Global number field 36.0.14319849782617254182563766395402434634362109577782479558283362304.1

• Label: 36.0.14319849782...2304.1
• Degree: $36$
• Signature: $[0, 18]$
• Discriminant: $2^{36}\cdot 37^{34}$
• Root discriminant: $60.55$
• Ramified primes: $2, 37$
• Class number: $130473$ (GRH)
• Class group: $[130473]$ (GRH)
• Galois group: $C_2\times C_{18}$ (as 36T2)

Normalized defining polynomial

\( x^{36} + 35 x^{34} + 561 x^{32} + 5456 x^{30} + 35960 x^{28} + 169911 x^{26} + 593775 x^{24} + 1560780 x^{22} + 3108105 x^{20} + 4686825 x^{18} + 5311735 x^{16} + 4457400 x^{14} + 2704156 x^{12} + 1144066 x^{10} + 319770 x^{8} + 54264 x^{6} + 4845 x^{4} + 171 x^{2} + 1 \)

Invariants

Degree:		$36$
Signature:		$[0, 18]$
Discriminant:		\(14319849782617254182563766395402434634362109577782479558283362304=2^{36}\cdot 37^{34}\)
Root discriminant:		$60.55$
Ramified primes:		$2, 37$
This field is Galois and abelian over $\Q$.
Conductor:		\(148=2^{2}\cdot 37\)
Dirichlet character group:		$\lbrace$$\chi_{148}(1,·)$, $\chi_{148}(3,·)$, $\chi_{148}(7,·)$, $\chi_{148}(9,·)$, $\chi_{148}(11,·)$, $\chi_{148}(141,·)$, $\chi_{148}(145,·)$, $\chi_{148}(147,·)$, $\chi_{148}(21,·)$, $\chi_{148}(25,·)$, $\chi_{148}(27,·)$, $\chi_{148}(33,·)$, $\chi_{148}(41,·)$, $\chi_{148}(47,·)$, $\chi_{148}(49,·)$, $\chi_{148}(53,·)$, $\chi_{148}(137,·)$, $\chi_{148}(63,·)$, $\chi_{148}(65,·)$, $\chi_{148}(67,·)$, $\chi_{148}(71,·)$, $\chi_{148}(73,·)$, $\chi_{148}(75,·)$, $\chi_{148}(77,·)$, $\chi_{148}(81,·)$, $\chi_{148}(83,·)$, $\chi_{148}(139,·)$, $\chi_{148}(85,·)$, $\chi_{148}(95,·)$, $\chi_{148}(99,·)$, $\chi_{148}(101,·)$, $\chi_{148}(107,·)$, $\chi_{148}(115,·)$, $\chi_{148}(121,·)$, $\chi_{148}(123,·)$, $\chi_{148}(127,·)$$\rbrace$
This is a CM field.

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

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

Class group and class number

$C_{130473}$, which has order $130473$ (assuming GRH)

Unit group

Rank:		$17$
Torsion generator:		\( a^{35} + 34 a^{33} + 528 a^{31} + 4960 a^{29} + 31465 a^{27} + 142506 a^{25} + 475020 a^{23} + 1184040 a^{21} + 2220075 a^{19} + 3124550 a^{17} + 3268760 a^{15} + 2496144 a^{13} + 1352078 a^{11} + 497420 a^{9} + 116280 a^{7} + 15504 a^{5} + 969 a^{3} + 18 a \) (order $4$)
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:		\( 3171872728760.222 \) (assuming GRH)

Galois group

$C_2\times C_{18}$ (as 36T2):

An abelian group of order 36

The 36 conjugacy class representatives for $C_2\times C_{18}$

Character table for $C_2\times C_{18}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-37}) \), \(\Q(\sqrt{37}) \), 3.3.1369.1, \(\Q(i, \sqrt{37})\), 6.0.119946304.1, 6.0.4438013248.1, 6.6.69343957.1, 9.9.3512479453921.1, 12.0.19695961589423509504.1, 18.0.3234204723240544858872018632704.1, 18.0.119665574759900159778264689410048.1, \(\Q(\zeta_{37})^+\)

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	R	$18^{2}$	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$	$18^{2}$	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{18}$	R	${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{18}$	${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/53.9.0.1}{9} }^{4}$	$18^{2}$

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
2	Data not computed
37	Data not computed