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Degree $n$ Residue characteristic $p$ Discriminant exponent $c$ Galois group Num. automorphisms
Ramification index $e$ Residue field degree $f$ Galois tame degree $t$ Galois unram. degree $u$ Intermediate field
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Results (3 matches)

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Label Polynomial $p$ $f$ $e$ $c$ Galois group $u$ $t$ Visible Artin slopes Visible Swan slopes Artin slope content Swan slope content Hidden Artin slopes Hidden Swan slopes Unram. Ext. Eisen. Poly. Ind. of Insep. Assoc. Inertia Resid. Poly Jump Set
3.3.3.12a7.1 $( x^{3} + 2 x + 1 )^{3} + \left(3 x^{2} + 6 x + 6\right) ( x^{3} + 2 x + 1 )^{2} + 3$ $3$ $3$ $3$ $12$ $C_9:C_3$ (as 9T6) $3$ $1$ $[2]$ $[1]$ $[2, 2]^{3}$ $[1,1]^{3}$ $[2]$ $[1]$ $t^{3} + 2 t + 1$ $x^{3} + \left(3 t^{2} + 6 t + 6\right) x^{2} + 18 t + 3$ $[2, 0]$ $[1]$ $z^2 + (t^2 + 2 t + 2)$ undefined
3.3.3.12a7.2 $( x^{3} + 2 x + 1 )^{3} + \left(3 x^{2} + 6 x + 6\right) ( x^{3} + 2 x + 1 )^{2} + 9 x + 3$ $3$ $3$ $3$ $12$ $C_3^2:C_3$ (as 9T7) $3$ $1$ $[2]$ $[1]$ $[2, 2]^{3}$ $[1,1]^{3}$ $[2]$ $[1]$ $t^{3} + 2 t + 1$ $x^{3} + \left(3 t^{2} + 3 t + 6\right) x^{2} + 3$ $[2, 0]$ $[1]$ $z^2 + (t^2 + 2 t + 2)$ undefined
3.3.3.12a7.3 $( x^{3} + 2 x + 1 )^{3} + \left(3 x^{2} + 6 x + 6\right) ( x^{3} + 2 x + 1 )^{2} + 18 x + 3$ $3$ $3$ $3$ $12$ $C_9:C_3$ (as 9T6) $3$ $1$ $[2]$ $[1]$ $[2, 2]^{3}$ $[1,1]^{3}$ $[2]$ $[1]$ $t^{3} + 2 t + 1$ $x^{3} + \left(3 t^{2} + 3\right) x^{2} + 9 t + 3$ $[2, 0]$ $[1]$ $z^2 + (t^2 + 2 t + 2)$ undefined
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