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The results below are complete, since the LMFDB contains all families of p-adic fields of degree at most 47 and residue characteristic at most 199

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Residue characteristic $p$ Degree $n$ Ramification index $e$ Residue field degree $f$ Discriminant exponent $c$
Mass Ambiguity Field count Wild segments Herbrand invariant
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Results (22 matches)

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Label $p$ $n$ $f$ $e$ $c$ Abs. Artin slopes Swan slopes Means Rams Generic poly Ambiguity Field count Mass Num. Packets
7.14.1.0a $7$ $14$ $14$ $1$ $0$ $[\ ]$ $[ ]$ $\langle \rangle$ $( )$ $x$ $14$ $1$ $1$ $1$
7.7.2.7a $7$ $14$ $7$ $2$ $7$ $[\ ]$ $[ ]$ $\langle \rangle$ $( )$ $x^2 + 7 d_{0}$ $14$ $2$ $1$ $1$
7.2.7.14a $7$ $14$ $2$ $7$ $14$ $[\frac{7}{6}]$ $[\frac{1}{6}]$ $\langle\frac{1}{7}\rangle$ $(\frac{1}{6})$ $x^7 + 7 a_{1} x + 7$ $2$ $27$ $48$ $27$
7.2.7.16a $7$ $14$ $2$ $7$ $16$ $[\frac{4}{3}]$ $[\frac{1}{3}]$ $\langle\frac{2}{7}\rangle$ $(\frac{1}{3})$ $x^7 + 7 a_{2} x^2 + 7$ $2$ $27$ $48$ $27$
7.2.7.18a $7$ $14$ $2$ $7$ $18$ $[\frac{3}{2}]$ $[\frac{1}{2}]$ $\langle\frac{3}{7}\rangle$ $(\frac{1}{2})$ $x^7 + 7 a_{3} x^3 + 7$ $2$ $27$ $48$ $27$
7.2.7.20a $7$ $14$ $2$ $7$ $20$ $[\frac{5}{3}]$ $[\frac{2}{3}]$ $\langle\frac{4}{7}\rangle$ $(\frac{2}{3})$ $x^7 + 7 a_{4} x^4 + 7$ $2$ $27$ $48$ $27$
7.2.7.22a $7$ $14$ $2$ $7$ $22$ $[\frac{11}{6}]$ $[\frac{5}{6}]$ $\langle\frac{5}{7}\rangle$ $(\frac{5}{6})$ $x^7 + 7 a_{5} x^5 + 7$ $2$ $27$ $48$ $27$
7.2.7.24a $7$ $14$ $2$ $7$ $24$ $[2]$ $[1]$ $\langle\frac{6}{7}\rangle$ $(1)$ $x^7 + 7 a_{6} x^6 + 49 c_{7} + 7$ $14$ $54$ $48$ $28$
7.2.7.26a $7$ $14$ $2$ $7$ $26$ $[\frac{13}{6}]$ $[\frac{7}{6}]$ $\langle1\rangle$ $(\frac{7}{6})$ $x^7 + 49 b_{8} x + 7$ $2$ $28$ $49$ $2$
7.1.14.14a $7$ $14$ $1$ $14$ $14$ $[\frac{13}{12}]$ $[\frac{1}{12}]$ $\langle\frac{1}{14}\rangle$ $(\frac{1}{6})$ $x^{14} + 7 a_{1} x + 7 d_{0}$ $2$ $6$ $6$ $1$
7.1.14.15a $7$ $14$ $1$ $14$ $15$ $[\frac{7}{6}]$ $[\frac{1}{6}]$ $\langle\frac{1}{7}\rangle$ $(\frac{1}{3})$ $x^{14} + 7 a_{2} x^2 + 7 d_{0}$ $2$ $12$ $6$ $2$
7.1.14.16a $7$ $14$ $1$ $14$ $16$ $[\frac{5}{4}]$ $[\frac{1}{4}]$ $\langle\frac{3}{14}\rangle$ $(\frac{1}{2})$ $x^{14} + 7 a_{3} x^3 + 7 d_{0}$ $2$ $6$ $6$ $3$
7.1.14.17a $7$ $14$ $1$ $14$ $17$ $[\frac{4}{3}]$ $[\frac{1}{3}]$ $\langle\frac{2}{7}\rangle$ $(\frac{2}{3})$ $x^{14} + 7 a_{4} x^4 + 7 d_{0}$ $2$ $12$ $6$ $2$
7.1.14.18a $7$ $14$ $1$ $14$ $18$ $[\frac{17}{12}]$ $[\frac{5}{12}]$ $\langle\frac{5}{14}\rangle$ $(\frac{5}{6})$ $x^{14} + 7 a_{5} x^5 + 7 d_{0}$ $2$ $6$ $6$ $1$
7.1.14.19a $7$ $14$ $1$ $14$ $19$ $[\frac{3}{2}]$ $[\frac{1}{2}]$ $\langle\frac{3}{7}\rangle$ $(1)$ $x^{14} + 7 c_{7} x^7 + 7 a_{6} x^6 + 7 d_{0}$ $14$ $18$ $6$ $7$
7.1.14.21a $7$ $14$ $1$ $14$ $21$ $[\frac{5}{3}]$ $[\frac{2}{3}]$ $\langle\frac{4}{7}\rangle$ $(\frac{4}{3})$ $x^{14} + 7 b_{9} x^9 + 7 a_{8} x^8 + 7 d_{0}$ $2$ $48$ $42$ $4$
7.1.14.22a $7$ $14$ $1$ $14$ $22$ $[\frac{7}{4}]$ $[\frac{3}{4}]$ $\langle\frac{9}{14}\rangle$ $(\frac{3}{2})$ $x^{14} + 7 b_{10} x^{10} + 7 a_{9} x^9 + 7 d_{0}$ $2$ $42$ $42$ $3$
7.1.14.23a $7$ $14$ $1$ $14$ $23$ $[\frac{11}{6}]$ $[\frac{5}{6}]$ $\langle\frac{5}{7}\rangle$ $(\frac{5}{3})$ $x^{14} + 7 b_{11} x^{11} + 7 a_{10} x^{10} + 7 d_{0}$ $2$ $48$ $42$ $4$
7.1.14.24a $7$ $14$ $1$ $14$ $24$ $[\frac{23}{12}]$ $[\frac{11}{12}]$ $\langle\frac{11}{14}\rangle$ $(\frac{11}{6})$ $x^{14} + 7 b_{12} x^{12} + 7 a_{11} x^{11} + 7 d_{0}$ $2$ $42$ $42$ $1$
7.1.14.25a $7$ $14$ $1$ $14$ $25$ $[2]$ $[1]$ $\langle\frac{6}{7}\rangle$ $(2)$ $x^{14} + 7 b_{13} x^{13} + 7 a_{12} x^{12} + 7 d_{0} + 49 c_{14}$ $14$ $96$ $42$ $12$
7.1.14.26a $7$ $14$ $1$ $14$ $26$ $[\frac{25}{12}]$ $[\frac{13}{12}]$ $\langle\frac{13}{14}\rangle$ $(\frac{13}{6})$ $x^{14} + 7 a_{13} x^{13} + 49 b_{15} x + 7 d_{0}$ $2$ $42$ $42$ $1$
7.1.14.27a $7$ $14$ $1$ $14$ $27$ $[\frac{13}{6}]$ $[\frac{7}{6}]$ $\langle1\rangle$ $(\frac{7}{3})$ $x^{14} + 49 b_{16} x^2 + 49 b_{15} x + 7 d_{0}$ $2$ $56$ $49$ $4$
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