↑

Properties

Label 5.2.5.18a
Base 5.1.1.0a1.1
Degree \(10\)
e \(5\)
f \(2\)
c \(18\)

Related objects

Downloads

Learn more

Defining polynomial over unramified subextension

$x^{5} + 25b_{6} x + 5$

Invariants

Residue field characteristic: $5$
Degree: $10$
Base field: $\Q_{5}$
Ramification index $e$: $5$
Residue field degree $f$: $2$
Discriminant exponent $c$: $18$
Artin slopes: $[\frac{9}{4}]$
Swan slopes: $[\frac{5}{4}]$
Means: $\langle1\rangle$
Rams: $(\frac{5}{4})$
Field count: $15$ (complete)
Ambiguity: $2$
Mass: $25$
Absolute Mass: $25/2$

Diagrams

Varying

Indices of inseparability: $[5,0]$
Associated inertia: $[1]$
Jump Set: undefined

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 15

  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
5.2.5.18a1.1 5 $( x^{2} + 4 x + 2 )^{5} + 5$ $F_{5}\times C_2$ (as 10T5) $40$ $2$ $[\frac{9}{4}]_{4}^{2}$ $[\frac{5}{4}]_{4}^{2}$ $[\ ]_{4}$ $[\ ]_{4}$ $[5, 0]$ $[1]$ $z + (3 t + 1)$ undefined
5.2.5.18a1.2 10 $( x^{2} + 4 x + 2 )^{5} + 25 x ( x^{2} + 4 x + 2 ) + 5$ $(C_5^2 : C_4) : C_2$ (as 10T17) $200$ $1$ $[\frac{5}{4}, \frac{9}{4}]_{4}^{2}$ $[\frac{1}{4},\frac{5}{4}]_{4}^{2}$ $[\frac{5}{4}]_{4}$ $[\frac{1}{4}]_{4}$ $[5, 0]$ $[1]$ $z + (3 t + 1)$ undefined
5.2.5.18a1.3 10 $( x^{2} + 4 x + 2 )^{5} + 50 x ( x^{2} + 4 x + 2 ) + 5$ $(C_5^2 : C_4) : C_2$ (as 10T17) $200$ $1$ $[\frac{5}{4}, \frac{9}{4}]_{4}^{2}$ $[\frac{1}{4},\frac{5}{4}]_{4}^{2}$ $[\frac{5}{4}]_{4}$ $[\frac{1}{4}]_{4}$ $[5, 0]$ $[1]$ $z + (3 t + 1)$ undefined
5.2.5.18a1.4 5 $( x^{2} + 4 x + 2 )^{5} + 25 ( x^{2} + 4 x + 2 ) + 5$ $F_{5}\times C_2$ (as 10T5) $40$ $2$ $[\frac{9}{4}]_{4}^{2}$ $[\frac{5}{4}]_{4}^{2}$ $[\ ]_{4}$ $[\ ]_{4}$ $[5, 0]$ $[1]$ $z + (3 t + 1)$ undefined
5.2.5.18a1.5 10 $( x^{2} + 4 x + 2 )^{5} + \left(25 x + 25\right) ( x^{2} + 4 x + 2 ) + 5$ $(C_5^2 : C_4) : C_2$ (as 10T17) $200$ $1$ $[\frac{5}{4}, \frac{9}{4}]_{4}^{2}$ $[\frac{1}{4},\frac{5}{4}]_{4}^{2}$ $[\frac{5}{4}]_{4}$ $[\frac{1}{4}]_{4}$ $[5, 0]$ $[1]$ $z + (3 t + 1)$ undefined
5.2.5.18a1.6 10 $( x^{2} + 4 x + 2 )^{5} + \left(50 x + 25\right) ( x^{2} + 4 x + 2 ) + 5$ $(C_5^2 : C_4) : C_2$ (as 10T17) $200$ $1$ $[\frac{5}{4}, \frac{9}{4}]_{4}^{2}$ $[\frac{1}{4},\frac{5}{4}]_{4}^{2}$ $[\frac{5}{4}]_{4}$ $[\frac{1}{4}]_{4}$ $[5, 0]$ $[1]$ $z + (3 t + 1)$ undefined
