Number field 35.35.876564456148583685580741416193317498080031692578600918378223281.1: \(\Q(\zeta_{71})^+\)

• Label: 35.35.876...281.1
• Degree: $35$
• Signature: $[35, 0]$
• Discriminant: $8.766\times 10^{62}$
• Root discriminant: $62.86$
• Ramified prime: $71$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{35}$ (as 35T1)

Normalized defining polynomial

\( x^{35} - x^{34} - 34 x^{33} + 33 x^{32} + 528 x^{31} - 496 x^{30} - 4960 x^{29} + 4495 x^{28} + 31465 x^{27} - 27405 x^{26} - 142506 x^{25} + 118755 x^{24} + 475020 x^{23} - 376740 x^{22} - 1184040 x^{21} + 888030 x^{20} + 2220075 x^{19} - 1562275 x^{18} - 3124550 x^{17} + 2042975 x^{16} + 3268760 x^{15} - 1961256 x^{14} - 2496144 x^{13} + 1352078 x^{12} + 1352078 x^{11} - 646646 x^{10} - 497420 x^{9} + 203490 x^{8} + 116280 x^{7} - 38760 x^{6} - 15504 x^{5} + 3876 x^{4} + 969 x^{3} - 153 x^{2} - 18 x + 1 \)

Invariants

Degree:		$35$
Signature:		$[35, 0]$
Discriminant:		\(876\!\cdots\!281\)\(\medspace = 71^{34}\)
Root discriminant:		$62.86$
Ramified primes:		$71$
$|\Gal(K/\Q)|$:		$35$
This field is Galois and abelian over $\Q$.
Conductor:		\(71\)
Dirichlet character group:		$\lbrace$$\chi_{71}(1,·)$, $\chi_{71}(2,·)$, $\chi_{71}(3,·)$, $\chi_{71}(4,·)$, $\chi_{71}(5,·)$, $\chi_{71}(6,·)$, $\chi_{71}(8,·)$, $\chi_{71}(9,·)$, $\chi_{71}(10,·)$, $\chi_{71}(12,·)$, $\chi_{71}(15,·)$, $\chi_{71}(16,·)$, $\chi_{71}(18,·)$, $\chi_{71}(19,·)$, $\chi_{71}(20,·)$, $\chi_{71}(24,·)$, $\chi_{71}(25,·)$, $\chi_{71}(27,·)$, $\chi_{71}(29,·)$, $\chi_{71}(30,·)$, $\chi_{71}(32,·)$, $\chi_{71}(36,·)$, $\chi_{71}(37,·)$, $\chi_{71}(38,·)$, $\chi_{71}(40,·)$, $\chi_{71}(43,·)$, $\chi_{71}(45,·)$, $\chi_{71}(48,·)$, $\chi_{71}(49,·)$, $\chi_{71}(50,·)$, $\chi_{71}(54,·)$, $\chi_{71}(57,·)$, $\chi_{71}(58,·)$, $\chi_{71}(60,·)$, $\chi_{71}(64,·)$$\rbrace$
This is not a CM field.

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

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

Class group and class number

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

Unit group

Rank:		$34$
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:		\( 190122458612916620000 \) (assuming GRH)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{35}\cdot(2\pi)^{0}\cdot 190122458612916620000 \cdot 1}{2\sqrt{876564456148583685580741416193317498080031692578600918378223281}}\approx 0.110321801792764$ (assuming GRH)

Galois group

$C_{35}$ (as 35T1):

A cyclic group of order 35

The 35 conjugacy class representatives for $C_{35}$

Character table for $C_{35}$ is not computed

Intermediate fields

5.5.25411681.1, 7.7.128100283921.1

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	$35$	$35$	${\href{/LocalNumberField/5.5.0.1}{5} }^{7}$	$35$	$35$	$35$	${\href{/LocalNumberField/17.5.0.1}{5} }^{7}$	$35$	${\href{/LocalNumberField/23.7.0.1}{7} }^{5}$	$35$	$35$	${\href{/LocalNumberField/37.7.0.1}{7} }^{5}$	${\href{/LocalNumberField/41.7.0.1}{7} }^{5}$	$35$	$35$	$35$	$35$

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
71	Data not computed