Number field 30.10.14414591053629993500878399466353926864896.1

• Label: 30.10.144...896.1
• Degree: $30$
• Signature: $[10, 10]$
• Discriminant: $1.441\times 10^{40}$
• Root discriminant: $21.81$
• Ramified primes: $2, 11$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{10}\times S_3$ (as 30T12)

Normalized defining polynomial

\( x^{30} - 15 x^{28} + 104 x^{26} - 482 x^{24} + 1708 x^{22} - 4753 x^{20} + 10608 x^{18} - 18925 x^{16} + 26563 x^{14} - 28460 x^{12} + 22463 x^{10} - 12716 x^{8} + 5049 x^{6} - 1331 x^{4} + 198 x^{2} - 11 \)

Invariants

Degree:		$30$
Signature:		$[10, 10]$
Discriminant:		\(14414591053629993500878399466353926864896\)\(\medspace = 2^{40}\cdot 11^{27}\)
Root discriminant:		$21.81$
Ramified primes:		$2, 11$
$|\Aut(K/\Q)|$:		$10$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{26} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{587409483542016518} a^{28} - \frac{103020836976652849}{587409483542016518} a^{26} - \frac{6816620393891934}{293704741771008259} a^{24} + \frac{149570912943989}{5389077830660702} a^{22} - \frac{47659317464097223}{293704741771008259} a^{20} + \frac{8081375902375867}{587409483542016518} a^{18} + \frac{123471128679091941}{587409483542016518} a^{16} - \frac{16960726100570672}{293704741771008259} a^{14} + \frac{126667923291359885}{587409483542016518} a^{12} - \frac{1}{2} a^{11} - \frac{76716716275769853}{587409483542016518} a^{10} - \frac{1}{2} a^{9} - \frac{237814649392225551}{587409483542016518} a^{8} - \frac{1}{2} a^{7} - \frac{127991483737504008}{293704741771008259} a^{6} - \frac{1}{2} a^{5} - \frac{143232585408898121}{293704741771008259} a^{4} - \frac{55036218279611367}{587409483542016518} a^{2} - \frac{1}{2} a + \frac{208974501016961141}{587409483542016518}$, $\frac{1}{587409483542016518} a^{29} - \frac{103020836976652849}{587409483542016518} a^{27} - \frac{6816620393891934}{293704741771008259} a^{25} + \frac{149570912943989}{5389077830660702} a^{23} - \frac{47659317464097223}{293704741771008259} a^{21} + \frac{8081375902375867}{587409483542016518} a^{19} + \frac{123471128679091941}{587409483542016518} a^{17} - \frac{16960726100570672}{293704741771008259} a^{15} + \frac{126667923291359885}{587409483542016518} a^{13} - \frac{1}{2} a^{12} - \frac{76716716275769853}{587409483542016518} a^{11} - \frac{1}{2} a^{10} - \frac{237814649392225551}{587409483542016518} a^{9} - \frac{1}{2} a^{8} - \frac{127991483737504008}{293704741771008259} a^{7} - \frac{1}{2} a^{6} - \frac{143232585408898121}{293704741771008259} a^{5} - \frac{55036218279611367}{587409483542016518} a^{3} - \frac{1}{2} a^{2} + \frac{208974501016961141}{587409483542016518} a$

Class group and class number

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

Unit group

Rank:		$19$
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:		\( 397294072.2356694 \) (assuming GRH)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{10}\cdot(2\pi)^{10}\cdot 397294072.2356694 \cdot 1}{2\sqrt{14414591053629993500878399466353926864896}}\approx 0.162472389195522$ (assuming GRH)

Galois group

$C_{10}\times S_3$ (as 30T12):

A solvable group of order 60

The 30 conjugacy class representatives for $C_{10}\times S_3$

Character table for $C_{10}\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), 3.1.44.1, \(\Q(\zeta_{11})^+\), 6.2.340736.1, \(\Q(\zeta_{44})^+\), 15.5.35351257235385344.1

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

Sibling fields

Degree 30 sibling:

