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:		\(-74\!\cdots\!504\)\(\medspace = -\,2^{42}\cdot 7^{76}\)
Root discriminant:		$67.65$
Ramified primes:		$2, 7$
$|\Gal(K/\Q)|$:		$42$
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.

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

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

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:		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:		\( 1776855897760068.5 \) (assuming GRH)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{21}\cdot 1776855897760068.5 \cdot 1923461}{4\sqrt{74252462132603256348231837398371002884673933378885582779211491265789772693504}}\approx 0.181174640338776$ (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	$42$	${\href{/LocalNumberField/13.7.0.1}{7} }^{6}$	$21^{2}$	${\href{/LocalNumberField/19.6.0.1}{6} }^{7}$	$42$	${\href{/LocalNumberField/29.7.0.1}{7} }^{6}$	${\href{/LocalNumberField/31.6.0.1}{6} }^{7}$	$21^{2}$	${\href{/LocalNumberField/41.7.0.1}{7} }^{6}$	${\href{/LocalNumberField/43.14.0.1}{14} }^{3}$	$42$	$21^{2}$	$42$

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	Data not computed
7	Data not computed