Global number field 15.15.10943023107606534329121.1

• Label: 15.15.1094302310...9121.1
• Degree: $15$
• Signature: $[15, 0]$
• Discriminant: $3^{20}\cdot 11^{12}$
• Root discriminant: $29.46$
• Ramified primes: $3, 11$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{15}$ (as 15T1)

Normalized defining polynomial

\( x^{15} - 27 x^{13} - 4 x^{12} + 252 x^{11} + 60 x^{10} - 976 x^{9} - 288 x^{8} + 1473 x^{7} + 384 x^{6} - 765 x^{5} - 168 x^{4} + 150 x^{3} + 27 x^{2} - 9 x - 1 \)

Invariants

Degree:		$15$
Signature:		$[15, 0]$
Discriminant:		\(10943023107606534329121=3^{20}\cdot 11^{12}\)
Root discriminant:		$29.46$
Ramified primes:		$3, 11$
This field is Galois and abelian over $\Q$.
Conductor:		\(99=3^{2}\cdot 11\)
Dirichlet character group:		$\lbrace$$\chi_{99}(64,·)$, $\chi_{99}(1,·)$, $\chi_{99}(34,·)$, $\chi_{99}(67,·)$, $\chi_{99}(4,·)$, $\chi_{99}(37,·)$, $\chi_{99}(70,·)$, $\chi_{99}(97,·)$, $\chi_{99}(16,·)$, $\chi_{99}(49,·)$, $\chi_{99}(82,·)$, $\chi_{99}(25,·)$, $\chi_{99}(58,·)$, $\chi_{99}(91,·)$, $\chi_{99}(31,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{164276239249} a^{14} + \frac{32330953608}{164276239249} a^{13} + \frac{74618147066}{164276239249} a^{12} - \frac{7544382288}{164276239249} a^{11} - \frac{50914705859}{164276239249} a^{10} + \frac{3571539703}{164276239249} a^{9} - \frac{62434792324}{164276239249} a^{8} - \frac{80812000837}{164276239249} a^{7} - \frac{13866789831}{164276239249} a^{6} - \frac{48159141661}{164276239249} a^{5} + \frac{60633542357}{164276239249} a^{4} - \frac{855515036}{164276239249} a^{3} - \frac{76391229334}{164276239249} a^{2} + \frac{263409277}{164276239249} a + \frac{11630783081}{164276239249}$

Class group and class number

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

Unit group

Rank:		$14$
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:		\( 967645.576239 \) (assuming GRH)

Galois group

$C_{15}$ (as 15T1):

A cyclic group of order 15

The 15 conjugacy class representatives for $C_{15}$

Character table for $C_{15}$

Intermediate fields

\(\Q(\zeta_{9})^+\), \(\Q(\zeta_{11})^+\)

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	$15$	R	$15$	$15$	R	$15$	${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$	${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$	${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$	$15$	$15$	${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$	$15$	${\href{/LocalNumberField/43.3.0.1}{3} }^{5}$	$15$	${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$	$15$

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
3	Data not computed
11	Data not computed

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.3e2.3t1.1c1	$1$	$ 3^{2}$	$x^{3} - 3 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.3e2.3t1.1c2	$1$	$ 3^{2}$	$x^{3} - 3 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.11.5t1.1c1	$1$	$ 11 $	$x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$	$C_5$ (as 5T1)	$0$	$1$
*	1.3e2_11.15t1.1c1	$1$	$ 3^{2} \cdot 11 $	$x^{15} - 27 x^{13} - 4 x^{12} + 252 x^{11} + 60 x^{10} - 976 x^{9} - 288 x^{8} + 1473 x^{7} + 384 x^{6} - 765 x^{5} - 168 x^{4} + 150 x^{3} + 27 x^{2} - 9 x - 1$	$C_{15}$ (as 15T1)	$0$	$1$
*	1.3e2_11.15t1.1c2	$1$	$ 3^{2} \cdot 11 $	$x^{15} - 27 x^{13} - 4 x^{12} + 252 x^{11} + 60 x^{10} - 976 x^{9} - 288 x^{8} + 1473 x^{7} + 384 x^{6} - 765 x^{5} - 168 x^{4} + 150 x^{3} + 27 x^{2} - 9 x - 1$	$C_{15}$ (as 15T1)	$0$	$1$
*	1.11.5t1.1c2	$1$	$ 11 $	$x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$	$C_5$ (as 5T1)	$0$	$1$
*	1.3e2_11.15t1.1c3	$1$	$ 3^{2} \cdot 11 $	$x^{15} - 27 x^{13} - 4 x^{12} + 252 x^{11} + 60 x^{10} - 976 x^{9} - 288 x^{8} + 1473 x^{7} + 384 x^{6} - 765 x^{5} - 168 x^{4} + 150 x^{3} + 27 x^{2} - 9 x - 1$	$C_{15}$ (as 15T1)	$0$	$1$
*	1.3e2_11.15t1.1c4	$1$	$ 3^{2} \cdot 11 $	$x^{15} - 27 x^{13} - 4 x^{12} + 252 x^{11} + 60 x^{10} - 976 x^{9} - 288 x^{8} + 1473 x^{7} + 384 x^{6} - 765 x^{5} - 168 x^{4} + 150 x^{3} + 27 x^{2} - 9 x - 1$	$C_{15}$ (as 15T1)	$0$	$1$
*	1.11.5t1.1c3	$1$	$ 11 $	$x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$	$C_5$ (as 5T1)	$0$	$1$
*	1.3e2_11.15t1.1c5	$1$	$ 3^{2} \cdot 11 $	$x^{15} - 27 x^{13} - 4 x^{12} + 252 x^{11} + 60 x^{10} - 976 x^{9} - 288 x^{8} + 1473 x^{7} + 384 x^{6} - 765 x^{5} - 168 x^{4} + 150 x^{3} + 27 x^{2} - 9 x - 1$	$C_{15}$ (as 15T1)	$0$	$1$
*	1.3e2_11.15t1.1c6	$1$	$ 3^{2} \cdot 11 $	$x^{15} - 27 x^{13} - 4 x^{12} + 252 x^{11} + 60 x^{10} - 976 x^{9} - 288 x^{8} + 1473 x^{7} + 384 x^{6} - 765 x^{5} - 168 x^{4} + 150 x^{3} + 27 x^{2} - 9 x - 1$	$C_{15}$ (as 15T1)	$0$	$1$
*	1.11.5t1.1c4	$1$	$ 11 $	$x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$	$C_5$ (as 5T1)	$0$	$1$
*	1.3e2_11.15t1.1c7	$1$	$ 3^{2} \cdot 11 $	$x^{15} - 27 x^{13} - 4 x^{12} + 252 x^{11} + 60 x^{10} - 976 x^{9} - 288 x^{8} + 1473 x^{7} + 384 x^{6} - 765 x^{5} - 168 x^{4} + 150 x^{3} + 27 x^{2} - 9 x - 1$	$C_{15}$ (as 15T1)	$0$	$1$
*	1.3e2_11.15t1.1c8	$1$	$ 3^{2} \cdot 11 $	$x^{15} - 27 x^{13} - 4 x^{12} + 252 x^{11} + 60 x^{10} - 976 x^{9} - 288 x^{8} + 1473 x^{7} + 384 x^{6} - 765 x^{5} - 168 x^{4} + 150 x^{3} + 27 x^{2} - 9 x - 1$	$C_{15}$ (as 15T1)	$0$	$1$

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 *.