30.0.1310417368511817590988945406032175169536.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	$30$	$15^{2}$	${\href{/LocalNumberField/7.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$	R	${\href{/LocalNumberField/13.10.0.1}{10} }^{3}$	${\href{/LocalNumberField/17.10.0.1}{10} }^{3}$	${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$	${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$	${\href{/LocalNumberField/29.10.0.1}{10} }^{3}$	$30$	$15^{2}$	${\href{/LocalNumberField/41.10.0.1}{10} }^{3}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{10}$	${\href{/LocalNumberField/47.10.0.1}{10} }^{3}$	${\href{/LocalNumberField/53.5.0.1}{5} }^{6}$	$30$

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
11	Data not computed

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.4.2t1.a.a	$1$	$ 2^{2}$	$x^{2} + 1$	$C_2$ (as 2T1)	$1$	$-1$
*	1.44.2t1.a.a	$1$	$ 2^{2} \cdot 11 $	$x^{2} - 11$	$C_2$ (as 2T1)	$1$	$1$
	1.11.2t1.a.a	$1$	$ 11 $	$x^{2} - x + 3$	$C_2$ (as 2T1)	$1$	$-1$
*	1.44.10t1.b.a	$1$	$ 2^{2} \cdot 11 $	$x^{10} - 11 x^{8} + 44 x^{6} - 77 x^{4} + 55 x^{2} - 11$	$C_{10}$ (as 10T1)	$0$	$1$
	1.44.10t1.a.a	$1$	$ 2^{2} \cdot 11 $	$x^{10} + 9 x^{8} + 28 x^{6} + 35 x^{4} + 15 x^{2} + 1$	$C_{10}$ (as 10T1)	$0$	$-1$
	1.11.10t1.a.a	$1$	$ 11 $	$x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$	$C_{10}$ (as 10T1)	$0$	$-1$
*	1.11.5t1.a.a	$1$	$ 11 $	$x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$	$C_5$ (as 5T1)	$0$	$1$
	1.11.10t1.a.b	$1$	$ 11 $	$x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$	$C_{10}$ (as 10T1)	$0$	$-1$
*	1.44.10t1.b.b	$1$	$ 2^{2} \cdot 11 $	$x^{10} - 11 x^{8} + 44 x^{6} - 77 x^{4} + 55 x^{2} - 11$	$C_{10}$ (as 10T1)	$0$	$1$
	1.11.10t1.a.c	$1$	$ 11 $	$x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$	$C_{10}$ (as 10T1)	$0$	$-1$
	1.44.10t1.a.b	$1$	$ 2^{2} \cdot 11 $	$x^{10} + 9 x^{8} + 28 x^{6} + 35 x^{4} + 15 x^{2} + 1$	$C_{10}$ (as 10T1)	$0$	$-1$
	1.11.10t1.a.d	$1$	$ 11 $	$x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$	$C_{10}$ (as 10T1)	$0$	$-1$
*	1.11.5t1.a.b	$1$	$ 11 $	$x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$	$C_5$ (as 5T1)	$0$	$1$
*	1.11.5t1.a.c	$1$	$ 11 $	$x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$	$C_5$ (as 5T1)	$0$	$1$
*	1.44.10t1.b.c	$1$	$ 2^{2} \cdot 11 $	$x^{10} - 11 x^{8} + 44 x^{6} - 77 x^{4} + 55 x^{2} - 11$	$C_{10}$ (as 10T1)	$0$	$1$
*	1.11.5t1.a.d	$1$	$ 11 $	$x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$	$C_5$ (as 5T1)	$0$	$1$
	1.44.10t1.a.c	$1$	$ 2^{2} \cdot 11 $	$x^{10} + 9 x^{8} + 28 x^{6} + 35 x^{4} + 15 x^{2} + 1$	$C_{10}$ (as 10T1)	$0$	$-1$
