Number field 42.0.9380082945933081406113456619151991432292083579779389915131296484043.1: \(\Q(\zeta_{43})\)

• Label: 42.0.938...043.1
• Degree: $42$
• Signature: $[0, 21]$
• Discriminant: $-9.380\times 10^{66}$
• Root discriminant: \(39.32\)
• Ramified prime: $43$
• Class number: $211$ (GRH)
• Class group: [211] (GRH)
• Galois group: $C_{42}$ (as 42T1)

Normalized defining polynomial

x^42 - x^41 + x^40 - x^39 + x^38 - x^37 + x^36 - x^35 + x^34 - x^33 + x^32 - x^31 + x^30 - x^29 + x^28 - x^27 + x^26 - x^25 + x^24 - x^23 + x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1

Invariants

Degree:		$42$
Signature:		$[0, 21]$
Discriminant:		\(-9380082945933081406113456619151991432292083579779389915131296484043\) \(\medspace = -\,43^{41}\)
Root discriminant:		\(39.32\)
Galois root discriminant:		$43^{41/42}\approx 39.31663579527138$
Ramified primes:		\(43\)
Discriminant root field:		\(\Q(\sqrt{-43}) \)
$\card{ \Gal(K/\Q) }$:		$42$
This field is Galois and abelian over $\Q$.
Conductor:		\(43\)
Dirichlet character group:		$\lbrace$$\chi_{43}(1,·)$, $\chi_{43}(2,·)$, $\chi_{43}(3,·)$, $\chi_{43}(4,·)$, $\chi_{43}(5,·)$, $\chi_{43}(6,·)$, $\chi_{43}(7,·)$, $\chi_{43}(8,·)$, $\chi_{43}(9,·)$, $\chi_{43}(10,·)$, $\chi_{43}(11,·)$, $\chi_{43}(12,·)$, $\chi_{43}(13,·)$, $\chi_{43}(14,·)$, $\chi_{43}(15,·)$, $\chi_{43}(16,·)$, $\chi_{43}(17,·)$, $\chi_{43}(18,·)$, $\chi_{43}(19,·)$, $\chi_{43}(20,·)$, $\chi_{43}(21,·)$, $\chi_{43}(22,·)$, $\chi_{43}(23,·)$, $\chi_{43}(24,·)$, $\chi_{43}(25,·)$, $\chi_{43}(26,·)$, $\chi_{43}(27,·)$, $\chi_{43}(28,·)$, $\chi_{43}(29,·)$, $\chi_{43}(30,·)$, $\chi_{43}(31,·)$, $\chi_{43}(32,·)$, $\chi_{43}(33,·)$, $\chi_{43}(34,·)$, $\chi_{43}(35,·)$, $\chi_{43}(36,·)$, $\chi_{43}(37,·)$, $\chi_{43}(38,·)$, $\chi_{43}(39,·)$, $\chi_{43}(40,·)$, $\chi_{43}(41,·)$, $\chi_{43}(42,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{1048576}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$

Monogenic:		Yes
Index:		$1$
Inessential primes:		None

Class group and class number

$C_{211}$, which has order $211$ (assuming GRH)

Unit group

Rank:		$20$
Torsion generator:		\( a \)  (order $86$)
Fundamental units:		$a-1$, $a^{26}-a^{5}$, $a^{24}+a^{4}$, $a^{24}-a^{3}+a^{2}$, $a^{3}-1$, $a^{2}+1$, $a^{4}+1$, $a^{28}-a^{17}$, $a^{14}+a^{12}+a^{10}+a^{8}+a^{6}+a^{4}+a^{2}+1$, $a^{41}-a^{40}-a^{38}+a^{37}+a^{35}-a^{34}-a^{32}+a^{29}-a^{26}+a^{23}+a^{21}-a^{20}-a^{18}+a^{17}+a^{15}-a^{14}-a^{12}+a^{9}-a^{6}+a^{3}+a-1$, $a^{10}+a^{8}+a^{6}+a^{4}+a^{2}+1$, $a^{4}-a^{3}+a^{2}-a+1$, $a^{39}-a^{10}+a^{3}$, $a^{29}+a^{15}+a$, $a^{30}-a^{15}+1$, $a^{41}+a^{39}+a^{37}+a^{35}+a^{33}+a^{31}+a^{29}+a^{27}+a^{25}+a^{23}+a^{21}+a^{19}+a^{17}+a^{15}+a^{13}$, $a^{41}+a^{21}+a$, $a^{16}+a^{14}+a^{12}+a^{10}+a^{8}+a^{6}+a^{4}+a^{2}+1$, $a^{6}-a^{3}+1$, $a^{41}+a^{37}-a^{32}+a^{27}-a^{22}-a^{18}+a^{13}-a^{8}+a^{3}$ (assuming GRH)
Regulator:		\( 2748021948787771.5 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{21}\cdot 2748021948787771.5 \cdot 211}{86\cdot\sqrt{9380082945933081406113456619151991432292083579779389915131296484043}}\cr\approx \mathstrut & 0.127197359940775 \end{aligned}\] (assuming GRH)

Galois group

$C_{42}$ (as 42T1):

A cyclic group of order 42

The 42 conjugacy class representatives for $C_{42}$

Character table for $C_{42}$

Intermediate fields

\(\Q(\sqrt{-43}) \), 3.3.1849.1, 6.0.147008443.1, 7.7.6321363049.1, 14.0.1718264124282290785243.1, \(\Q(\zeta_{43})^+\)

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	${\href{/padicField/2.14.0.1}{14} }^{3}$	$42$	$42$	${\href{/padicField/7.6.0.1}{6} }^{7}$	${\href{/padicField/11.7.0.1}{7} }^{6}$	$21^{2}$	$21^{2}$	$42$	$21^{2}$	$42$	$21^{2}$	${\href{/padicField/37.6.0.1}{6} }^{7}$	${\href{/padicField/41.7.0.1}{7} }^{6}$	R	${\href{/padicField/47.7.0.1}{7} }^{6}$	$21^{2}$	${\href{/padicField/59.7.0.1}{7} }^{6}$

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
\(43\)		Deg $42$	$42$	$1$	$41$