LMFDB - $p$-adic family 5.1.10.15a

Defining polynomial

$x^{10} + 5 b_{7} x^{7} + 5 a_{6} x^{6} + 5 d_{0}$

Invariants

Residue field characteristic:	$5$
Degree:	$10$
Base field:	$\Q_{5}$
Ramification index $e$:	$10$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$15$
Artin slopes:	$[\frac{7}{4}]$
Swan slopes:	$[\frac{3}{4}]$
Means:	$\langle\frac{3}{5}\rangle$
Rams:	$(\frac{3}{2})$
Field count:	$24$ (complete)
Ambiguity:	$2$
Mass:	$20$
Absolute Mass:	$20$

Diagrams

Varying

Indices of inseparability:	$[6,0]$
Associated inertia:	$[1,1]$ (show 12), $[1,2]$ (show 12)
Jump Set:	undefined

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 20
		$F_5$ (as 10T4)
hidden slopes	$[\ ]_{2}$	4

		Galois groups of order 40
		$C_2\times F_5$ (as 10T5)
hidden slopes	$[\ ]^{2}_{2}$	4

		Galois groups of order 100
		$C_5:F_5$ (as 10T10)
hidden slopes	$[\frac{5}{4}]_{2}$	8

		Galois groups of order 200
		$D_5:F_5$ (as 10T17)
hidden slopes	$[\frac{5}{4}]^{2}_{2}$	8

