LMFDB - $p$-adic family 3.1.18.27b

Defining polynomial

$x^{18} + 3 b_{13} x^{13} + 3 b_{11} x^{11} + 3 a_{10} x^{10} + 3 a_{3} x^{3} + 3 d_{0}$

Invariants

Residue field characteristic:	$3$
Degree:	$18$
Base field:	$\Q_{3}$
Ramification index $e$:	$18$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$27$
Artin slopes:	$[\frac{5}{4},\frac{7}{4}]$
Swan slopes:	$[\frac{1}{4},\frac{3}{4}]$
Means:	$\langle\frac{1}{6},\frac{5}{9}\rangle$
Rams:	$(\frac{1}{2},\frac{7}{2})$
Field count:	$36$ (complete)
Ambiguity:	$2$
Mass:	$36$
Absolute Mass:	$36$

Diagrams

Varying

Indices of inseparability:	$[10,3,0]$
Associated inertia:	$[1,1,1]$
Jump Set:	undefined (show 18), $[1,4,19]$ (show 18)

Fields

Galois group	Galois Artin	Indices

Assoc. Inertia	Jump Set

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Showing all 36

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Label	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
3.1.18.27b1.1	$x^{18} + 6 x^{10} + 3 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z + 1$	$[1, 4, 19]$
3.1.18.27b1.2	$x^{18} + 3 x^{13} + 6 x^{10} + 3 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z + 1$	$[1, 4, 19]$
3.1.18.27b1.3	$x^{18} + 6 x^{13} + 6 x^{10} + 3 x^{3} + 3$	$C_3^4.\SOPlus(4,2)$ (as 18T505)	$5832$	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z + 1$	$[1, 4, 19]$
3.1.18.27b1.4	$x^{18} + 3 x^{11} + 6 x^{10} + 3 x^{3} + 3$	$C_3^3:\SOPlus(4,2)$ (as 18T356)	$1944$	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z + 1$	$[1, 4, 19]$
3.1.18.27b1.5	$x^{18} + 3 x^{13} + 3 x^{11} + 6 x^{10} + 3 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z + 1$	$[1, 4, 19]$
3.1.18.27b1.6	$x^{18} + 6 x^{13} + 3 x^{11} + 6 x^{10} + 3 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z + 1$	$[1, 4, 19]$
3.1.18.27b1.7	$x^{18} + 6 x^{11} + 6 x^{10} + 3 x^{3} + 3$	$C_3^3.\SOPlus(4,2)$ (as 18T352)	$1944$	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z + 1$	$[1, 4, 19]$
3.1.18.27b1.8	$x^{18} + 3 x^{13} + 6 x^{11} + 6 x^{10} + 3 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z + 1$	$[1, 4, 19]$
3.1.18.27b1.9	$x^{18} + 6 x^{13} + 6 x^{11} + 6 x^{10} + 3 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z + 1$	$[1, 4, 19]$
3.1.18.27b1.10	$x^{18} + 3 x^{10} + 3 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z + 1$	undefined
3.1.18.27b1.11	$x^{18} + 3 x^{13} + 3 x^{10} + 3 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z + 1$	undefined
3.1.18.27b1.12	$x^{18} + 6 x^{13} + 3 x^{10} + 3 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z + 1$	undefined
3.1.18.27b1.13	$x^{18} + 3 x^{11} + 3 x^{10} + 3 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z + 1$	undefined
3.1.18.27b1.14	$x^{18} + 3 x^{13} + 3 x^{11} + 3 x^{10} + 3 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z + 1$	undefined
3.1.18.27b1.15	$x^{18} + 6 x^{13} + 3 x^{11} + 3 x^{10} + 3 x^{3} + 6$	$C_3^3:\SOPlus(4,2)$ (as 18T356)	$1944$	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z + 1$	undefined
3.1.18.27b1.16	$x^{18} + 6 x^{11} + 3 x^{10} + 3 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z + 1$	undefined
3.1.18.27b1.17	$x^{18} + 3 x^{13} + 6 x^{11} + 3 x^{10} + 3 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z + 1$	undefined
