Properties

Label 3.1.4.3a1.2-1.9.75a
Base 3.1.4.3a1.2
Degree \(9\)
e \(9\)
f \(1\)
c \(75\)

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Defining polynomial

$x^{9} + \left(b_{80} \pi^{9} + b_{71} \pi^{8}\right) x^{8} + \left(b_{79} \pi^{9} + b_{70} \pi^{8}\right) x^{7} + \left(b_{42} \pi^{5} + a_{33} \pi^{4}\right) x^{6} + \left(b_{77} \pi^{9} + b_{68} \pi^{8}\right) x^{5} + \left(b_{76} \pi^{9} + a_{67} \pi^{8}\right) x^{4} + \left(c_{84} \pi^{10} + b_{48} \pi^{6} + b_{39} \pi^{5}\right) x^{3} + \left(b_{83} \pi^{10} + b_{74} \pi^{9}\right) x^{2} + \left(b_{82} \pi^{10} + b_{73} \pi^{9}\right) x + \pi$

Invariants

Residue field characteristic: $3$
Degree: $9$
Base field: 3.1.4.3a1.2
Ramification index $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $75$
Absolute Artin slopes: $[\frac{19}{8},\frac{10}{3}]$
Swan slopes: $[\frac{11}{2},\frac{28}{3}]$
Means: $\langle\frac{11}{3},\frac{67}{9}\rangle$
Rams: $(\frac{11}{2},17)$
Field count: $0$ (incomplete)
Ambiguity: $3$
Mass: $19131876$
Absolute Mass: $9565938$ ($0$ currently in the LMFDB)

Diagrams

The LMFDB does not contain any fields from this family.