Normalized defining polynomial

\( x^{42} - 43 x^{40} + 860 x^{38} - 10621 x^{36} + 90687 x^{34} - 567987 x^{32} + 2701776 x^{30} + \cdots - 43 \) x^42 - 43*x^40 + 860*x^38 - 10621*x^36 + 90687*x^34 - 567987*x^32 + 2701776*x^30 - 9970840*x^28 + 28915436*x^26 - 66335412*x^24 + 120609840*x^22 - 173376645*x^20 + 195747825*x^18 - 171655785*x^16 + 115000920*x^14 - 57500460*x^12 + 20764055*x^10 - 5167525*x^8 + 826804*x^6 - 76153*x^4 + 3311*x^2 - 43

Invariants

Degree:		$42$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[42, 0]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(41254041074227118944013302420247071185885898029927512658248841159725538158313472\) 41254041074227118944013302420247071185885898029927512658248841159725538158313472 \(\medspace = 2^{42}\cdot 43^{41}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(78.63\)	sage: (K.disc().abs())^(1./K.degree())   gp: abs(K.disc)^(1/poldegree(K.pol))   magma: Abs(Discriminant(OK))^(1/Degree(K));   oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:		$2\cdot 43^{41/42}\approx 78.63327159054276$
Ramified primes:		\(2\), \(43\) 2, 43	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q(\sqrt{43}) \)
$\card{ \Gal(K/\Q) }$:		$42$	sage: K.automorphisms()   magma: Automorphisms(K);   oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:		\(172=2^{2}\cdot 43\)
Dirichlet character group:		$\lbrace$$\chi_{172}(1,·)$, $\chi_{172}(3,·)$, $\chi_{172}(133,·)$, $\chi_{172}(7,·)$, $\chi_{172}(9,·)$, $\chi_{172}(13,·)$, $\chi_{172}(115,·)$, $\chi_{172}(17,·)$, $\chi_{172}(19,·)$, $\chi_{172}(21,·)$, $\chi_{172}(151,·)$, $\chi_{172}(25,·)$, $\chi_{172}(155,·)$, $\chi_{172}(159,·)$, $\chi_{172}(27,·)$, $\chi_{172}(165,·)$, $\chi_{172}(39,·)$, $\chi_{172}(41,·)$, $\chi_{172}(171,·)$, $\chi_{172}(49,·)$, $\chi_{172}(51,·)$, $\chi_{172}(53,·)$, $\chi_{172}(55,·)$, $\chi_{172}(57,·)$, $\chi_{172}(63,·)$, $\chi_{172}(71,·)$, $\chi_{172}(75,·)$, $\chi_{172}(81,·)$, $\chi_{172}(163,·)$, $\chi_{172}(169,·)$, $\chi_{172}(91,·)$, $\chi_{172}(97,·)$, $\chi_{172}(131,·)$, $\chi_{172}(101,·)$, $\chi_{172}(145,·)$, $\chi_{172}(109,·)$, $\chi_{172}(147,·)$, $\chi_{172}(117,·)$, $\chi_{172}(119,·)$, $\chi_{172}(153,·)$, $\chi_{172}(121,·)$, $\chi_{172}(123,·)$$\rbrace$
This is not a CM field.

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

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

not computed

Unit group

Rank:		$41$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( -1 \) -1  (order $2$)	sage: UK.torsion_generator()   gp: K.tu[2]   magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);   oscar: torsion_units_generator(OK)
Fundamental units:		not computed	sage: UK.fundamental_units()   gp: K.fu   magma: [K|fUK(g): g in Generators(UK)];   oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:		not computed	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

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):

Intermediate fields

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

Frobenius cycle types

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
\(2\) 2	2.14.14.15	$x^{14} + 14 x^{13} + 126 x^{12} + 784 x^{11} + 4300 x^{10} + 19592 x^{9} + 80680 x^{8} + 276608 x^{7} + 822832 x^{6} + 1982880 x^{5} + 3998112 x^{4} + 6222080 x^{3} + 7679040 x^{2} + 6275456 x + 3453824$	$2$	$7$	$14$	$C_{14}$	$[2]^{7}$
	2.14.14.15	$x^{14} + 14 x^{13} + 126 x^{12} + 784 x^{11} + 4300 x^{10} + 19592 x^{9} + 80680 x^{8} + 276608 x^{7} + 822832 x^{6} + 1982880 x^{5} + 3998112 x^{4} + 6222080 x^{3} + 7679040 x^{2} + 6275456 x + 3453824$	$2$	$7$	$14$	$C_{14}$	$[2]^{7}$
	2.14.14.15	$x^{14} + 14 x^{13} + 126 x^{12} + 784 x^{11} + 4300 x^{10} + 19592 x^{9} + 80680 x^{8} + 276608 x^{7} + 822832 x^{6} + 1982880 x^{5} + 3998112 x^{4} + 6222080 x^{3} + 7679040 x^{2} + 6275456 x + 3453824$	$2$	$7$	$14$	$C_{14}$	$[2]^{7}$
\(43\) 43		Deg $42$	$42$	$1$	$41$