Normalized defining polynomial

\( x^{16} - 8x^{14} + 37x^{12} - 110x^{10} + 190x^{8} - 176x^{6} + 80x^{4} - 16x^{2} + 1 \) x^16 - 8*x^14 + 37*x^12 - 110*x^10 + 190*x^8 - 176*x^6 + 80*x^4 - 16*x^2 + 1

Invariants

Degree:		$16$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[12, 2]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(1186620610969600000000\) 1186620610969600000000 \(\medspace = 2^{32}\cdot 5^{8}\cdot 29^{4}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(20.76\)	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:		$2^{39/16}5^{1/2}29^{1/2}\approx 65.22958500917873$
Ramified primes:		\(2\), \(5\), \(29\) 2, 5, 29	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q\)
$\card{ \Aut(K/\Q) }$:		$4$	sage: K.automorphisms()   magma: Automorphisms(K);   oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$ 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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$13$	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:		$4a^{14}-28a^{12}+121a^{10}-325a^{8}+459a^{6}-301a^{4}+73a^{2}-3$, $5a^{12}-30a^{10}+122a^{8}-288a^{6}+301a^{4}-106a^{2}+9$, $a^{12}-6a^{10}+24a^{8}-56a^{6}+54a^{4}-12a^{2}$, $a$, $a^{15}-3a^{13}+6a^{11}+17a^{9}-118a^{7}+165a^{5}-56a^{3}+a$, $4a^{14}-28a^{12}+122a^{10}-330a^{8}+479a^{6}-341a^{4}+103a^{2}-8$, $a-1$, $9a^{15}+4a^{14}-67a^{13}-28a^{12}+298a^{11}+121a^{10}-838a^{9}-325a^{8}+1300a^{7}+459a^{6}-995a^{5}-301a^{4}+313a^{3}+73a^{2}-28a-3$, $a^{15}-4a^{14}-8a^{13}+28a^{12}+37a^{11}-121a^{10}-110a^{9}+325a^{8}+190a^{7}-459a^{6}-176a^{5}+301a^{4}+80a^{3}-73a^{2}-15a+2$, $13a^{15}+a^{14}-95a^{13}-7a^{12}+418a^{11}+30a^{10}-1157a^{9}-80a^{8}+1735a^{7}+110a^{6}-1240a^{5}-66a^{4}+332a^{3}+14a^{2}-19a-2$, $a^{15}+4a^{14}-7a^{13}-28a^{12}+30a^{11}+122a^{10}-80a^{9}-330a^{8}+110a^{7}+479a^{6}-66a^{5}-341a^{4}+14a^{3}+103a^{2}-3a-8$, $9a^{15}+4a^{14}-63a^{13}-28a^{12}+273a^{11}+122a^{10}-735a^{9}-330a^{8}+1048a^{7}+479a^{6}-708a^{5}-341a^{4}+189a^{3}+103a^{2}-13a-10$, $9a^{15}-68a^{13}+304a^{11}-862a^{9}+1356a^{7}-1049a^{5}+325a^{3}-29a+1$ 4*a^14 - 28*a^12 + 121*a^10 - 325*a^8 + 459*a^6 - 301*a^4 + 73*a^2 - 3, 5*a^12 - 30*a^10 + 122*a^8 - 288*a^6 + 301*a^4 - 106*a^2 + 9, a^12 - 6*a^10 + 24*a^8 - 56*a^6 + 54*a^4 - 12*a^2, a, a^15 - 3*a^13 + 6*a^11 + 17*a^9 - 118*a^7 + 165*a^5 - 56*a^3 + a, 4*a^14 - 28*a^12 + 122*a^10 - 330*a^8 + 479*a^6 - 341*a^4 + 103*a^2 - 8, a - 1, 9*a^15 + 4*a^14 - 67*a^13 - 28*a^12 + 298*a^11 + 121*a^10 - 838*a^9 - 325*a^8 + 1300*a^7 + 459*a^6 - 995*a^5 - 301*a^4 + 313*a^3 + 73*a^2 - 28*a - 3, a^15 - 4*a^14 - 8*a^13 + 28*a^12 + 37*a^11 - 121*a^10 - 110*a^9 + 325*a^8 + 190*a^7 - 459*a^6 - 176*a^5 + 301*a^4 + 80*a^3 - 73*a^2 - 15*a + 2, 13*a^15 + a^14 - 95*a^13 - 7*a^12 + 418*a^11 + 30*a^10 - 1157*a^9 - 80*a^8 + 1735*a^7 + 110*a^6 - 1240*a^5 - 66*a^4 + 332*a^3 + 14*a^2 - 19*a - 2, a^15 + 4*a^14 - 7*a^13 - 28*a^12 + 30*a^11 + 122*a^10 - 80*a^9 - 330*a^8 + 110*a^7 + 479*a^6 - 66*a^5 - 341*a^4 + 14*a^3 + 103*a^2 - 3*a - 8, 9*a^15 + 4*a^14 - 63*a^13 - 28*a^12 + 273*a^11 + 122*a^10 - 735*a^9 - 330*a^8 + 1048*a^7 + 479*a^6 - 708*a^5 - 341*a^4 + 189*a^3 + 103*a^2 - 13*a - 10, 9*a^15 - 68*a^13 + 304*a^11 - 862*a^9 + 1356*a^7 - 1049*a^5 + 325*a^3 - 29*a + 1 (assuming GRH)	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:		\( 68808.2956194 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{2}\cdot 68808.2956194 \cdot 1}{2\cdot\sqrt{1186620610969600000000}}\cr\approx \mathstrut & 0.161500750839 \end{aligned}\] (assuming GRH)

Galois group

$D_4^2:C_2^2$ (as 16T509):

Intermediate fields

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

Sibling fields

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		Deg $16$	$4$	$4$	$32$
\(5\) 5	5.8.4.1	$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	5.8.4.1	$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
\(29\) 29	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.4.2.1	$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	29.4.2.1	$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$