Number field 9.1.33860761.1

• Label: 9.1.33860761.1
• Degree: $9$
• Signature: $[1, 4]$
• Discriminant: $33860761$
• Root discriminant: $6.86$
• Ramified primes: $11, 23$
• Class number: $1$
• Class group: trivial
• Galois group: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30)

Normalized defining polynomial

\(x^{9} - 2 x^{8} + 2 x^{7} - 2 x^{5} + 2 x^{4} - x + 1\)  

Invariants

Degree:		$9$
Signature:		$[1, 4]$
Discriminant:		\(33860761\)\(\medspace = 11^{2}\cdot 23^{4}\)
Root discriminant:		$6.86$
Ramified primes:		$11, 23$
$|\Aut(K/\Q)|$:		$1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$  

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$4$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( a \),  \( a^{8} - a^{7} + a^{6} - a^{4} + a - 1 \),  \( a^{8} - a^{7} + a^{6} - a^{4} + a^{3} + a \),  \( a - 1 \)
Regulator:		\( 0.744153660637 \)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{4}\cdot 0.744153660637 \cdot 1}{2\sqrt{33860761}}\approx 0.1993121338299$

Galois group

$(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30):

A solvable group of order 648

The 14 conjugacy class representatives for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$

Character table for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$

Intermediate fields

3.1.23.1

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

Sibling fields

Degree 12 siblings:	data not computed
Degree 18 siblings:	data not computed
Degree 24 siblings:	data not computed
Degree 27 siblings:	data not computed
Degree 36 siblings:	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	${\href{/LocalNumberField/2.9.0.1}{9} }$	${\href{/LocalNumberField/3.9.0.1}{9} }$	${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$	${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$	R	${\href{/LocalNumberField/13.9.0.1}{9} }$	${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$	${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$	R	${\href{/LocalNumberField/29.3.0.1}{3} }^{3}$	${\href{/LocalNumberField/31.9.0.1}{9} }$	${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$	${\href{/LocalNumberField/41.9.0.1}{9} }$	${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$	${\href{/LocalNumberField/47.9.0.1}{9} }$	${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$	${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$11$	11.2.0.1	$x^{2} - x + 7$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	11.3.0.1	$x^{3} - x + 3$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	11.4.2.2	$x^{4} - 11 x^{2} + 847$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$
$23$	$\Q_{23}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	23.2.0.1	$x^{2} - x + 7$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	23.2.1.2	$x^{2} + 46$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	23.4.3.1	$x^{4} + 46$	$4$	$1$	$3$	$D_{4}$	$[\ ]_{4}^{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
	1.23.2t1.a.a	$1$	$ 23 $	\(\Q(\sqrt{-23}) \)	$C_2$ (as 2T1)	$1$	$-1$
*	2.23.3t2.b.a	$2$	$ 23 $	3.1.23.1	$S_3$ (as 3T2)	$1$	$0$
	3.64009.6t8.a.a	$3$	$ 11^{2} \cdot 23^{2}$	4.2.1472207.1	$S_4$ (as 4T5)	$1$	$-1$
	3.1472207.4t5.a.a	$3$	$ 11^{2} \cdot 23^{3}$	4.2.1472207.1	$S_4$ (as 4T5)	$1$	$1$
	6.778797503.18t217.a.a	$6$	$ 11^{2} \cdot 23^{5}$	9.1.33860761.1	$(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30)	$1$	$0$
*	6.1472207.9t30.a.a	$6$	$ 11^{2} \cdot 23^{3}$	9.1.33860761.1	$(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30)	$1$	$0$
	6.94234497863.36t1121.a.a	$6$	$ 11^{4} \cdot 23^{5}$	9.1.33860761.1	$(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30)	$1$	$0$
	6.94234497863.36t1121.a.b	$6$	$ 11^{4} \cdot 23^{5}$	9.1.33860761.1	$(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30)	$1$	$0$
	8.216...849.12t177.a.a	$8$	$ 11^{4} \cdot 23^{6}$	9.1.33860761.1	$(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30)	$1$	$0$
	8.216...849.12t177.b.a	$8$	$ 11^{4} \cdot 23^{6}$	9.1.33860761.1	$(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30)	$1$	$0$
	8.216...849.12t178.a.a	$8$	$ 11^{4} \cdot 23^{6}$	9.1.33860761.1	$(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30)	$1$	$0$
	12.319...743.36t1123.a.a	$12$	$ 11^{6} \cdot 23^{9}$	9.1.33860761.1	$(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30)	$1$	$-2$
	12.319...743.18t218.a.a	$12$	$ 11^{6} \cdot 23^{9}$	9.1.33860761.1	$(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30)	$1$	$2$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.