Normalized defining polynomial

\( x^{17} - 4 x^{16} - 11 x^{15} + 53 x^{14} + 44 x^{13} - 271 x^{12} - 81 x^{11} + 676 x^{10} + 78 x^{9} + \cdots - 1 \) x^17 - 4*x^16 - 11*x^15 + 53*x^14 + 44*x^13 - 271*x^12 - 81*x^11 + 676*x^10 + 78*x^9 - 866*x^8 - 56*x^7 + 563*x^6 + 36*x^5 - 176*x^4 - 12*x^3 + 23*x^2 + x - 1

Invariants

Degree:		$17$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[17, 0]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(2349760091653551409013119601402609\) 2349760091653551409013119601402609 \(\medspace = 19501\cdot 163667513\cdot 736214146077100976893\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(91.83\)	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:		$19501^{1/2}163667513^{1/2}736214146077100976893^{1/2}\approx 4.847432404534953e+16$
Ramified primes:		\(19501\), \(163667513\), \(736214146077100976893\) 19501, 163667513, 736214146077100976893	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		$\Q(\sqrt{23497\!\cdots\!02609}$)
$\card{ \Aut(K/\Q) }$:		$1$	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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$ 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

Class group and class number

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

Unit group

Rank:		$16$	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:		$a$, $a^{16}-4a^{15}-11a^{14}+53a^{13}+44a^{12}-271a^{11}-81a^{10}+676a^{9}+78a^{8}-866a^{7}-56a^{6}+563a^{5}+36a^{4}-176a^{3}-12a^{2}+22a+2$, $a^{16}-4a^{15}-11a^{14}+53a^{13}+44a^{12}-271a^{11}-81a^{10}+676a^{9}+78a^{8}-866a^{7}-56a^{6}+563a^{5}+35a^{4}-174a^{3}-8a^{2}+18a-2$, $a^{16}-4a^{15}-11a^{14}+53a^{13}+44a^{12}-271a^{11}-81a^{10}+676a^{9}+78a^{8}-866a^{7}-56a^{6}+563a^{5}+37a^{4}-177a^{3}-17a^{2}+23a+4$, $5a^{16}-24a^{15}-37a^{14}+299a^{13}-5a^{12}-1408a^{11}+655a^{10}+3134a^{9}-1951a^{8}-3397a^{7}+2186a^{6}+1714a^{5}-969a^{4}-358a^{3}+140a^{2}+22a-5$, $a^{16}-5a^{15}-6a^{14}+59a^{13}-15a^{12}-256a^{11}+175a^{10}+501a^{9}-423a^{8}-443a^{7}+387a^{6}+176a^{5}-140a^{4}-37a^{3}+26a^{2}+a-2$, $7a^{16}-30a^{15}-67a^{14}+384a^{13}+185a^{12}-1872a^{11}+a^{10}+4357a^{9}-679a^{8}-5011a^{7}+867a^{6}+2734a^{5}-335a^{4}-638a^{3}+36a^{2}+45a$, $a^{16}-4a^{15}-11a^{14}+53a^{13}+44a^{12}-271a^{11}-81a^{10}+676a^{9}+78a^{8}-866a^{7}-56a^{6}+562a^{5}+37a^{4}-170a^{3}-14a^{2}+17a+1$, $3a^{16}-14a^{15}-26a^{14}+185a^{13}+36a^{12}-950a^{11}+265a^{10}+2412a^{9}-1071a^{8}-3207a^{7}+1531a^{6}+2203a^{5}-977a^{4}-718a^{3}+270a^{2}+82a-25$, $a^{16}-4a^{15}-11a^{14}+53a^{13}+44a^{12}-271a^{11}-81a^{10}+676a^{9}+78a^{8}-866a^{7}-56a^{6}+563a^{5}+36a^{4}-176a^{3}-13a^{2}+22a+3$, $2a^{15}-9a^{14}-17a^{13}+113a^{12}+24a^{11}-532a^{10}+148a^{9}+1158a^{8}-551a^{7}-1158a^{6}+660a^{5}+458a^{4}-302a^{3}-41a^{2}+39a-3$, $a^{16}-4a^{15}-11a^{14}+53a^{13}+44a^{12}-271a^{11}-81a^{10}+676a^{9}+78a^{8}-866a^{7}-56a^{6}+563a^{5}+36a^{4}-175a^{3}-12a^{2}+18a$, $2a^{16}-17a^{15}+13a^{14}+205a^{13}-363a^{12}-945a^{11}+2039a^{10}+2147a^{9}-4897a^{8}-2630a^{7}+5419a^{6}+1805a^{5}-2553a^{4}-613a^{3}+408a^{2}+51a-19$, $2a^{16}-8a^{15}-19a^{14}+95a^{13}+53a^{12}-401a^{11}-2a^{10}+674a^{9}-230a^{8}-229a^{7}+438a^{6}-396a^{5}-366a^{4}+257a^{3}+115a^{2}-30a-9$, $9a^{16}-25a^{15}-145a^{14}+368a^{13}+983a^{12}-2095a^{11}-3573a^{10}+5788a^{9}+7241a^{8}-8046a^{7}-7852a^{6}+5289a^{5}+4172a^{4}-1340a^{3}-948a^{2}+58a+51$, $14a^{16}-62a^{15}-125a^{14}+785a^{13}+257a^{12}-3770a^{11}+540a^{10}+8584a^{9}-2572a^{8}-9525a^{7}+3062a^{6}+4873a^{5}-1318a^{4}-1006a^{3}+186a^{2}+53a-11$ a, a^16 - 4*a^15 - 11*a^14 + 53*a^13 + 44*a^12 - 271*a^11 - 81*a^10 + 676*a^9 + 78*a^8 - 866*a^7 - 56*a^6 + 563*a^5 + 36*a^4 - 176*a^3 - 12*a^2 + 22*a + 2, a^16 - 4*a^15 - 11*a^14 + 53*a^13 + 44*a^12 - 271*a^11 - 81*a^10 + 676*a^9 + 78*a^8 - 866*a^7 - 56*a^6 + 563*a^5 + 35*a^4 - 174*a^3 - 8*a^2 + 18*a - 2, a^16 - 4*a^15 - 11*a^14 + 53*a^13 + 44*a^12 - 271*a^11 - 81*a^10 + 676*a^9 + 78*a^8 - 866*a^7 - 56*a^6 + 563*a^5 + 37*a^4 - 177*a^3 - 17*a^2 + 23*a + 4, 5*a^16 - 24*a^15 - 37*a^14 + 299*a^13 - 5*a^12 - 1408*a^11 + 655*a^10 + 3134*a^9 - 1951*a^8 - 3397*a^7 + 2186*a^6 + 1714*a^5 - 969*a^4 - 358*a^3 + 140*a^2 + 22*a - 5, a^16 - 5*a^15 - 6*a^14 + 59*a^13 - 15*a^12 - 256*a^11 + 175*a^10 + 501*a^9 - 423*a^8 - 443*a^7 + 387*a^6 + 176*a^5 - 140*a^4 - 37*a^3 + 26*a^2 + a - 2, 7*a^16 - 30*a^15 - 67*a^14 + 384*a^13 + 185*a^12 - 1872*a^11 + a^10 + 4357*a^9 - 679*a^8 - 5011*a^7 + 867*a^6 + 2734*a^5 - 335*a^4 - 638*a^3 + 36*a^2 + 45*a, a^16 - 4*a^15 - 11*a^14 + 53*a^13 + 44*a^12 - 271*a^11 - 81*a^10 + 676*a^9 + 78*a^8 - 866*a^7 - 56*a^6 + 562*a^5 + 37*a^4 - 170*a^3 - 14*a^2 + 17*a + 1, 3*a^16 - 14*a^15 - 26*a^14 + 185*a^13 + 36*a^12 - 950*a^11 + 265*a^10 + 2412*a^9 - 1071*a^8 - 3207*a^7 + 1531*a^6 + 2203*a^5 - 977*a^4 - 718*a^3 + 270*a^2 + 82*a - 25, a^16 - 4*a^15 - 11*a^14 + 53*a^13 + 44*a^12 - 271*a^11 - 81*a^10 + 676*a^9 + 78*a^8 - 866*a^7 - 56*a^6 + 563*a^5 + 36*a^4 - 176*a^3 - 13*a^2 + 22*a + 3, 2*a^15 - 9*a^14 - 17*a^13 + 113*a^12 + 24*a^11 - 532*a^10 + 148*a^9 + 1158*a^8 - 551*a^7 - 1158*a^6 + 660*a^5 + 458*a^4 - 302*a^3 - 41*a^2 + 39*a - 3, a^16 - 4*a^15 - 11*a^14 + 53*a^13 + 44*a^12 - 271*a^11 - 81*a^10 + 676*a^9 + 78*a^8 - 866*a^7 - 56*a^6 + 563*a^5 + 36*a^4 - 175*a^3 - 12*a^2 + 18*a, 2*a^16 - 17*a^15 + 13*a^14 + 205*a^13 - 363*a^12 - 945*a^11 + 2039*a^10 + 2147*a^9 - 4897*a^8 - 2630*a^7 + 5419*a^6 + 1805*a^5 - 2553*a^4 - 613*a^3 + 408*a^2 + 51*a - 19, 2*a^16 - 8*a^15 - 19*a^14 + 95*a^13 + 53*a^12 - 401*a^11 - 2*a^10 + 674*a^9 - 230*a^8 - 229*a^7 + 438*a^6 - 396*a^5 - 366*a^4 + 257*a^3 + 115*a^2 - 30*a - 9, 9*a^16 - 25*a^15 - 145*a^14 + 368*a^13 + 983*a^12 - 2095*a^11 - 3573*a^10 + 5788*a^9 + 7241*a^8 - 8046*a^7 - 7852*a^6 + 5289*a^5 + 4172*a^4 - 1340*a^3 - 948*a^2 + 58*a + 51, 14*a^16 - 62*a^15 - 125*a^14 + 785*a^13 + 257*a^12 - 3770*a^11 + 540*a^10 + 8584*a^9 - 2572*a^8 - 9525*a^7 + 3062*a^6 + 4873*a^5 - 1318*a^4 - 1006*a^3 + 186*a^2 + 53*a - 11 (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:		\( 384409141351 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{17}\cdot(2\pi)^{0}\cdot 384409141351 \cdot 1}{2\cdot\sqrt{2349760091653551409013119601402609}}\cr\approx \mathstrut & 0.519710960054038 \end{aligned}\] (assuming GRH)

Galois group

$S_{17}$ (as 17T10):

Intermediate fields

Frobenius cycle types

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
\(19501\) 19501		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $3$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
		Deg $6$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
		Deg $6$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
\(163667513\) 163667513	$\Q_{163667513}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $12$	$1$	$12$	$0$	$C_{12}$	$[\ ]^{12}$
\(736\!\cdots\!893\) 736214146077100976893		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $15$	$1$	$15$	$0$	$C_{15}$	$[\ ]^{15}$