Properties

Label 2.11.2.33a
Base 2.1.1.0a1.1
Degree \(22\)
e \(2\)
f \(11\)
c \(33\)

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Defining polynomial over unramified subextension

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

Invariants

Residue field characteristic: $2$
Degree: $22$
Base field: $\Q_{2}$
Ramification index $e$: $2$
Residue field degree $f$: $11$
Discriminant exponent $c$: $33$
Artin slopes: $[3]$
Swan slopes: $[2]$
Means: $\langle1\rangle$
Rams: $(2)$
Field count: $376$ (complete)
Ambiguity: $22$
Mass: $2048$
Absolute Mass: $2048/11$

Diagrams

Varying

Indices of inseparability: $[2,0]$
Associated inertia: $[1]$
Jump Set: $[1,3]$

Fields


Showing 1-50 of 376

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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
2.11.2.33a1.1 $( x^{11} + x^{2} + 1 )^{2} + 2$ $C_{22}$ (as 22T1) $22$ $22$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.2 $( x^{11} + x^{2} + 1 )^{2} + 10$ not computed $ $not computed$ $ $22$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.3 $( x^{11} + x^{2} + 1 )^{2} + 4 x^{9} ( x^{11} + x^{2} + 1 ) + 2$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.4 $( x^{11} + x^{2} + 1 )^{2} + 4 x^{9} ( x^{11} + x^{2} + 1 ) + 10$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.5 $( x^{11} + x^{2} + 1 )^{2} + 4 x^{5} ( x^{11} + x^{2} + 1 ) + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.6 $( x^{11} + x^{2} + 1 )^{2} + 4 x^{5} ( x^{11} + x^{2} + 1 ) + 10$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.7 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{8} + 4 x^{5}\right) ( x^{11} + x^{2} + 1 ) + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.8 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{8} + 4 x^{5}\right) ( x^{11} + x^{2} + 1 ) + 10$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.9 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{6} + 4 x^{5}\right) ( x^{11} + x^{2} + 1 ) + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.10 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{6} + 4 x^{5}\right) ( x^{11} + x^{2} + 1 ) + 10$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.11 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{7} + 4 x^{4}\right) ( x^{11} + x^{2} + 1 ) + 2$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.12 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{7} + 4 x^{4}\right) ( x^{11} + x^{2} + 1 ) + 10$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.13 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{7} + 4 x^{6} + 4 x^{4}\right) ( x^{11} + x^{2} + 1 ) + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.14 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{7} + 4 x^{6} + 4 x^{4}\right) ( x^{11} + x^{2} + 1 ) + 10$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.15 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{5} + 4 x^{4}\right) ( x^{11} + x^{2} + 1 ) + 2$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.16 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{5} + 4 x^{4}\right) ( x^{11} + x^{2} + 1 ) + 10$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.17 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{6} + 4 x^{5} + 4 x^{4}\right) ( x^{11} + x^{2} + 1 ) + 2$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.18 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{6} + 4 x^{5} + 4 x^{4}\right) ( x^{11} + x^{2} + 1 ) + 10$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.19 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{4}\right) ( x^{11} + x^{2} + 1 ) + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.20 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{4}\right) ( x^{11} + x^{2} + 1 ) + 10$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.21 $( x^{11} + x^{2} + 1 )^{2} + 4 x^{3} ( x^{11} + x^{2} + 1 ) + 2$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.22 $( x^{11} + x^{2} + 1 )^{2} + 4 x^{3} ( x^{11} + x^{2} + 1 ) + 10$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.23 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{9} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.24 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{9} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 10$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.25 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{6} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 2$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.26 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{6} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 10$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.27 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{5} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 2$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.28 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{5} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 10$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.29 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{7} + 4 x^{5} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 2$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.30 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{7} + 4 x^{5} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 10$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.31 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{8} + 4 x^{7} + 4 x^{5} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.32 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{8} + 4 x^{7} + 4 x^{5} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 10$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.33 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{6} + 4 x^{5} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.34 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{6} + 4 x^{5} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 10$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.35 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{4} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.36 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{4} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 10$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.37 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{7} + 4 x^{4} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 2$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.38 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{7} + 4 x^{4} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 10$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.39 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{6} + 4 x^{4} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 2$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.40 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{6} + 4 x^{4} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 10$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.41 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{7} + 4 x^{6} + 4 x^{4} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.42 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{7} + 4 x^{6} + 4 x^{4} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 10$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.43 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{5} + 4 x^{4} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.44 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{5} + 4 x^{4} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 10$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.45 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.46 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 10$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.47 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 2$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.48 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3}\right) ( x^{11} + x^{2} + 1 ) + 10$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.49 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{6} + 4 x^{5} + 4 x^{2}\right) ( x^{11} + x^{2} + 1 ) + 2$ $C_2^{10}.C_{22}$ (as 22T28) $22528$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
2.11.2.33a1.50 $( x^{11} + x^{2} + 1 )^{2} + \left(4 x^{6} + 4 x^{5} + 4 x^{2}\right) ( x^{11} + x^{2} + 1 ) + 10$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1]$ $z + (t^9 + t^2 + 1)$ $[1, 3]$
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