Number field 42.0.74252462132603256348231837398371002884673933378885582779211491265789772693504.1

• Label: 42.0.742...504.1
• Degree: $42$
• Signature: $[0, 21]$
• Discriminant: $-7.425\times 10^{76}$
• Root discriminant: \(67.65\)
• Ramified primes: $2,7$
• Class number: $1923461$ (GRH)
• Class group: [1923461] (GRH)
• Galois group: $C_{42}$ (as 42T1)

Normalized defining polynomial

x^42 + 42*x^40 + 819*x^38 + 9842*x^36 + 81585*x^34 + 494802*x^32 + 2272424*x^30 + 8069423*x^28 + 22428224*x^26 + 49085050*x^24 + 84669739*x^22 + 114704933*x^20 + 121049551*x^18 + 98190708*x^16 + 60019861*x^14 + 26865216*x^12 + 8444436*x^10 + 1749188*x^8 + 215453*x^6 + 13181*x^4 + 294*x^2 + 1

Invariants

Degree:		$42$
Signature:		$[0, 21]$
Discriminant:		\(-74252462132603256348231837398371002884673933378885582779211491265789772693504\) \(\medspace = -\,2^{42}\cdot 7^{76}\)
Root discriminant:		\(67.65\)
Galois root discriminant:		$2\cdot 7^{38/21}\approx 67.6480092151321$
Ramified primes:		\(2\), \(7\)
Discriminant root field:		\(\Q(\sqrt{-1}) \)
$\card{ \Gal(K/\Q) }$:		$42$
This field is Galois and abelian over $\Q$.
Conductor:		\(196=2^{2}\cdot 7^{2}\)
Dirichlet character group:		$\lbrace$$\chi_{196}(1,·)$, $\chi_{196}(51,·)$, $\chi_{196}(135,·)$, $\chi_{196}(9,·)$, $\chi_{196}(11,·)$, $\chi_{196}(141,·)$, $\chi_{196}(15,·)$, $\chi_{196}(149,·)$, $\chi_{196}(23,·)$, $\chi_{196}(25,·)$, $\chi_{196}(155,·)$, $\chi_{196}(29,·)$, $\chi_{196}(163,·)$, $\chi_{196}(37,·)$, $\chi_{196}(65,·)$, $\chi_{196}(39,·)$, $\chi_{196}(169,·)$, $\chi_{196}(43,·)$, $\chi_{196}(177,·)$, $\chi_{196}(179,·)$, $\chi_{196}(53,·)$, $\chi_{196}(137,·)$, $\chi_{196}(57,·)$, $\chi_{196}(151,·)$, $\chi_{196}(191,·)$, $\chi_{196}(193,·)$, $\chi_{196}(67,·)$, $\chi_{196}(71,·)$, $\chi_{196}(183,·)$, $\chi_{196}(79,·)$, $\chi_{196}(81,·)$, $\chi_{196}(85,·)$, $\chi_{196}(93,·)$, $\chi_{196}(95,·)$, $\chi_{196}(165,·)$, $\chi_{196}(99,·)$, $\chi_{196}(107,·)$, $\chi_{196}(109,·)$, $\chi_{196}(113,·)$, $\chi_{196}(121,·)$, $\chi_{196}(123,·)$, $\chi_{196}(127,·)$$\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

$C_{1923461}$, which has order $1923461$ (assuming GRH)

