Global number field 22.0.233978328890422752873824182193069474777787737239153876898740585020329558016.1

• Label: 22.0.23397832889...8016.1
• Degree: $22$
• Signature: $[0, 11]$
• Discriminant: $-\,2^{45}\cdot 3^{28}\cdot 7^{4}\cdot 13\cdot 23^{4}\cdot 137^{16}\cdot 2161$
• Root discriminant: $2401.14$
• Ramified primes: $2, 3, 7, 13, 23, 137, 2161$
• Class number: Not computed
• Class group: Not computed
• Galois group: 22T52

Normalized defining polynomial

\( x^{22} + 58 x^{20} + 1489 x^{18} + 22320 x^{16} + 216663 x^{14} + 1424964 x^{12} + 6438231 x^{10} + 19757580 x^{8} + 39484992 x^{6} + 46601624 x^{4} + 25144376 x^{2} + 898976 \)

Invariants

Degree:		$22$
Signature:		$[0, 11]$
Discriminant:		\(-233978328890422752873824182193069474777787737239153876898740585020329558016=-\,2^{45}\cdot 3^{28}\cdot 7^{4}\cdot 13\cdot 23^{4}\cdot 137^{16}\cdot 2161\)
Root discriminant:		$2401.14$
Ramified primes:		$2, 3, 7, 13, 23, 137, 2161$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{12} a^{18} - \frac{1}{4} a^{14} - \frac{1}{2} a^{12} + \frac{1}{4} a^{10} - \frac{1}{2} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{3}$, $\frac{1}{12} a^{19} - \frac{1}{4} a^{15} - \frac{1}{2} a^{13} + \frac{1}{4} a^{11} - \frac{1}{2} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{3} a$, $\frac{1}{27000} a^{20} - \frac{467}{13500} a^{18} - \frac{129}{1000} a^{16} - \frac{77}{375} a^{14} + \frac{1937}{9000} a^{12} + \frac{69}{250} a^{10} + \frac{161}{1000} a^{8} + \frac{113}{2250} a^{6} + \frac{13}{150} a^{4} + \frac{43}{3375} a^{2} - \frac{1234}{3375}$, $\frac{1}{108000} a^{21} - \frac{1}{54000} a^{20} + \frac{1783}{54000} a^{19} + \frac{467}{27000} a^{18} - \frac{629}{4000} a^{17} - \frac{371}{2000} a^{16} + \frac{221}{3000} a^{15} - \frac{149}{375} a^{14} + \frac{15437}{36000} a^{13} + \frac{2563}{18000} a^{12} - \frac{7}{125} a^{11} + \frac{181}{500} a^{10} + \frac{661}{4000} a^{9} - \frac{661}{2000} a^{8} + \frac{619}{4500} a^{7} - \frac{113}{4500} a^{6} + \frac{11}{75} a^{5} - \frac{22}{75} a^{4} - \frac{6707}{13500} a^{3} - \frac{43}{6750} a^{2} - \frac{109}{13500} a - \frac{2141}{6750}$

Class group and class number

Not computed

Unit group

Rank:		$10$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Not computed
Regulator:		Not computed

Galois group

22T52:

A non-solvable group of order 40874803200

The 400 conjugacy class representatives for t22n52 are not computed

Character table for t22n52 is not computed

Intermediate fields

11.7.63019333158425674204677255696384.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 44 sibling:

data not computed

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	R	${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$	R	$22$	R	${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$	R	${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$	$22$	${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/41.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$	${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$	${\href{/LocalNumberField/47.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$	${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	2.8.21.22	$x^{8} + 2 x^{6} + 4 x^{2} + 10$	$8$	$1$	$21$	$C_2 \wr C_2\wr C_2$	$[2, 2, 3, 7/2, 7/2, 15/4]^{2}$
	2.12.24.388	$x^{12} + 2 x^{10} + 4 x^{9} - 2 x^{8} + 2 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} - 2 x^{2} + 4 x - 2$	$12$	$1$	$24$	12T149	$[4/3, 4/3, 2, 7/3, 7/3, 3]_{3}^{2}$
$3$	3.2.1.1	$x^{2} - 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.1	$x^{2} - 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.9.13.2	$x^{9} + 6 x^{5} + 3 x^{3} + 3$	$9$	$1$	$13$	$C_3^2 : D_{6} $	$[3/2, 3/2, 5/3]_{2}^{2}$
	3.9.13.2	$x^{9} + 6 x^{5} + 3 x^{3} + 3$	$9$	$1$	$13$	$C_3^2 : D_{6} $	$[3/2, 3/2, 5/3]_{2}^{2}$
7	Data not computed
$13$	$\Q_{13}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{13}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	13.2.1.2	$x^{2} + 26$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.9.0.1	$x^{9} - 2 x + 2$	$1$	$9$	$0$	$C_9$	$[\ ]^{9}$
	13.9.0.1	$x^{9} - 2 x + 2$	$1$	$9$	$0$	$C_9$	$[\ ]^{9}$
$23$	23.4.2.1	$x^{4} + 299 x^{2} + 25921$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	23.4.2.1	$x^{4} + 299 x^{2} + 25921$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	23.7.0.1	$x^{7} - x + 8$	$1$	$7$	$0$	$C_7$	$[\ ]^{7}$
	23.7.0.1	$x^{7} - x + 8$	$1$	$7$	$0$	$C_7$	$[\ ]^{7}$
137	Data not computed
2161	Data not computed