Global number field 20.0.2351977956823175708448011472615877.1

• Label: 20.0.23519779568...5877.1
• Degree: $20$
• Signature: $[0, 10]$
• Discriminant: $11^{16}\cdot 13^{15}$
• Root discriminant: $46.62$
• Ramified primes: $11, 13$
• Class number: $61$ (GRH)
• Class group: $[61]$ (GRH)
• Galois group: $C_{20}$ (as 20T1)

Normalized defining polynomial

\( x^{20} - x^{19} - 9 x^{18} + 31 x^{17} + 55 x^{16} - 48 x^{15} + 189 x^{14} + 352 x^{13} + 1680 x^{12} + 4302 x^{11} + 4614 x^{10} + 17042 x^{9} + 24675 x^{8} + 19061 x^{7} + 33622 x^{6} - 55350 x^{5} - 15945 x^{4} + 101736 x^{3} + 82138 x^{2} - 16471 x + 40283 \)

Invariants

Degree:		$20$
Signature:		$[0, 10]$
Discriminant:		\(2351977956823175708448011472615877=11^{16}\cdot 13^{15}\)
Root discriminant:		$46.62$
Ramified primes:		$11, 13$
This field is Galois and abelian over $\Q$.
Conductor:		\(143=11\cdot 13\)
Dirichlet character group:		$\lbrace$$\chi_{143}(64,·)$, $\chi_{143}(1,·)$, $\chi_{143}(5,·)$, $\chi_{143}(70,·)$, $\chi_{143}(135,·)$, $\chi_{143}(12,·)$, $\chi_{143}(14,·)$, $\chi_{143}(86,·)$, $\chi_{143}(25,·)$, $\chi_{143}(27,·)$, $\chi_{143}(92,·)$, $\chi_{143}(31,·)$, $\chi_{143}(34,·)$, $\chi_{143}(38,·)$, $\chi_{143}(103,·)$, $\chi_{143}(47,·)$, $\chi_{143}(53,·)$, $\chi_{143}(122,·)$, $\chi_{143}(60,·)$, $\chi_{143}(125,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{14} + \frac{1}{3} a^{12} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{1002552909232732911876759584880629716221031866388410333} a^{19} + \frac{13322468613009091424986291199443448538033041576481547}{334184303077577637292253194960209905407010622129470111} a^{18} + \frac{100367416746831160734708785725084141084484061717365446}{1002552909232732911876759584880629716221031866388410333} a^{17} - \frac{6887487268967052784399285571338552805765967138994746}{334184303077577637292253194960209905407010622129470111} a^{16} + \frac{51755272969180980571439087646114829029830396178310730}{334184303077577637292253194960209905407010622129470111} a^{15} - \frac{14209687856415122232857842505287132396567787698591129}{334184303077577637292253194960209905407010622129470111} a^{14} + \frac{312864643001654378912317626704563760863426671187864205}{1002552909232732911876759584880629716221031866388410333} a^{13} - \frac{61084222438149699733877602742405346393610567601123201}{334184303077577637292253194960209905407010622129470111} a^{12} - \frac{440486565097841447558041070371950277643183442686491787}{1002552909232732911876759584880629716221031866388410333} a^{11} + \frac{14884885638048560923001901904617025365845235489743052}{334184303077577637292253194960209905407010622129470111} a^{10} + \frac{173461174828237013495802562511869865191520590231158182}{1002552909232732911876759584880629716221031866388410333} a^{9} + \frac{115365782535210522934093463781545811712324428408476409}{334184303077577637292253194960209905407010622129470111} a^{8} + \frac{195691401029955052478875815641899730433829067667856525}{1002552909232732911876759584880629716221031866388410333} a^{7} - \frac{71940385383994937279447134941308584210540865466243010}{334184303077577637292253194960209905407010622129470111} a^{6} + \frac{121405721813249971216174226519753365525087393740598780}{334184303077577637292253194960209905407010622129470111} a^{5} - \frac{331849143495694748602477487853913233111324328750101878}{1002552909232732911876759584880629716221031866388410333} a^{4} + \frac{155082227053570771270413311462400629305879553300978650}{334184303077577637292253194960209905407010622129470111} a^{3} - \frac{150336049286177240065406483267365011827341337391936639}{334184303077577637292253194960209905407010622129470111} a^{2} + \frac{214298918132153023299728851434881485775393532406917238}{1002552909232732911876759584880629716221031866388410333} a - \frac{440552992461246321241642853073813072871847106051584809}{1002552909232732911876759584880629716221031866388410333}$

Class group and class number

$C_{61}$, which has order $61$ (assuming GRH)

Unit group

Rank:		$9$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
Regulator:		\( 2015201.7242 \) (assuming GRH)

Galois group

$C_{20}$ (as 20T1):

A cyclic group of order 20

The 20 conjugacy class representatives for $C_{20}$

Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, \(\Q(\zeta_{11})^+\), 10.10.79589952003133.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	$20$	${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$	$20$	$20$	R	R	${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$	$20$	${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$	${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$	$20$	$20$	$20$	${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$	$20$	${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$	$20$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
11	Data not computed
13	Data not computed