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Properties

Label 2.1.2.3a1.2-3.2.12a
Base 2.1.2.3a1.2
Degree \(6\)
e \(2\)
f \(3\)
c \(12\)

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Defining polynomial over unramified subextension

$x^{2} + \left(b_{5} \pi^{3} + a_{3} \pi^{2}\right) x + c_{6} \pi^{4} + \pi$

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 $c$: $12$
Absolute Artin slopes: $[3,\frac{7}{2}]$
Swan slopes: $[3]$
Means: $\langle\frac{3}{2}\rangle$
Rams: $(3)$
Field count: $20$ (complete)
Ambiguity: $6$
Mass: $56$
Absolute Mass: $28/3$

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 2), $D_4\times A_4$ (show 2), $C_2^3:A_4$ (show 2), $C_2^4:A_4$ (show 6), $C_2\wr C_6$ (show 8)
Hidden Artin slopes: $[2,2,2,\frac{7}{2},\frac{7}{2}]$ (show 8), $[2,2,\frac{7}{2}]^{2}$ (show 2), $[2,2,2]$ (show 2), $[2,2,2,\frac{7}{2}]$ (show 4), $[2]$ (show 2), $[2,2,\frac{7}{2}]$ (show 2)
Indices of inseparability: $[7,4,0]$
Associated inertia: $[1,1]$
Jump Set: $[1,3,7]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 20

