 2, 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t^2*x^6 + t^2*x^4 + t^2*x^3 + t 10 1 18 (C24 : C5) : C4 (10T24) 320 80 [ 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t^2*x^6 + t^2*x^5 + t^2*x^4 + t^2*x^3 + t 10 1 18 (C24 : C5) : C4 (10T24) 320 80 [ 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + (t + t^2)*x^8 + t^2*x^6 + t^2*x + t 10 1 18 (C24 : C5) : C4 (10T24) 320 80 [ 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t^2*x^7 + t^2*x^6 + t^2*x^4 + t^2*x^3 + t 10 1 18 (C24 : C5) : C4 (10T24) 320 80 [ 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + (t + t^2)*x^8 + t^2*x^6 + t^2*x^5 + t^2*x + t 10 1 18 (C24 : C5) : C4 (10T24) 320 80 [ 14/5, 14/5, 14/5, 14/5 ] 5 4
x^10 + t*x^9 + t^2*x^8 + t^2*x^6 + t^2*x^4 + t^2*x^3 + t 10 1 18 (C24 : C5) : C4 (10T25) 320 80 [ 14/5, 14/5, 14/5, 14/5 ] 5 4
inserted by FC2 system