Unit group

Rank:		$20$
Torsion generator:		\( a^{35} + 35 a^{33} + 560 a^{31} + 5425 a^{29} + 35525 a^{27} + 166257 a^{25} + 573300 a^{23} + 1480049 a^{21} + 2877854 a^{19} + 4205936 a^{17} + 4575312 a^{15} + 3637270 a^{13} + 2051777 a^{11} + 784343 a^{9} + 188653 a^{7} + 25060 a^{5} + 1414 a^{3} + 21 a \)  (order $4$)
Fundamental units:		$a^{14}+14a^{12}+77a^{10}+210a^{8}+294a^{6}+196a^{4}+49a^{2}+1$, $a^{35}+35a^{33}+560a^{31}+5425a^{29}+35525a^{27}+166257a^{25}+573300a^{23}+1480049a^{21}+2877854a^{19}+4205936a^{17}+4575312a^{15}+3637270a^{13}+2051777a^{11}+784343a^{9}+188652a^{7}+25053a^{5}+1400a^{3}+14a$, $a^{11}+11a^{9}+44a^{7}+77a^{5}+55a^{3}+11a$, $a^{10}+10a^{8}+35a^{6}+50a^{4}+25a^{2}+2$, $a^{5}+5a^{3}+5a$, $a^{27}+27a^{25}+324a^{23}+2277a^{21}+10395a^{19}+32319a^{17}+69768a^{15}+104652a^{13}+107406a^{11}+72930a^{9}+30888a^{7}+7371a^{5}+819a^{3}+27a$, $a^{30}+30a^{28}+405a^{26}+3250a^{24}+17250a^{22}+63756a^{20}+168245a^{18}+319770a^{16}+436050a^{14}+419900a^{12}+277134a^{10}+119340a^{8}+30940a^{6}+4200a^{4}+225a^{2}+2$, $a^{34}+34a^{32}+527a^{30}+4930a^{28}+31059a^{26}+139230a^{24}+457470a^{22}+1118260a^{20}+2042975a^{18}+2778446a^{16}+2778446a^{14}+1998724a^{12}+999362a^{10}+329460a^{8}+65892a^{6}+6936a^{4}+289a^{2}+2$, $a^{17}+17a^{15}+119a^{13}+442a^{11}+935a^{9}+1122a^{7}+714a^{5}+204a^{3}+17a$, $a^{33}+33a^{31}+495a^{29}+4466a^{27}+27027a^{25}+115830a^{23}+361790a^{21}+834900a^{19}+1427679a^{17}+1797818a^{15}+1641486a^{13}+1058148a^{11}+461890a^{9}+127908a^{7}+20196a^{5}+1496a^{3}+33a$, $a^{8}+8a^{6}+20a^{4}+16a^{2}+2$, $a^{4}+4a^{2}+2$, $a^{24}+24a^{22}+252a^{20}+1520a^{18}+5814a^{16}+14688a^{14}+24752a^{12}+27456a^{10}+19305a^{8}+8008a^{6}+1716a^{4}+144a^{2}+2$, $a^{31}+31a^{29}+434a^{27}+3627a^{25}+20150a^{23}+78430a^{21}+219604a^{19}+447051a^{17}+660858a^{15}+700910a^{13}+520676a^{11}+260338a^{9}+82212a^{7}+14756a^{5}+1240a^{3}+31a$, $a^{36}+35a^{34}+560a^{32}+5425a^{30}+35525a^{28}+166257a^{26}+573300a^{24}+1480049a^{22}+2877854a^{20}+4205936a^{18}+4575312a^{16}+3637270a^{14}+2051777a^{12}+784343a^{10}+188653a^{8}+25060a^{6}+1414a^{4}+21a^{2}$, $a^{35}+35a^{33}+560a^{31}+5425a^{29}+35525a^{27}+166258a^{25}+573325a^{23}+1480324a^{21}+2879604a^{19}+4213061a^{17}+4594692a^{15}+3672970a^{13}+2095977a^{11}+820093a^{9}+206528a^{7}+30065a^{5}+2064a^{3}+47a$, $a^{40}+40a^{38}+740a^{36}+8400a^{34}+65450a^{32}+371008a^{30}+1582240a^{28}+5178240a^{26}+13147875a^{24}+26013000a^{22}+40060020a^{20}+47720400a^{18}+43459650a^{16}+29716000a^{14}+14858000a^{12}+5230016a^{10}+1225785a^{8}+175560a^{6}+13300a^{4}+400a^{2}+2$, $a^{20}+20a^{18}+170a^{16}+800a^{14}+2275a^{12}+4004a^{10}+4290a^{8}+2640a^{6}+825a^{4}+100a^{2}+2$, $a^{6}+6a^{4}+9a^{2}+2$, $a^{2}+2$ (assuming GRH)
Regulator:		\( 1776855897760068.5 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{21}\cdot 1776855897760068.5 \cdot 1923461}{4\cdot\sqrt{74252462132603256348231837398371002884673933378885582779211491265789772693504}}\cr\approx \mathstrut & 0.181174640338776 \end{aligned}\] (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 $C_{42}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\zeta_{7})^+\), 6.0.153664.1, 7.7.13841287201.1, 14.0.3138866894939200133545984.1, \(\Q(\zeta_{49})^+\)

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	R	$42$	$21^{2}$	R	$42$	${\href{/padicField/13.7.0.1}{7} }^{6}$	$21^{2}$	${\href{/padicField/19.6.0.1}{6} }^{7}$	$42$	${\href{/padicField/29.7.0.1}{7} }^{6}$	${\href{/padicField/31.6.0.1}{6} }^{7}$	$21^{2}$	${\href{/padicField/41.7.0.1}{7} }^{6}$	${\href{/padicField/43.14.0.1}{14} }^{3}$	$42$	$21^{2}$	$42$

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
\(2\)		Deg $42$	$2$	$21$	$42$
\(7\)		Deg $42$	$21$	$2$	$76$