5.2.5.18a1.7 5 $( x^{2} + 4 x + 2 )^{5} + 50 ( x^{2} + 4 x + 2 ) + 5$ $F_{5}\times C_2$ (as 10T5) $40$ $2$ $[\frac{9}{4}]_{4}^{2}$ $[\frac{5}{4}]_{4}^{2}$ $[\ ]_{4}$ $[\ ]_{4}$ $[5, 0]$ $[1]$ $z + (3 t + 1)$ undefined
5.2.5.18a1.8 10 $( x^{2} + 4 x + 2 )^{5} + \left(25 x + 50\right) ( x^{2} + 4 x + 2 ) + 5$ $(C_5^2 : C_4) : C_2$ (as 10T17) $200$ $1$ $[\frac{5}{4}, \frac{9}{4}]_{4}^{2}$ $[\frac{1}{4},\frac{5}{4}]_{4}^{2}$ $[\frac{5}{4}]_{4}$ $[\frac{1}{4}]_{4}$ $[5, 0]$ $[1]$ $z + (3 t + 1)$ undefined
5.2.5.18a1.9 10 $( x^{2} + 4 x + 2 )^{5} + \left(50 x + 50\right) ( x^{2} + 4 x + 2 ) + 5$ $(C_5^2 : C_4) : C_2$ (as 10T17) $200$ $1$ $[\frac{5}{4}, \frac{9}{4}]_{4}^{2}$ $[\frac{1}{4},\frac{5}{4}]_{4}^{2}$ $[\frac{5}{4}]_{4}$ $[\frac{1}{4}]_{4}$ $[5, 0]$ $[1]$ $z + (3 t + 1)$ undefined
5.2.5.18a1.10 5 $( x^{2} + 4 x + 2 )^{5} + 75 ( x^{2} + 4 x + 2 ) + 5$ $F_{5}\times C_2$ (as 10T5) $40$ $2$ $[\frac{9}{4}]_{4}^{2}$ $[\frac{5}{4}]_{4}^{2}$ $[\ ]_{4}$ $[\ ]_{4}$ $[5, 0]$ $[1]$ $z + (3 t + 1)$ undefined
5.2.5.18a1.11 10 $( x^{2} + 4 x + 2 )^{5} + \left(25 x + 75\right) ( x^{2} + 4 x + 2 ) + 5$ $(C_5^2 : C_4) : C_2$ (as 10T17) $200$ $1$ $[\frac{5}{4}, \frac{9}{4}]_{4}^{2}$ $[\frac{1}{4},\frac{5}{4}]_{4}^{2}$ $[\frac{5}{4}]_{4}$ $[\frac{1}{4}]_{4}$ $[5, 0]$ $[1]$ $z + (3 t + 1)$ undefined
5.2.5.18a1.12 10 $( x^{2} + 4 x + 2 )^{5} + \left(50 x + 75\right) ( x^{2} + 4 x + 2 ) + 5$ $(C_5^2 : C_4) : C_2$ (as 10T17) $200$ $1$ $[\frac{5}{4}, \frac{9}{4}]_{4}^{2}$ $[\frac{1}{4},\frac{5}{4}]_{4}^{2}$ $[\frac{5}{4}]_{4}$ $[\frac{1}{4}]_{4}$ $[5, 0]$ $[1]$ $z + (3 t + 1)$ undefined
5.2.5.18a1.13 5 $( x^{2} + 4 x + 2 )^{5} + 100 ( x^{2} + 4 x + 2 ) + 5$ $F_{5}\times C_2$ (as 10T5) $40$ $2$ $[\frac{9}{4}]_{4}^{2}$ $[\frac{5}{4}]_{4}^{2}$ $[\ ]_{4}$ $[\ ]_{4}$ $[5, 0]$ $[1]$ $z + (3 t + 1)$ undefined
5.2.5.18a1.14 10 $( x^{2} + 4 x + 2 )^{5} + \left(25 x + 100\right) ( x^{2} + 4 x + 2 ) + 5$ $(C_5^2 : C_4) : C_2$ (as 10T17) $200$ $1$ $[\frac{5}{4}, \frac{9}{4}]_{4}^{2}$ $[\frac{1}{4},\frac{5}{4}]_{4}^{2}$ $[\frac{5}{4}]_{4}$ $[\frac{1}{4}]_{4}$ $[5, 0]$ $[1]$ $z + (3 t + 1)$ undefined
5.2.5.18a1.15 10 $( x^{2} + 4 x + 2 )^{5} + \left(50 x + 100\right) ( x^{2} + 4 x + 2 ) + 5$ $(C_5^2 : C_4) : C_2$ (as 10T17) $200$ $1$ $[\frac{5}{4}, \frac{9}{4}]_{4}^{2}$ $[\frac{1}{4},\frac{5}{4}]_{4}^{2}$ $[\frac{5}{4}]_{4}$ $[\frac{1}{4}]_{4}$ $[5, 0]$ $[1]$ $z + (3 t + 1)$ undefined
  displayed columns for results