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Degree Residue characteristic Galois group Galois unramified degree Assoc. Inertia
Ramification index Discriminant exponent Inertia subgroup Galois tame degree Indices
Residue field degree Top slope Wild Visible Wild inertia subgroup
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Results (1-50 of 59 matches)

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Label Polynomial $p$ $e$ $f$ $c$ Galois group Visible slopes Slope content Unram. Ext. Eisen. Poly.
2.4.0.1 $x^{4} + x + 1$ $2$ $1$ $4$ $0$ $C_4$ (as 4T1) $[\ ]$ $[\ ]^{4}$ $t^{4} + t + 1$ $x - 2$
2.4.4.1 $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ $2$ $2$ $2$ $4$ $C_2^2$ (as 4T2) $[2]$ $[2]^{2}$ $t^{2} + t + 1$ $x^{2} + 2 x + 2$
2.4.4.2 $x^{4} + 4 x^{3} + 4 x^{2} + 12$ $2$ $2$ $2$ $4$ $C_4$ (as 4T1) $[2]$ $[2]^{2}$ $t^{2} + t + 1$ $x^{2} + 2 x + 4 t + 2$
2.4.4.3 $x^{4} + 2 x^{3} + 4 x^{2} + 12 x + 12$ $2$ $2$ $2$ $4$ $D_{4}$ (as 4T3) $[2]$ $[2, 2]^{2}$ $t^{2} + t + 1$ $x^{2} + \left(2 t + 2\right) x + 4 t + 2$
2.4.4.4 $x^{4} - 2 x^{3} + 4 x^{2} + 12 x + 12$ $2$ $2$ $2$ $4$ $D_{4}$ (as 4T3) $[2]$ $[2, 2]^{2}$ $t^{2} + t + 1$ $x^{2} + 2 t x + 4 t + 2$
2.4.4.5 $x^{4} + 2 x + 2$ $2$ $4$ $1$ $4$ $S_4$ (as 4T5) $[4/3, 4/3]$ $[4/3, 4/3]_{3}^{2}$ $t + 1$ $x^{4} + 2 x + 2$
2.4.6.1 $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ $2$ $2$ $2$ $6$ $C_2^2$ (as 4T2) $[3]$ $[3]^{2}$ $t^{2} + t + 1$ $x^{2} + 14$
2.4.6.2 $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ $2$ $2$ $2$ $6$ $C_2^2$ (as 4T2) $[3]$ $[3]^{2}$ $t^{2} + t + 1$ $x^{2} + \left(4 t + 4\right) x + 4 t + 2$
2.4.6.3 $x^{4} + 8 x^{3} + 28 x^{2} + 48 x + 84$ $2$ $2$ $2$ $6$ $C_4$ (as 4T1) $[3]$ $[3]^{2}$ $t^{2} + t + 1$ $x^{2} + 4 x + 8 t + 10$
2.4.6.4 $x^{4} + 4 x^{3} + 24 x^{2} + 88 x + 124$ $2$ $2$ $2$ $6$ $C_4$ (as 4T1) $[3]$ $[3]^{2}$ $t^{2} + t + 1$ $x^{2} + \left(4 t + 4\right) x + 12 t + 10$
2.4.6.5 $x^{4} - 4 x^{3} + 36 x^{2} + 8 x + 148$ $2$ $2$ $2$ $6$ $D_{4}$ (as 4T3) $[3]$ $[2, 3]^{2}$ $t^{2} + t + 1$ $x^{2} + 4 t x + 8 t + 14$
2.4.6.6 $x^{4} - 4 x^{3} + 28 x^{2} - 24 x + 36$ $2$ $2$ $2$ $6$ $D_{4}$ (as 4T3) $[3]$ $[2, 3]^{2}$ $t^{2} + t + 1$ $x^{2} + 4 t x + 6$
2.4.6.7 $x^{4} + 2 x^{3} + 2 x^{2} + 2$ $2$ $4$ $1$ $6$ $A_4$ (as 4T4) $[2, 2]$ $[2, 2]^{3}$ $t + 1$ $x^{4} + 2 x^{3} + 2 x^{2} + 2$
