Number field 42.0.1614301355762283810837865300933983862308309471908336718662196745400668275448050914569211.1

• Label: 42.0.161...211.1
• Degree: $42$
• Signature: $[0, 21]$
• Discriminant: $-1.614\times 10^{87}$
• Root discriminant: \(119.23\)
• Ramified primes: $11,43$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{42}$ (as 42T1)

Normalized defining polynomial

x^42 - 19*x^41 + 193*x^40 - 1368*x^39 + 7730*x^38 - 37504*x^37 + 163424*x^36 - 653089*x^35 + 2424227*x^34 - 8429227*x^33 + 27684879*x^32 - 86328882*x^31 + 256735509*x^30 - 730024655*x^29 + 1991090346*x^28 - 5216701163*x^27 + 13158377790*x^26 - 31970357867*x^25 + 74948907297*x^24 - 169511696295*x^23 + 370383530887*x^22 - 781186383780*x^21 + 1592472532020*x^20 - 3132457376500*x^19 + 5954059663605*x^18 - 10905928130092*x^17 + 19286342092183*x^16 - 32785066852903*x^15 + 53727191837034*x^14 - 84287557742257*x^13 + 127210697307990*x^12 - 182579457912758*x^11 + 251464793643596*x^10 - 325756069207898*x^9 + 403964380869043*x^8 - 462110196994188*x^7 + 505689982566923*x^6 - 491465579236069*x^5 + 459935510967285*x^4 - 350571727520514*x^3 + 266244504765895*x^2 - 126760076648507*x + 70799148954451

Invariants

Degree:		$42$
Signature:		$[0, 21]$
Discriminant:		-1614301355762283810837865300933983862308309471908336718662196745400668275448050914569211
Root discriminant:		\(119.23\)
Galois root discriminant:		$11^{1/2}43^{20/21}\approx 119.22863886148409$
Ramified primes:		\(11\), \(43\)
Discriminant root field:		\(\Q(\sqrt{-11}) \)
$\card{ \Gal(K/\Q) }$:		$42$
This field is Galois and abelian over $\Q$.
Conductor:		\(473=11\cdot 43\)
Dirichlet character group:		$\lbrace$$\chi_{473}(384,·)$, $\chi_{473}(1,·)$, $\chi_{473}(133,·)$, $\chi_{473}(10,·)$, $\chi_{473}(397,·)$, $\chi_{473}(142,·)$, $\chi_{473}(144,·)$, $\chi_{473}(274,·)$, $\chi_{473}(21,·)$, $\chi_{473}(23,·)$, $\chi_{473}(408,·)$, $\chi_{473}(153,·)$, $\chi_{473}(164,·)$, $\chi_{473}(296,·)$, $\chi_{473}(298,·)$, $\chi_{473}(428,·)$, $\chi_{473}(307,·)$, $\chi_{473}(54,·)$, $\chi_{473}(439,·)$, $\chi_{473}(56,·)$, $\chi_{473}(441,·)$, $\chi_{473}(186,·)$, $\chi_{473}(188,·)$, $\chi_{473}(318,·)$, $\chi_{473}(67,·)$, $\chi_{473}(197,·)$, $\chi_{473}(461,·)$, $\chi_{473}(78,·)$, $\chi_{473}(208,·)$, $\chi_{473}(210,·)$, $\chi_{473}(342,·)$, $\chi_{473}(87,·)$, $\chi_{473}(219,·)$, $\chi_{473}(221,·)$, $\chi_{473}(353,·)$, $\chi_{473}(100,·)$, $\chi_{473}(230,·)$, $\chi_{473}(232,·)$, $\chi_{473}(109,·)$, $\chi_{473}(111,·)$, $\chi_{473}(375,·)$, $\chi_{473}(122,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{1048576}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $\frac{1}{13\!\cdots\!59}a^{41}-\frac{55\!\cdots\!41}{13\!\cdots\!59}a^{40}-\frac{18\!\cdots\!88}{13\!\cdots\!59}a^{39}+\frac{45\!\cdots\!96}{13\!\cdots\!59}a^{38}-\frac{14\!\cdots\!06}{13\!\cdots\!59}a^{37}+\frac{50\!\cdots\!69}{13\!\cdots\!59}a^{36}+\frac{35\!\cdots\!10}{13\!\cdots\!59}a^{35}-\frac{13\!\cdots\!52}{53\!\cdots\!87}a^{34}-\frac{59\!\cdots\!36}{13\!\cdots\!59}a^{33}-\frac{42\!\cdots\!71}{13\!\cdots\!59}a^{32}-\frac{40\!\cdots\!39}{13\!\cdots\!59}a^{31}+\frac{23\!\cdots\!78}{13\!\cdots\!59}a^{30}-\frac{64\!\cdots\!50}{13\!\cdots\!59}a^{29}+\frac{52\!\cdots\!61}{13\!\cdots\!59}a^{28}-\frac{10\!\cdots\!49}{13\!\cdots\!59}a^{27}-\frac{51\!\cdots\!76}{13\!\cdots\!59}a^{26}-\frac{23\!\cdots\!71}{13\!\cdots\!59}a^{25}+\frac{60\!\cdots\!19}{13\!\cdots\!59}a^{24}+\frac{35\!\cdots\!76}{13\!\cdots\!59}a^{23}-\frac{47\!\cdots\!15}{13\!\cdots\!59}a^{22}-\frac{31\!\cdots\!48}{13\!\cdots\!59}a^{21}+\frac{45\!\cdots\!75}{13\!\cdots\!59}a^{20}-\frac{43\!\cdots\!97}{13\!\cdots\!59}a^{19}+\frac{42\!\cdots\!03}{13\!\cdots\!59}a^{18}+\frac{32\!\cdots\!40}{13\!\cdots\!59}a^{17}+\frac{65\!\cdots\!22}{13\!\cdots\!59}a^{16}-\frac{68\!\cdots\!20}{13\!\cdots\!59}a^{15}-\frac{10\!\cdots\!34}{13\!\cdots\!59}a^{14}-\frac{18\!\cdots\!87}{13\!\cdots\!59}a^{13}+\frac{40\!\cdots\!37}{13\!\cdots\!59}a^{12}-\frac{20\!\cdots\!31}{13\!\cdots\!59}a^{11}-\frac{81\!\cdots\!27}{13\!\cdots\!59}a^{10}-\frac{38\!\cdots\!32}{13\!\cdots\!59}a^{9}-\frac{55\!\cdots\!26}{13\!\cdots\!59}a^{8}-\frac{35\!\cdots\!14}{13\!\cdots\!59}a^{7}+\frac{18\!\cdots\!76}{13\!\cdots\!59}a^{6}-\frac{19\!\cdots\!59}{13\!\cdots\!59}a^{5}-\frac{26\!\cdots\!71}{13\!\cdots\!59}a^{4}-\frac{59\!\cdots\!50}{13\!\cdots\!59}a^{3}-\frac{65\!\cdots\!11}{13\!\cdots\!59}a^{2}+\frac{38\!\cdots\!08}{13\!\cdots\!59}a+\frac{26\!\cdots\!21}{13\!\cdots\!59}$