Fields

Galois group	Galois Artin	Indices

Assoc. Inertia	Jump Set	Fields per packet

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Showing all 24

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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
5.1.10.15a1.1	4	$x^{10} + 5 x^{6} + 5$	$F_{5}\times C_2$ (as 10T5)	$40$	$2$	$[\frac{7}{4}]_{4}^{2}$	$[\frac{3}{4}]_{4}^{2}$	$[\ ]^{2}_{2}$	$[\ ]^{2}_{2}$	$[6, 0]$	$[1, 2]$	$z^5 + 2,2 z^2 + 4$	undefined
5.1.10.15a1.2	8	$x^{10} + 5 x^{7} + 5 x^{6} + 5$	$(C_5^2 : C_4) : C_2$ (as 10T17)	$200$	$1$	$[\frac{5}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{1}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{5}{4}]^{2}_{2}$	$[\frac{1}{4}]^{2}_{2}$	$[6, 0]$	$[1, 2]$	$z^5 + 2,2 z^2 + 4$	undefined
5.1.10.15a1.3	8	$x^{10} + 10 x^{7} + 5 x^{6} + 5$	$(C_5^2 : C_4) : C_2$ (as 10T17)	$200$	$1$	$[\frac{5}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{1}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{5}{4}]^{2}_{2}$	$[\frac{1}{4}]^{2}_{2}$	$[6, 0]$	$[1, 2]$	$z^5 + 2,2 z^2 + 4$	undefined
5.1.10.15a1.4	4	$x^{10} + 20 x^{6} + 5$	$F_{5}\times C_2$ (as 10T5)	$40$	$2$	$[\frac{7}{4}]_{4}^{2}$	$[\frac{3}{4}]_{4}^{2}$	$[\ ]^{2}_{2}$	$[\ ]^{2}_{2}$	$[6, 0]$	$[1, 2]$	$z^5 + 2,2 z^2 + 1$	undefined
5.1.10.15a1.5	8	$x^{10} + 5 x^{7} + 20 x^{6} + 5$	$(C_5^2 : C_4) : C_2$ (as 10T17)	$200$	$1$	$[\frac{5}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{1}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{5}{4}]^{2}_{2}$	$[\frac{1}{4}]^{2}_{2}$	$[6, 0]$	$[1, 2]$	$z^5 + 2,2 z^2 + 1$	undefined
5.1.10.15a1.6	8	$x^{10} + 10 x^{7} + 20 x^{6} + 5$	$(C_5^2 : C_4) : C_2$ (as 10T17)	$200$	$1$	$[\frac{5}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{1}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{5}{4}]^{2}_{2}$	$[\frac{1}{4}]^{2}_{2}$	$[6, 0]$	$[1, 2]$	$z^5 + 2,2 z^2 + 1$	undefined
5.1.10.15a1.7	4	$x^{10} + 10 x^{6} + 10$	$F_{5}\times C_2$ (as 10T5)	$40$	$2$	$[\frac{7}{4}]_{4}^{2}$	$[\frac{3}{4}]_{4}^{2}$	$[\ ]^{2}_{2}$	$[\ ]^{2}_{2}$	$[6, 0]$	$[1, 2]$	$z^5 + 2,2 z^2 + 4$	undefined
5.1.10.15a1.8	8	$x^{10} + 5 x^{7} + 10 x^{6} + 10$	$(C_5^2 : C_4) : C_2$ (as 10T17)	$200$	$1$	$[\frac{5}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{1}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{5}{4}]^{2}_{2}$	$[\frac{1}{4}]^{2}_{2}$	$[6, 0]$	$[1, 2]$	$z^5 + 2,2 z^2 + 4$	undefined
5.1.10.15a1.9	8	$x^{10} + 10 x^{7} + 10 x^{6} + 10$	$(C_5^2 : C_4) : C_2$ (as 10T17)	$200$	$1$	$[\frac{5}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{1}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{5}{4}]^{2}_{2}$	$[\frac{1}{4}]^{2}_{2}$	$[6, 0]$	$[1, 2]$	$z^5 + 2,2 z^2 + 4$	undefined
5.1.10.15a1.10	4	$x^{10} + 15 x^{6} + 10$	$F_{5}\times C_2$ (as 10T5)	$40$	$2$	$[\frac{7}{4}]_{4}^{2}$	$[\frac{3}{4}]_{4}^{2}$	$[\ ]^{2}_{2}$	$[\ ]^{2}_{2}$	$[6, 0]$	$[1, 2]$	$z^5 + 2,2 z^2 + 1$	undefined
5.1.10.15a1.11	8	$x^{10} + 5 x^{7} + 15 x^{6} + 10$	$(C_5^2 : C_4) : C_2$ (as 10T17)	$200$	$1$	$[\frac{5}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{1}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{5}{4}]^{2}_{2}$	$[\frac{1}{4}]^{2}_{2}$	$[6, 0]$	$[1, 2]$	$z^5 + 2,2 z^2 + 1$	undefined
