Global number field 39.39.1287743804278744050410620426954739687963064854495168753870500853746064161.1: \(\Q(\zeta_{79})^+\)

• Label: 39.39.1287743804...4161.1
• Degree: $39$
• Signature: $[39, 0]$
• Discriminant: $79^{38}$
• Root discriminant: $70.63$
• Ramified prime: $79$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{39}$ (as 39T1)

Normalized defining polynomial

\( x^{39} - x^{38} - 38 x^{37} + 37 x^{36} + 666 x^{35} - 630 x^{34} - 7140 x^{33} + 6545 x^{32} + 52360 x^{31} - 46376 x^{30} - 278256 x^{29} + 237336 x^{28} + 1107568 x^{27} - 906192 x^{26} - 3365856 x^{25} + 2629575 x^{24} + 7888725 x^{23} - 5852925 x^{22} - 14307150 x^{21} + 10015005 x^{20} + 20030010 x^{19} - 13123110 x^{18} - 21474180 x^{17} + 13037895 x^{16} + 17383860 x^{15} - 9657700 x^{14} - 10400600 x^{13} + 5200300 x^{12} + 4457400 x^{11} - 1961256 x^{10} - 1307504 x^{9} + 490314 x^{8} + 245157 x^{7} - 74613 x^{6} - 26334 x^{5} + 5985 x^{4} + 1330 x^{3} - 190 x^{2} - 20 x + 1 \)

Invariants

Degree:		$39$
Signature:		$[39, 0]$
Discriminant:		\(1287743804278744050410620426954739687963064854495168753870500853746064161=79^{38}\)
Root discriminant:		$70.63$
Ramified primes:		$79$
This field is Galois and abelian over $\Q$.
Conductor:		\(79\)
Dirichlet character group:		$\lbrace$$\chi_{79}(1,·)$, $\chi_{79}(2,·)$, $\chi_{79}(4,·)$, $\chi_{79}(5,·)$, $\chi_{79}(8,·)$, $\chi_{79}(9,·)$, $\chi_{79}(10,·)$, $\chi_{79}(11,·)$, $\chi_{79}(13,·)$, $\chi_{79}(16,·)$, $\chi_{79}(18,·)$, $\chi_{79}(19,·)$, $\chi_{79}(20,·)$, $\chi_{79}(21,·)$, $\chi_{79}(22,·)$, $\chi_{79}(23,·)$, $\chi_{79}(25,·)$, $\chi_{79}(26,·)$, $\chi_{79}(31,·)$, $\chi_{79}(32,·)$, $\chi_{79}(36,·)$, $\chi_{79}(38,·)$, $\chi_{79}(40,·)$, $\chi_{79}(42,·)$, $\chi_{79}(44,·)$, $\chi_{79}(45,·)$, $\chi_{79}(46,·)$, $\chi_{79}(49,·)$, $\chi_{79}(50,·)$, $\chi_{79}(51,·)$, $\chi_{79}(52,·)$, $\chi_{79}(55,·)$, $\chi_{79}(62,·)$, $\chi_{79}(64,·)$, $\chi_{79}(65,·)$, $\chi_{79}(67,·)$, $\chi_{79}(72,·)$, $\chi_{79}(73,·)$, $\chi_{79}(76,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$38$
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:		\( 536916737153376600000000 \) (assuming GRH)

Galois group

$C_{39}$ (as 39T1):

A cyclic group of order 39

The 39 conjugacy class representatives for $C_{39}$

Character table for $C_{39}$ is not computed

Intermediate fields

3.3.6241.1, 13.13.59091511031674153381441.1

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	$39$	$39$	$39$	$39$	$39$	$39$	${\href{/LocalNumberField/17.13.0.1}{13} }^{3}$	$39$	${\href{/LocalNumberField/23.3.0.1}{3} }^{13}$	$39$	$39$	$39$	${\href{/LocalNumberField/41.13.0.1}{13} }^{3}$	$39$	$39$	$39$	$39$

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
79	Data not computed