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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
2.3.4.30a1.29 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 10$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[2, 3, \frac{7}{2}]^{3}$ $[1,2,\frac{5}{2}]^{3}$ $[2]$ $[1]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + 1$ $[1, 3, 7]$
2.3.4.30a1.30 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{3} + 12 ( x^{3} + x + 1 )^{2} + 10$ $D_4 \times C_3$ (as 12T14) $24$ $6$ $[2, 3, \frac{7}{2}]^{3}$ $[1,2,\frac{5}{2}]^{3}$ $[2]$ $[1]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + 1$ $[1, 3, 7]$
2.3.4.30a1.31 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 x ( x^{3} + x + 1 ) + 10$ $D_4\times A_4$ (as 12T51) $96$ $2$ $[2, 2, 2, 3, \frac{7}{2}]^{3}$ $[1,1,1,2,\frac{5}{2}]^{3}$ $[2,2,2]$ $[1,1,1]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + 1$ $[1, 3, 7]$
2.3.4.30a1.32 $( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{3} + 12 ( x^{3} + x + 1 )^{2} + 8 x ( x^{3} + x + 1 ) + 10$ $D_4\times A_4$ (as 12T51) $96$ $2$ $[2, 2, 2, 3, \frac{7}{2}]^{3}$ $[1,1,1,2,\frac{5}{2}]^{3}$ $[2,2,2]$ $[1,1,1]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + 1$ $[1, 3, 7]$
2.3.4.30a2.25 $( x^{3} + x + 1 )^{4} + 4 x ( x^{3} + x + 1 )^{3} + 4 x^{2} ( x^{3} + x + 1 )^{2} + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2}]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + t$ $[1, 3, 7]$
2.3.4.30a2.26 $( x^{3} + x + 1 )^{4} + 4 x ( x^{3} + x + 1 )^{3} + \left(4 x^{2} + 8\right) ( x^{3} + x + 1 )^{2} + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2}]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + t$ $[1, 3, 7]$
2.3.4.30a2.27 $( x^{3} + x + 1 )^{4} + 4 x ( x^{3} + x + 1 )^{3} + 4 x^{2} ( x^{3} + x + 1 )^{2} + 8 x^{2} ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2}]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + t$ $[1, 3, 7]$
2.3.4.30a2.28 $( x^{3} + x + 1 )^{4} + 4 x ( x^{3} + x + 1 )^{3} + \left(4 x^{2} + 8\right) ( x^{3} + x + 1 )^{2} + 8 x^{2} ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2}]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + t$ $[1, 3, 7]$
2.3.4.30a2.29 $( x^{3} + x + 1 )^{4} + 4 x ( x^{3} + x + 1 )^{3} + 4 x^{2} ( x^{3} + x + 1 )^{2} + 8 ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2}]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + t$ $[1, 3, 7]$
2.3.4.30a2.30 $( x^{3} + x + 1 )^{4} + 4 x ( x^{3} + x + 1 )^{3} + \left(4 x^{2} + 8\right) ( x^{3} + x + 1 )^{2} + 8 ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2}]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + t$ $[1, 3, 7]$
2.3.4.30a2.31 $( x^{3} + x + 1 )^{4} + 4 x ( x^{3} + x + 1 )^{3} + 4 x^{2} ( x^{3} + x + 1 )^{2} + \left(8 x^{2} + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2}]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + t$ $[1, 3, 7]$
2.3.4.30a2.32 $( x^{3} + x + 1 )^{4} + 4 x ( x^{3} + x + 1 )^{3} + \left(4 x^{2} + 8\right) ( x^{3} + x + 1 )^{2} + \left(8 x^{2} + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2\wr C_6$ (as 12T134) $384$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2}]^{3}$ $[2,2,2,\frac{7}{2},\frac{7}{2}]$ $[1,1,1,\frac{5}{2},\frac{5}{2}]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + t$ $[1, 3, 7]$
2.3.4.30a3.89 $( x^{3} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + \left(4 x^{2} + 4\right) ( x^{3} + x + 1 )^{2} + 10$ $C_2^3:A_4$ (as 12T58) $96$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{3}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t + 1)$ $[1, 3, 7]$
2.3.4.30a3.90 $( x^{3} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + \left(12 x^{2} + 4\right) ( x^{3} + x + 1 )^{2} + 10$ $C_2^4:A_4$ (as 12T87) $192$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{6}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{6}$ $[2,2,\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2}]^{2}$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t + 1)$ $[1, 3, 7]$
2.3.4.30a3.91 $( x^{3} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + \left(4 x^{2} + 4\right) ( x^{3} + x + 1 )^{2} + 8 x^{2} ( x^{3} + x + 1 ) + 10$ $C_2^4:A_4$ (as 12T87) $192$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2}]^{3}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t + 1)$ $[1, 3, 7]$
2.3.4.30a3.92 $( x^{3} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + \left(12 x^{2} + 4\right) ( x^{3} + x + 1 )^{2} + 8 x^{2} ( x^{3} + x + 1 ) + 10$ $C_2^4:A_4$ (as 12T87) $192$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2}]^{3}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t + 1)$ $[1, 3, 7]$
2.3.4.30a3.93 $( x^{3} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + \left(4 x^{2} + 4\right) ( x^{3} + x + 1 )^{2} + 8 ( x^{3} + x + 1 ) + 10$ $C_2^3:A_4$ (as 12T58) $96$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{3}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{3}$ $[2,2,\frac{7}{2}]$ $[1,1,\frac{5}{2}]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t + 1)$ $[1, 3, 7]$
2.3.4.30a3.94 $( x^{3} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + \left(12 x^{2} + 4\right) ( x^{3} + x + 1 )^{2} + 8 ( x^{3} + x + 1 ) + 10$ $C_2^4:A_4$ (as 12T87) $192$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{6}$ $[1,1,2,\frac{5}{2},\frac{5}{2}]^{6}$ $[2,2,\frac{7}{2}]^{2}$ $[1,1,\frac{5}{2}]^{2}$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t + 1)$ $[1, 3, 7]$
2.3.4.30a3.95 $( x^{3} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + \left(4 x^{2} + 4\right) ( x^{3} + x + 1 )^{2} + \left(8 x^{2} + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2^4:A_4$ (as 12T87) $192$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2}]^{3}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t + 1)$ $[1, 3, 7]$
2.3.4.30a3.96 $( x^{3} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{3} + x + 1 )^{3} + \left(12 x^{2} + 4\right) ( x^{3} + x + 1 )^{2} + \left(8 x^{2} + 8\right) ( x^{3} + x + 1 ) + 10$ $C_2^4:A_4$ (as 12T87) $192$ $2$ $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{3}$ $[1,1,1,2,\frac{5}{2},\frac{5}{2}]^{3}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $[7, 4, 0]$ $[1, 1]$ $z^2 + (t + 1),(t + 1) z + (t + 1)$ $[1, 3, 7]$
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