5.1.10.15a1.12	8	$x^{10} + 10 x^{7} + 15 x^{6} + 10$	$(C_5^2 : C_4) : C_2$ (as 10T17)	$200$	$1$	$[\frac{5}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{1}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{5}{4}]^{2}_{2}$	$[\frac{1}{4}]^{2}_{2}$	$[6, 0]$	$[1, 2]$	$z^5 + 2,2 z^2 + 1$	undefined
5.1.10.15a2.1	4	$x^{10} + 10 x^{6} + 5$	$F_5$ (as 10T4)	$20$	$2$	$[\frac{7}{4}]_{4}$	$[\frac{3}{4}]_{4}$	$[\ ]_{2}$	$[\ ]_{2}$	$[6, 0]$	$[1, 1]$	$z^5 + 2,2 z^2 + 3$	undefined
5.1.10.15a2.2	8	$x^{10} + 5 x^{7} + 10 x^{6} + 5$	$C_5^2 : C_4$ (as 10T10)	$100$	$1$	$[\frac{5}{4}, \frac{7}{4}]_{4}$	$[\frac{1}{4},\frac{3}{4}]_{4}$	$[\frac{5}{4}]_{2}$	$[\frac{1}{4}]_{2}$	$[6, 0]$	$[1, 1]$	$z^5 + 2,2 z^2 + 3$	undefined
5.1.10.15a2.3	8	$x^{10} + 10 x^{7} + 10 x^{6} + 5$	$C_5^2 : C_4$ (as 10T10)	$100$	$1$	$[\frac{5}{4}, \frac{7}{4}]_{4}$	$[\frac{1}{4},\frac{3}{4}]_{4}$	$[\frac{5}{4}]_{2}$	$[\frac{1}{4}]_{2}$	$[6, 0]$	$[1, 1]$	$z^5 + 2,2 z^2 + 3$	undefined
5.1.10.15a2.4	4	$x^{10} + 15 x^{6} + 5$	$F_5$ (as 10T4)	$20$	$2$	$[\frac{7}{4}]_{4}$	$[\frac{3}{4}]_{4}$	$[\ ]_{2}$	$[\ ]_{2}$	$[6, 0]$	$[1, 1]$	$z^5 + 2,2 z^2 + 2$	undefined
5.1.10.15a2.5	8	$x^{10} + 5 x^{7} + 15 x^{6} + 5$	$C_5^2 : C_4$ (as 10T10)	$100$	$1$	$[\frac{5}{4}, \frac{7}{4}]_{4}$	$[\frac{1}{4},\frac{3}{4}]_{4}$	$[\frac{5}{4}]_{2}$	$[\frac{1}{4}]_{2}$	$[6, 0]$	$[1, 1]$	$z^5 + 2,2 z^2 + 2$	undefined
5.1.10.15a2.6	8	$x^{10} + 10 x^{7} + 15 x^{6} + 5$	$C_5^2 : C_4$ (as 10T10)	$100$	$1$	$[\frac{5}{4}, \frac{7}{4}]_{4}$	$[\frac{1}{4},\frac{3}{4}]_{4}$	$[\frac{5}{4}]_{2}$	$[\frac{1}{4}]_{2}$	$[6, 0]$	$[1, 1]$	$z^5 + 2,2 z^2 + 2$	undefined
5.1.10.15a2.7	4	$x^{10} + 5 x^{6} + 10$	$F_5$ (as 10T4)	$20$	$2$	$[\frac{7}{4}]_{4}$	$[\frac{3}{4}]_{4}$	$[\ ]_{2}$	$[\ ]_{2}$	$[6, 0]$	$[1, 1]$	$z^5 + 2,2 z^2 + 2$	undefined
5.1.10.15a2.8	8	$x^{10} + 5 x^{7} + 5 x^{6} + 10$	$C_5^2 : C_4$ (as 10T10)	$100$	$1$	$[\frac{5}{4}, \frac{7}{4}]_{4}$	$[\frac{1}{4},\frac{3}{4}]_{4}$	$[\frac{5}{4}]_{2}$	$[\frac{1}{4}]_{2}$	$[6, 0]$	$[1, 1]$	$z^5 + 2,2 z^2 + 2$	undefined
5.1.10.15a2.9	8	$x^{10} + 10 x^{7} + 5 x^{6} + 10$	$C_5^2 : C_4$ (as 10T10)	$100$	$1$	$[\frac{5}{4}, \frac{7}{4}]_{4}$	$[\frac{1}{4},\frac{3}{4}]_{4}$	$[\frac{5}{4}]_{2}$	$[\frac{1}{4}]_{2}$	$[6, 0]$	$[1, 1]$	$z^5 + 2,2 z^2 + 2$	undefined
5.1.10.15a2.10	4	$x^{10} + 20 x^{6} + 10$	$F_5$ (as 10T4)	$20$	$2$	$[\frac{7}{4}]_{4}$	$[\frac{3}{4}]_{4}$	$[\ ]_{2}$	$[\ ]_{2}$	$[6, 0]$	$[1, 1]$	$z^5 + 2,2 z^2 + 3$	undefined
5.1.10.15a2.11	8	$x^{10} + 5 x^{7} + 20 x^{6} + 10$	$C_5^2 : C_4$ (as 10T10)	$100$	$1$	$[\frac{5}{4}, \frac{7}{4}]_{4}$	$[\frac{1}{4},\frac{3}{4}]_{4}$	$[\frac{5}{4}]_{2}$	$[\frac{1}{4}]_{2}$	$[6, 0]$	$[1, 1]$	$z^5 + 2,2 z^2 + 3$	undefined
5.1.10.15a2.12	8	$x^{10} + 10 x^{7} + 20 x^{6} + 10$	$C_5^2 : C_4$ (as 10T10)	$100$	$1$	$[\frac{5}{4}, \frac{7}{4}]_{4}$	$[\frac{1}{4},\frac{3}{4}]_{4}$	$[\frac{5}{4}]_{2}$	$[\frac{1}{4}]_{2}$	$[6, 0]$	$[1, 1]$	$z^5 + 2,2 z^2 + 3$	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	5.1.10.15a
Base	5.1.1.0a1.1
Degree	\(10\)
e	\(10\)
f	\(1\)
c	\(15\)

Introduction

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Defining polynomial

Invariants

Diagrams

Varying

Galois groups and Hidden Artin slopes

Fields

Showing all 24

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