Global number field 41.41.57959375186337998161464929843210464026538099255933595673241672975683189751201.1: \(\Q(\zeta_{83})^+\)

• Label: 41.41.5795937518...1201.1
• Degree: $41$
• Signature: $[41, 0]$
• Discriminant: $83^{40}$
• Root discriminant: $74.52$
• Ramified prime: $83$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{41}$ (as 41T1)

Normalized defining polynomial

\( x^{41} - x^{40} - 40 x^{39} + 39 x^{38} + 741 x^{37} - 703 x^{36} - 8436 x^{35} + 7770 x^{34} + 66045 x^{33} - 58905 x^{32} - 376992 x^{31} + 324632 x^{30} + 1623160 x^{29} - 1344904 x^{28} - 5379616 x^{27} + 4272048 x^{26} + 13884156 x^{25} - 10518300 x^{24} - 28048800 x^{23} + 20160075 x^{22} + 44352165 x^{21} - 30045015 x^{20} - 54627300 x^{19} + 34597290 x^{18} + 51895935 x^{17} - 30421755 x^{16} - 37442160 x^{15} + 20058300 x^{14} + 20058300 x^{13} - 9657700 x^{12} - 7726160 x^{11} + 3268760 x^{10} + 2042975 x^{9} - 735471 x^{8} - 346104 x^{7} + 100947 x^{6} + 33649 x^{5} - 7315 x^{4} - 1540 x^{3} + 210 x^{2} + 21 x - 1 \)

Invariants

Degree:		$41$
Signature:		$[41, 0]$
Discriminant:		\(57959375186337998161464929843210464026538099255933595673241672975683189751201=83^{40}\)
Root discriminant:		$74.52$
Ramified primes:		$83$
This field is Galois and abelian over $\Q$.
Conductor:		\(83\)
Dirichlet character group:		$\lbrace$$\chi_{83}(1,·)$, $\chi_{83}(3,·)$, $\chi_{83}(4,·)$, $\chi_{83}(7,·)$, $\chi_{83}(9,·)$, $\chi_{83}(10,·)$, $\chi_{83}(11,·)$, $\chi_{83}(12,·)$, $\chi_{83}(16,·)$, $\chi_{83}(17,·)$, $\chi_{83}(21,·)$, $\chi_{83}(23,·)$, $\chi_{83}(25,·)$, $\chi_{83}(26,·)$, $\chi_{83}(27,·)$, $\chi_{83}(28,·)$, $\chi_{83}(29,·)$, $\chi_{83}(30,·)$, $\chi_{83}(31,·)$, $\chi_{83}(33,·)$, $\chi_{83}(36,·)$, $\chi_{83}(37,·)$, $\chi_{83}(38,·)$, $\chi_{83}(40,·)$, $\chi_{83}(41,·)$, $\chi_{83}(44,·)$, $\chi_{83}(48,·)$, $\chi_{83}(49,·)$, $\chi_{83}(51,·)$, $\chi_{83}(59,·)$, $\chi_{83}(61,·)$, $\chi_{83}(63,·)$, $\chi_{83}(64,·)$, $\chi_{83}(65,·)$, $\chi_{83}(68,·)$, $\chi_{83}(69,·)$, $\chi_{83}(70,·)$, $\chi_{83}(75,·)$, $\chi_{83}(77,·)$, $\chi_{83}(78,·)$, $\chi_{83}(81,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$

Class group and class number

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

Unit group

Rank:		$40$
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:		\( 29837391421490950000000000 \) (assuming GRH)

Galois group

$C_{41}$ (as 41T1):

A cyclic group of order 41

The 41 conjugacy class representatives for $C_{41}$

Character table for $C_{41}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$

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
83	Data not computed