2.4.6.8 $x^{4} + 2 x^{3} + 2$ $2$ $4$ $1$ $6$ $D_{4}$ (as 4T3) $[2, 2]$ $[2, 2]^{2}$ $t + 1$ $x^{4} + 2 x^{3} + 2$
2.4.6.9 $x^{4} + 2 x^{3} + 6$ $2$ $4$ $1$ $6$ $D_{4}$ (as 4T3) $[2, 2]$ $[2, 2]^{2}$ $t + 1$ $x^{4} + 2 x^{3} + 6$
2.4.8.1 $x^{4} + 2 x^{2} + 4 x + 10$ $2$ $4$ $1$ $8$ $C_2^2$ (as 4T2) $[2, 3]$ $[2, 3]$ $t + 1$ $x^{4} + 2 x^{2} + 4 x + 10$
2.4.8.2 $x^{4} + 2 x^{2} + 4 x + 2$ $2$ $4$ $1$ $8$ $C_2^2$ (as 4T2) $[2, 3]$ $[2, 3]$ $t + 1$ $x^{4} + 2 x^{2} + 4 x + 2$
2.4.8.3 $x^{4} + 6 x^{2} + 4 x + 14$ $2$ $4$ $1$ $8$ $C_2^2$ (as 4T2) $[2, 3]$ $[2, 3]$ $t + 1$ $x^{4} + 6 x^{2} + 4 x + 14$
2.4.8.4 $x^{4} + 6 x^{2} + 4 x + 6$ $2$ $4$ $1$ $8$ $C_2^2$ (as 4T2) $[2, 3]$ $[2, 3]$ $t + 1$ $x^{4} + 6 x^{2} + 4 x + 6$
2.4.8.5 $x^{4} + 2 x^{2} + 4 x + 6$ $2$ $4$ $1$ $8$ $D_{4}$ (as 4T3) $[2, 3]$ $[2, 3]^{2}$ $t + 1$ $x^{4} + 2 x^{2} + 4 x + 6$
2.4.8.6 $x^{4} + 6 x^{2} + 4 x + 10$ $2$ $4$ $1$ $8$ $D_{4}$ (as 4T3) $[2, 3]$ $[2, 3]^{2}$ $t + 1$ $x^{4} + 6 x^{2} + 4 x + 10$
2.4.8.7 $x^{4} + 4 x^{2} + 4 x + 2$ $2$ $4$ $1$ $8$ $S_4$ (as 4T5) $[8/3, 8/3]$ $[8/3, 8/3]_{3}^{2}$ $t + 1$ $x^{4} + 4 x^{2} + 4 x + 2$
2.4.8.8 $x^{4} + 4 x + 2$ $2$ $4$ $1$ $8$ $S_4$ (as 4T5) $[8/3, 8/3]$ $[8/3, 8/3]_{3}^{2}$ $t + 1$ $x^{4} + 4 x + 2$
2.4.9.1 $x^{4} + 10 x^{2} + 2$ $2$ $4$ $1$ $9$ $D_{4}$ (as 4T3) $[2, 7/2]$ $[2, 3, 7/2]$ $t + 1$ $x^{4} + 10 x^{2} + 2$
2.4.9.2 $x^{4} + 2 x^{2} + 2$ $2$ $4$ $1$ $9$ $D_{4}$ (as 4T3) $[2, 7/2]$ $[2, 3, 7/2]$ $t + 1$ $x^{4} + 2 x^{2} + 2$
2.4.9.3 $x^{4} + 4 x^{3} + 2 x^{2} + 2$ $2$ $4$ $1$ $9$ $D_{4}$ (as 4T3) $[2, 7/2]$ $[2, 3, 7/2]$ $t + 1$ $x^{4} + 4 x^{3} + 2 x^{2} + 2$
2.4.9.4 $x^{4} + 10 x^{2} + 8 x + 2$ $2$ $4$ $1$ $9$ $D_{4}$ (as 4T3) $[2, 7/2]$ $[2, 3, 7/2]$ $t + 1$ $x^{4} + 10 x^{2} + 8 x + 2$
2.4.9.5 $x^{4} + 10 x^{2} + 8 x + 6$ $2$ $4$ $1$ $9$ $D_{4}$ (as 4T3) $[2, 7/2]$ $[2, 3, 7/2]$ $t + 1$ $x^{4} + 10 x^{2} + 8 x + 6$
2.4.9.6 $x^{4} + 4 x^{3} + 10 x^{2} + 14$ $2$ $4$ $1$ $9$ $D_{4}$ (as 4T3) $[2, 7/2]$ $[2, 3, 7/2]$ $t + 1$ $x^{4} + 4 x^{3} + 10 x^{2} + 14$
2.4.9.7 $x^{4} + 2 x^{2} + 6$ $2$ $4$ $1$ $9$ $D_{4}$ (as 4T3) $[2, 7/2]$ $[2, 3, 7/2]$ $t + 1$ $x^{4} + 2 x^{2} + 6$
2.4.9.8 $x^{4} + 10 x^{2} + 6$ $2$ $4$ $1$ $9$ $D_{4}$ (as 4T3) $[2, 7/2]$ $[2, 3, 7/2]$ $t + 1$ $x^{4} + 10 x^{2} + 6$
2.4.10.1 $x^{4} + 4 x^{3} + 10$ $2$ $4$ $1$ $10$ $D_{4}$ (as 4T3) $[3, 7/2]$ $[2, 3, 7/2]$ $t + 1$ $x^{4} + 4 x^{3} + 10$