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

not computed

Unit group

Rank:		$20$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		not computed
Regulator:		not computed

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

Galois group

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

A cyclic group of order 42

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

Character table for $C_{42}$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.3.1849.1, 6.0.4550424131.1, 7.7.6321363049.1, 14.0.778700158443122158226851571.1, \(\Q(\zeta_{43})^+\)

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	${\href{/padicField/2.14.0.1}{14} }^{3}$	$21^{2}$	$21^{2}$	${\href{/padicField/7.6.0.1}{6} }^{7}$	R	$42$	$42$	$42$	$21^{2}$	$42$	$21^{2}$	${\href{/padicField/37.3.0.1}{3} }^{14}$	${\href{/padicField/41.14.0.1}{14} }^{3}$	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
\(11\)	11.14.7.2	$x^{14} + 77 x^{12} + 2541 x^{10} + 46593 x^{8} + 18 x^{7} + 511203 x^{6} - 4158 x^{5} + 3382071 x^{4} + 76230 x^{3} + 12550015 x^{2} - 167634 x + 19370300$	$2$	$7$	$7$	$C_{14}$	$[\ ]_{2}^{7}$
	11.14.7.2	$x^{14} + 77 x^{12} + 2541 x^{10} + 46593 x^{8} + 18 x^{7} + 511203 x^{6} - 4158 x^{5} + 3382071 x^{4} + 76230 x^{3} + 12550015 x^{2} - 167634 x + 19370300$	$2$	$7$	$7$	$C_{14}$	$[\ ]_{2}^{7}$
	11.14.7.2	$x^{14} + 77 x^{12} + 2541 x^{10} + 46593 x^{8} + 18 x^{7} + 511203 x^{6} - 4158 x^{5} + 3382071 x^{4} + 76230 x^{3} + 12550015 x^{2} - 167634 x + 19370300$	$2$	$7$	$7$	$C_{14}$	$[\ ]_{2}^{7}$
\(43\)		Deg $42$	$21$	$2$	$40$