↑

Properties

Label 2.1.4.8b
Base 2.1.1.0a1.1
Degree \(4\)
e \(4\)
f \(1\)
c \(8\)

Related objects

Downloads

Learn more

Defining polynomial

$x^{4} + 4 b_{7} x^{3} + 2 a_{2} x^{2} + 4 a_{5} x + 4 c_{4} + 8 c_{8} + 2$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: $\Q_{2}$
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
Artin slopes: $[2,3]$
Swan slopes: $[1,2]$
Means: $\langle\frac{1}{2},\frac{5}{4}\rangle$
Rams: $(1,3)$
Field count: $6$ (complete)
Ambiguity: $4$
Mass: $2$
Absolute Mass: $2$

Diagrams

Varying

Indices of inseparability: $[5,2,0]$
Associated inertia: $[1,1]$
Jump Set: $[1,2,4]$ (show 1), $[1,2,8]$ (show 3), $[1,7,11]$ (show 1), $[1,8,12]$ (show 1)

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
Sort order Select

Showing all 6

  displayed columns for results
Label Packet size Polynomial Galois group Galois degree $\#\Aut(K/\Q_p)$ Artin slope content Swan slope content Hidden Artin slopes Hidden Swan slopes Ind. of Insep. Assoc. Inertia Resid. Poly Jump Set
2.1.4.8b1.1 4 $x^{4} + 2 x^{2} + 4 x + 2$ $C_2^2$ (as 4T2) $4$ $4$ $[2, 3]$ $[1,2]$ $[\ ]$ $[\ ]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 4]$
2.1.4.8b1.2 4 $x^{4} + 2 x^{2} + 4 x + 10$ $C_2^2$ (as 4T2) $4$ $4$ $[2, 3]$ $[1,2]$ $[\ ]$ $[\ ]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 8, 12]$
2.1.4.8b1.3 2 $x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 2$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3]^{2}$ $[1,2]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[5, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 7, 11]$
2.1.4.8b1.4 2 $x^{4} + 2 x^{2} + 4 x + 6$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3]^{2}$ $[1,2]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[5, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 8]$
2.1.4.8b1.5 4 $x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 6$ $C_2^2$ (as 4T2) $4$ $4$ $[2, 3]$ $[1,2]$ $[\ ]$ $[\ ]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 8]$
2.1.4.8b1.6 4 $x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 14$ $C_2^2$ (as 4T2) $4$ $4$ $[2, 3]$ $[1,2]$ $[\ ]$ $[\ ]$ $[5, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 8]$
  displayed columns for results