3.1.18.27b1.18	$x^{18} + 6 x^{13} + 6 x^{11} + 3 x^{10} + 3 x^{3} + 6$	$C_3^3.\SOPlus(4,2)$ (as 18T352)	$1944$	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z + 1$	undefined
3.1.18.27b2.1	$x^{18} + 3 x^{10} + 3 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z + 2$	$[1, 4, 19]$
3.1.18.27b2.2	$x^{18} + 3 x^{13} + 3 x^{10} + 3 x^{3} + 3$	$D_9^2:S_3$ (as 18T353)	$1944$	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z + 2$	$[1, 4, 19]$
3.1.18.27b2.3	$x^{18} + 6 x^{13} + 3 x^{10} + 3 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z + 2$	$[1, 4, 19]$
3.1.18.27b2.4	$x^{18} + 3 x^{11} + 3 x^{10} + 3 x^{3} + 3$	$D_9^2:C_6$ (as 18T338)	$1944$	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z + 2$	$[1, 4, 19]$
3.1.18.27b2.5	$x^{18} + 3 x^{13} + 3 x^{11} + 3 x^{10} + 3 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z + 2$	$[1, 4, 19]$
3.1.18.27b2.6	$x^{18} + 6 x^{13} + 3 x^{11} + 3 x^{10} + 3 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z + 2$	$[1, 4, 19]$
3.1.18.27b2.7	$x^{18} + 6 x^{11} + 3 x^{10} + 3 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z + 2$	$[1, 4, 19]$
3.1.18.27b2.8	$x^{18} + 3 x^{13} + 6 x^{11} + 3 x^{10} + 3 x^{3} + 3$	$\He_3^2:D_4$ (as 18T493)	$5832$	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z + 2$	$[1, 4, 19]$
3.1.18.27b2.9	$x^{18} + 6 x^{13} + 6 x^{11} + 3 x^{10} + 3 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z + 2$	$[1, 4, 19]$
3.1.18.27b2.10	$x^{18} + 6 x^{10} + 3 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z + 2$	undefined
3.1.18.27b2.11	$x^{18} + 3 x^{13} + 6 x^{10} + 3 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z + 2$	undefined
3.1.18.27b2.12	$x^{18} + 6 x^{13} + 6 x^{10} + 3 x^{3} + 6$	$C_3^4.\SOPlus(4,2)$ (as 18T491)	$5832$	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z + 2$	undefined
3.1.18.27b2.13	$x^{18} + 3 x^{11} + 6 x^{10} + 3 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z + 2$	undefined
3.1.18.27b2.14	$x^{18} + 3 x^{13} + 3 x^{11} + 6 x^{10} + 3 x^{3} + 6$	$D_9^2:C_6$ (as 18T338)	$1944$	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z + 2$	undefined
3.1.18.27b2.15	$x^{18} + 6 x^{13} + 3 x^{11} + 6 x^{10} + 3 x^{3} + 6$	$D_9\wr C_2$ (as 18T211)	$648$	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z + 2$	undefined
3.1.18.27b2.16	$x^{18} + 6 x^{11} + 6 x^{10} + 3 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z + 2$	undefined
3.1.18.27b2.17	$x^{18} + 3 x^{13} + 6 x^{11} + 6 x^{10} + 3 x^{3} + 6$	$\He_3^2:D_4$ (as 18T493)	$5832$	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z + 2$	undefined
3.1.18.27b2.18	$x^{18} + 6 x^{13} + 6 x^{11} + 6 x^{10} + 3 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[10, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z + 2$	undefined

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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Label	3.1.18.27b
Base	3.1.1.0a1.1
Degree	\(18\)
e	\(18\)
f	\(1\)
c	\(27\)

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Defining polynomial

Invariants

Diagrams

Varying

Fields

Showing all 36

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.