Global number field 36.0.114573059505387793044837364496233492772337802886962890625.1

• Label: 36.0.11457305950...0625.1
• Degree: $36$
• Signature: $[0, 18]$
• Discriminant: $5^{18}\cdot 19^{34}$
• Root discriminant: $36.07$
• Ramified primes: $5, 19$
• Class number: $76$ (GRH)
• Class group: $[76]$ (GRH)
• Galois group: $C_2\times C_{18}$ (as 36T2)

Normalized defining polynomial

\( x^{36} - x^{35} + 2 x^{34} - 3 x^{33} + 5 x^{32} - 8 x^{31} + 13 x^{30} - 21 x^{29} + 34 x^{28} - 55 x^{27} + 89 x^{26} - 144 x^{25} + 233 x^{24} - 377 x^{23} + 610 x^{22} - 987 x^{21} + 1597 x^{20} - 2584 x^{19} + 4181 x^{18} + 2584 x^{17} + 1597 x^{16} + 987 x^{15} + 610 x^{14} + 377 x^{13} + 233 x^{12} + 144 x^{11} + 89 x^{10} + 55 x^{9} + 34 x^{8} + 21 x^{7} + 13 x^{6} + 8 x^{5} + 5 x^{4} + 3 x^{3} + 2 x^{2} + x + 1 \)

Invariants

Degree:		$36$
Signature:		$[0, 18]$
Discriminant:		\(114573059505387793044837364496233492772337802886962890625=5^{18}\cdot 19^{34}\)
Root discriminant:		$36.07$
Ramified primes:		$5, 19$
This field is Galois and abelian over $\Q$.
Conductor:		\(95=5\cdot 19\)
Dirichlet character group:		$\lbrace$$\chi_{95}(1,·)$, $\chi_{95}(4,·)$, $\chi_{95}(6,·)$, $\chi_{95}(9,·)$, $\chi_{95}(11,·)$, $\chi_{95}(14,·)$, $\chi_{95}(16,·)$, $\chi_{95}(21,·)$, $\chi_{95}(24,·)$, $\chi_{95}(26,·)$, $\chi_{95}(29,·)$, $\chi_{95}(31,·)$, $\chi_{95}(34,·)$, $\chi_{95}(36,·)$, $\chi_{95}(39,·)$, $\chi_{95}(41,·)$, $\chi_{95}(44,·)$, $\chi_{95}(46,·)$, $\chi_{95}(49,·)$, $\chi_{95}(51,·)$, $\chi_{95}(54,·)$, $\chi_{95}(56,·)$, $\chi_{95}(59,·)$, $\chi_{95}(61,·)$, $\chi_{95}(64,·)$, $\chi_{95}(66,·)$, $\chi_{95}(69,·)$, $\chi_{95}(71,·)$, $\chi_{95}(74,·)$, $\chi_{95}(79,·)$, $\chi_{95}(81,·)$, $\chi_{95}(84,·)$, $\chi_{95}(86,·)$, $\chi_{95}(89,·)$, $\chi_{95}(91,·)$, $\chi_{95}(94,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{4181} a^{19} - \frac{1597}{4181}$, $\frac{1}{4181} a^{20} - \frac{1597}{4181} a$, $\frac{1}{4181} a^{21} - \frac{1597}{4181} a^{2}$, $\frac{1}{4181} a^{22} - \frac{1597}{4181} a^{3}$, $\frac{1}{4181} a^{23} - \frac{1597}{4181} a^{4}$, $\frac{1}{4181} a^{24} - \frac{1597}{4181} a^{5}$, $\frac{1}{4181} a^{25} - \frac{1597}{4181} a^{6}$, $\frac{1}{4181} a^{26} - \frac{1597}{4181} a^{7}$, $\frac{1}{4181} a^{27} - \frac{1597}{4181} a^{8}$, $\frac{1}{4181} a^{28} - \frac{1597}{4181} a^{9}$, $\frac{1}{4181} a^{29} - \frac{1597}{4181} a^{10}$, $\frac{1}{4181} a^{30} - \frac{1597}{4181} a^{11}$, $\frac{1}{4181} a^{31} - \frac{1597}{4181} a^{12}$, $\frac{1}{4181} a^{32} - \frac{1597}{4181} a^{13}$, $\frac{1}{4181} a^{33} - \frac{1597}{4181} a^{14}$, $\frac{1}{4181} a^{34} - \frac{1597}{4181} a^{15}$, $\frac{1}{4181} a^{35} - \frac{1597}{4181} a^{16}$

Class group and class number

$C_{76}$, which has order $76$ (assuming GRH)

Unit group

Rank:		$17$
Torsion generator:		\( -\frac{987}{4181} a^{35} - \frac{9227465}{4181} a^{16} \) (order $38$)
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:		\( 3431432369157.3267 \) (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

\(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{5}) \), 3.3.361.1, \(\Q(\sqrt{5}, \sqrt{-19})\), 6.0.2476099.1, 6.0.309512375.1, 6.6.16290125.1, \(\Q(\zeta_{19})^+\), 12.0.95797910278140625.1, \(\Q(\zeta_{19})\), 18.0.10703880581610941769412109375.1, 18.18.563362135874260093126953125.1

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	$18^{2}$	$18^{2}$	R	${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$	$18^{2}$	$18^{2}$	R	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$	$18^{2}$	$18^{2}$	$18^{2}$	$18^{2}$	$18^{2}$

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
5	Data not computed
19	Data not computed