↑

Properties

Label 2.1.2.3a1.2-3.2.15a
Base 2.1.2.3a1.2
Degree \(6\)
e \(2\)
f \(3\)
c \(15\)

Related objects

Downloads

Learn more

Defining polynomial over unramified subextension

$x^{2} + \left(b_{7} \pi^{4} + b_{5} \pi^{3}\right) x + c_{8} \pi^{5} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $6$
Base field: $\Q_{2}(\sqrt{-2\cdot 5})$
Ramification index $e$: $2$
Residue field degree $f$: $3$
Discriminant exponent $c$: $15$
Absolute Artin slopes: $[3,4]$
Swan slopes: $[4]$
Means: $\langle2\rangle$
Rams: $(4)$
Field count: $24$ (complete)
Ambiguity: $6$
Mass: $64$
Absolute Mass: $32/3$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.

Galois group: $D_4 \times C_3$ (show 4), $D_4\times A_4$ (show 4), $C_2\wr C_6$ (show 16)
Hidden Artin slopes: $[2,2]^{2}$ (show 2), $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ (show 8), $[2,2,2,\frac{7}{2},\frac{7}{2}]$ (show 8), $[2]$ (show 2), $[2,2,2]$ (show 2), $[\ ]^{2}$ (show 2)
Indices of inseparability: $[8,4,0]$
Associated inertia: $[1,1]$
Jump Set: $[1,3,7]$

Fields

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

Showing all 24

  displayed columns for results
Label Polynomial $/ \Q_p$ Galois group $/ \Q_p$ Galois degree $/ \Q_p$ $\#\Aut(K/\Q_p)$ Artin slope content $/ \Q_p$ Swan slope content $/ \Q_p$ Hidden Artin slopes $/ \Q_p$ Hidden Swan slopes $/ \Q_p$ Ind. of Insep. $/ \Q_p$ Assoc. Inertia $/ \Q_p$ Resid. Poly Jump Set
2.3.4.33a1.25 $( x^{3} + x + 1 )^{4} + 10$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[2, 3, 4]^{3}$ $[1,2,3]^{3}$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.26 $( x^{3} + x + 1 )^{4} + 26$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[2, 3, 4]^{3}$ $[1,2,3]^{3}$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.27 $( x^{3} + x + 1 )^{4} + 8 x ( x^{3} + x + 1 )^{3} + 10$ $D_4\times A_4$ (as 12T51) $96$ $2$ $[2, 2, 2, 3, 4]^{3}$ $[1,1,1,2,3]^{3}$ $[2,2,2]$ $[1,1,1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.28 $( x^{3} + x + 1 )^{4} + 8 x ( x^{3} + x + 1 )^{3} + 26$ $D_4\times A_4$ (as 12T51) $96$ $2$ $[2, 2, 2, 3, 4]^{3}$ $[1,1,1,2,3]^{3}$ $[2,2,2]$ $[1,1,1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.29 $( x^{3} + x + 1 )^{4} + 8 x^{2} ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.30 $( x^{3} + x + 1 )^{4} + 8 x ( x^{3} + x + 1 )^{3} + 8 x^{2} ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.31 $( x^{3} + x + 1 )^{4} + 8 ( x^{3} + x + 1 )^{3} + 8 x^{2} ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.32 $( x^{3} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{3} + x + 1 )^{3} + 8 x^{2} ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.33 $( x^{3} + x + 1 )^{4} + 8 x ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.34 $( x^{3} + x + 1 )^{4} + 8 x^{2} ( x^{3} + x + 1 )^{3} + 8 x ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.35 $( x^{3} + x + 1 )^{4} + 8 ( x^{3} + x + 1 )^{3} + 8 x ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.36 $( x^{3} + x + 1 )^{4} + \left(8 x^{2} + 8\right) ( x^{3} + x + 1 )^{3} + 8 x ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.37 $( x^{3} + x + 1 )^{4} + 8 ( x^{3} + x + 1 ) + 10$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[3, 4]^{6}$ $[2,3]^{6}$ $[\ ]^{2}$ $[\ ]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.38 $( x^{3} + x + 1 )^{4} + 8 x ( x^{3} + x + 1 )^{3} + 8 ( x^{3} + x + 1 ) + 10$ $D_4\times A_4$ (as 12T51) $96$ $2$ $[2, 2, 3, 4]^{6}$ $[1,1,2,3]^{6}$ $[2,2]^{2}$ $[1,1]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.39 $( x^{3} + x + 1 )^{4} + 8 ( x^{3} + x + 1 )^{3} + 8 ( x^{3} + x + 1 ) + 10$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[3, 4]^{6}$ $[2,3]^{6}$ $[\ ]^{2}$ $[\ ]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.40 $( x^{3} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{3} + x + 1 )^{3} + 8 ( x^{3} + x + 1 ) + 10$ $D_4\times A_4$ (as 12T51) $96$ $2$ $[2, 2, 3, 4]^{6}$ $[1,1,2,3]^{6}$ $[2,2]^{2}$ $[1,1]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.41 $( x^{3} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{6}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{6}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.42 $( x^{3} + x + 1 )^{4} + 8 x^{2} ( x^{3} + x + 1 )^{3} + \left(8 x + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{6}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{6}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.43 $( x^{3} + x + 1 )^{4} + 8 x ( x^{3} + x + 1 )^{3} + \left(8 x + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{6}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{6}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.44 $( x^{3} + x + 1 )^{4} + \left(8 x^{2} + 8 x\right) ( x^{3} + x + 1 )^{3} + \left(8 x + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{6}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{6}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.45 $( x^{3} + x + 1 )^{4} + 8 ( x^{3} + x + 1 )^{3} + \left(8 x + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{6}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{6}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.46 $( x^{3} + x + 1 )^{4} + \left(8 x^{2} + 8\right) ( x^{3} + x + 1 )^{3} + \left(8 x + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{6}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{6}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.47 $( x^{3} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{3} + x + 1 )^{3} + \left(8 x + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{6}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{6}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
2.3.4.33a1.48 $( x^{3} + x + 1 )^{4} + \left(8 x^{2} + 8 x + 8\right) ( x^{3} + x + 1 )^{3} + \left(8 x + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{6}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3]^{6}$ $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2},\frac{5}{2}]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t^2 + 1)$ $[1, 3, 7]$
  displayed columns for results