	1.44.10t1.a.d	$1$	$ 2^{2} \cdot 11 $	$x^{10} + 9 x^{8} + 28 x^{6} + 35 x^{4} + 15 x^{2} + 1$	$C_{10}$ (as 10T1)	$0$	$-1$
*	1.44.10t1.b.d	$1$	$ 2^{2} \cdot 11 $	$x^{10} - 11 x^{8} + 44 x^{6} - 77 x^{4} + 55 x^{2} - 11$	$C_{10}$ (as 10T1)	$0$	$1$
*	2.176.6t3.b.a	$2$	$ 2^{4} \cdot 11 $	$x^{6} - 2 x^{5} + 2 x^{4} - 2 x^{3} + 4 x^{2} - 4 x + 2$	$D_{6}$ (as 6T3)	$1$	$0$
*	2.44.3t2.b.a	$2$	$ 2^{2} \cdot 11 $	$x^{3} - x^{2} + x + 1$	$S_3$ (as 3T2)	$1$	$0$
*	2.484.15t4.a.a	$2$	$ 2^{2} \cdot 11^{2}$	$x^{15} - 3 x^{13} - 5 x^{12} + 8 x^{11} + x^{10} + 14 x^{9} - 13 x^{8} - 17 x^{7} + 4 x^{6} + x^{5} + 22 x^{4} - 2 x^{3} - 8 x^{2} - 3 x - 1$	$S_3 \times C_5$ (as 15T4)	$0$	$0$
*	2.484.15t4.a.b	$2$	$ 2^{2} \cdot 11^{2}$	$x^{15} - 3 x^{13} - 5 x^{12} + 8 x^{11} + x^{10} + 14 x^{9} - 13 x^{8} - 17 x^{7} + 4 x^{6} + x^{5} + 22 x^{4} - 2 x^{3} - 8 x^{2} - 3 x - 1$	$S_3 \times C_5$ (as 15T4)	$0$	$0$
*	2.484.15t4.a.c	$2$	$ 2^{2} \cdot 11^{2}$	$x^{15} - 3 x^{13} - 5 x^{12} + 8 x^{11} + x^{10} + 14 x^{9} - 13 x^{8} - 17 x^{7} + 4 x^{6} + x^{5} + 22 x^{4} - 2 x^{3} - 8 x^{2} - 3 x - 1$	$S_3 \times C_5$ (as 15T4)	$0$	$0$
*	2.1936.30t12.b.a	$2$	$ 2^{4} \cdot 11^{2}$	$x^{30} - 15 x^{28} + 104 x^{26} - 482 x^{24} + 1708 x^{22} - 4753 x^{20} + 10608 x^{18} - 18925 x^{16} + 26563 x^{14} - 28460 x^{12} + 22463 x^{10} - 12716 x^{8} + 5049 x^{6} - 1331 x^{4} + 198 x^{2} - 11$	$C_{10}\times S_3$ (as 30T12)	$0$	$0$
*	2.1936.30t12.b.b	$2$	$ 2^{4} \cdot 11^{2}$	$x^{30} - 15 x^{28} + 104 x^{26} - 482 x^{24} + 1708 x^{22} - 4753 x^{20} + 10608 x^{18} - 18925 x^{16} + 26563 x^{14} - 28460 x^{12} + 22463 x^{10} - 12716 x^{8} + 5049 x^{6} - 1331 x^{4} + 198 x^{2} - 11$	$C_{10}\times S_3$ (as 30T12)	$0$	$0$
*	2.1936.30t12.b.c	$2$	$ 2^{4} \cdot 11^{2}$	$x^{30} - 15 x^{28} + 104 x^{26} - 482 x^{24} + 1708 x^{22} - 4753 x^{20} + 10608 x^{18} - 18925 x^{16} + 26563 x^{14} - 28460 x^{12} + 22463 x^{10} - 12716 x^{8} + 5049 x^{6} - 1331 x^{4} + 198 x^{2} - 11$	$C_{10}\times S_3$ (as 30T12)	$0$	$0$
*	2.484.15t4.a.d	$2$	$ 2^{2} \cdot 11^{2}$	$x^{15} - 3 x^{13} - 5 x^{12} + 8 x^{11} + x^{10} + 14 x^{9} - 13 x^{8} - 17 x^{7} + 4 x^{6} + x^{5} + 22 x^{4} - 2 x^{3} - 8 x^{2} - 3 x - 1$	$S_3 \times C_5$ (as 15T4)	$0$	$0$
*	2.1936.30t12.b.d	$2$	$ 2^{4} \cdot 11^{2}$	$x^{30} - 15 x^{28} + 104 x^{26} - 482 x^{24} + 1708 x^{22} - 4753 x^{20} + 10608 x^{18} - 18925 x^{16} + 26563 x^{14} - 28460 x^{12} + 22463 x^{10} - 12716 x^{8} + 5049 x^{6} - 1331 x^{4} + 198 x^{2} - 11$	$C_{10}\times S_3$ (as 30T12)	$0$	$0$

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 *.