2.4.10.2 $x^{4} + 4 x^{3} + 2$ $2$ $4$ $1$ $10$ $D_{4}$ (as 4T3) $[3, 7/2]$ $[2, 3, 7/2]$ $t + 1$ $x^{4} + 4 x^{3} + 2$
2.4.10.3 $x^{4} + 4 x^{3} + 8 x^{2} + 2$ $2$ $4$ $1$ $10$ $D_{4}$ (as 4T3) $[3, 7/2]$ $[2, 3, 7/2]$ $t + 1$ $x^{4} + 4 x^{3} + 8 x^{2} + 2$
2.4.10.4 $x^{4} + 4 x^{3} + 8 x^{2} + 10$ $2$ $4$ $1$ $10$ $D_{4}$ (as 4T3) $[3, 7/2]$ $[2, 3, 7/2]$ $t + 1$ $x^{4} + 4 x^{3} + 8 x^{2} + 10$
2.4.10.5 $x^{4} + 4 x^{3} + 4 x^{2} + 10$ $2$ $4$ $1$ $10$ $D_{4}$ (as 4T3) $[3, 7/2]$ $[2, 3, 7/2]$ $t + 1$ $x^{4} + 4 x^{3} + 4 x^{2} + 10$
2.4.10.6 $x^{4} + 4 x^{3} + 12 x^{2} + 10$ $2$ $4$ $1$ $10$ $D_{4}$ (as 4T3) $[3, 7/2]$ $[2, 3, 7/2]$ $t + 1$ $x^{4} + 4 x^{3} + 12 x^{2} + 10$
2.4.10.7 $x^{4} + 4 x^{3} + 4 x^{2} + 2$ $2$ $4$ $1$ $10$ $D_{4}$ (as 4T3) $[3, 7/2]$ $[2, 3, 7/2]$ $t + 1$ $x^{4} + 4 x^{3} + 4 x^{2} + 2$
2.4.10.8 $x^{4} + 4 x^{3} + 12 x^{2} + 2$ $2$ $4$ $1$ $10$ $D_{4}$ (as 4T3) $[3, 7/2]$ $[2, 3, 7/2]$ $t + 1$ $x^{4} + 4 x^{3} + 12 x^{2} + 2$
2.4.11.1 $x^{4} + 8 x^{3} + 4 x^{2} + 2$ $2$ $4$ $1$ $11$ $C_4$ (as 4T1) $[3, 4]$ $[3, 4]$ $t + 1$ $x^{4} + 8 x^{3} + 4 x^{2} + 2$
2.4.11.10 $x^{4} + 4 x^{2} + 26$ $2$ $4$ $1$ $11$ $C_4$ (as 4T1) $[3, 4]$ $[3, 4]$ $t + 1$ $x^{4} + 4 x^{2} + 26$
2.4.11.11 $x^{4} + 10$ $2$ $4$ $1$ $11$ $D_{4}$ (as 4T3) $[3, 4]$ $[2, 3, 4]$ $t + 1$ $x^{4} + 10$
2.4.11.12 $x^{4} + 26$ $2$ $4$ $1$ $11$ $D_{4}$ (as 4T3) $[3, 4]$ $[2, 3, 4]$ $t + 1$ $x^{4} + 26$
2.4.11.13 $x^{4} + 8 x^{3} + 8 x + 10$ $2$ $4$ $1$ $11$ $D_{4}$ (as 4T3) $[3, 4]$ $[3, 4]^{2}$ $t + 1$ $x^{4} + 8 x^{3} + 8 x + 10$
2.4.11.14 $x^{4} + 8 x + 10$ $2$ $4$ $1$ $11$ $D_{4}$ (as 4T3) $[3, 4]$ $[3, 4]^{2}$ $t + 1$ $x^{4} + 8 x + 10$
2.4.11.15 $x^{4} + 12 x^{2} + 8 x + 2$ $2$ $4$ $1$ $11$ $D_{4}$ (as 4T3) $[3, 4]$ $[2, 3, 4]$ $t + 1$ $x^{4} + 12 x^{2} + 8 x + 2$
2.4.11.16 $x^{4} + 12 x^{2} + 8 x + 18$ $2$ $4$ $1$ $11$ $D_{4}$ (as 4T3) $[3, 4]$ $[2, 3, 4]$ $t + 1$ $x^{4} + 12 x^{2} + 8 x + 18$
2.4.11.17 $x^{4} + 8 x^{3} + 8 x + 2$ $2$ $4$ $1$ $11$ $D_{4}$ (as 4T3) $[3, 4]$ $[3, 4]^{2}$ $t + 1$ $x^{4} + 8 x^{3} + 8 x + 2$
2.4.11.18 $x^{4} + 8 x + 18$ $2$ $4$ $1$ $11$ $D_{4}$ (as 4T3) $[3, 4]$ $[3, 4]^{2}$ $t + 1$ $x^{4} + 8 x + 18$
2.4.11.19 $x^{4} + 12 x^{2} + 8 x + 26$ $2$ $4$ $1$ $11$ $D_{4}$ (as 4T3) $[3, 4]$ $[2, 3, 4]$ $t + 1$ $x^{4} + 12 x^{2} + 8 x + 26$
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