Normalized defining polynomial

\( x^{17} - 34 x^{14} - 68 x^{13} + 17 x^{12} + 323 x^{11} + 884 x^{10} + 1241 x^{9} + 1394 x^{8} + \cdots + 57 \) x^17 - 34*x^14 - 68*x^13 + 17*x^12 + 323*x^11 + 884*x^10 + 1241*x^9 + 1394*x^8 + 1003*x^7 + 935*x^6 + 663*x^5 + 901*x^4 + 578*x^3 + 493*x^2 + 136*x + 57

Invariants

Degree:		$17$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[1, 8]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(92267897090016343666010049\) 92267897090016343666010049 \(\medspace = 3^{8}\cdot 17^{18}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(33.68\)	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:		$3^{1/2}17^{151/136}\approx 40.245824295791174$
Ramified primes:		\(3\), \(17\) 3, 17	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q\)
$\card{ \Aut(K/\Q) }$:		$1$	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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{26}a^{15}-\frac{1}{26}a^{14}-\frac{1}{26}a^{13}-\frac{5}{26}a^{12}+\frac{2}{13}a^{11}+\frac{5}{13}a^{10}+\frac{3}{13}a^{9}-\frac{1}{2}a^{8}-\frac{3}{13}a^{7}-\frac{2}{13}a^{6}+\frac{5}{26}a^{5}-\frac{11}{26}a^{4}-\frac{6}{13}a^{3}-\frac{1}{26}a^{2}-\frac{4}{13}a-\frac{2}{13}$, $\frac{1}{29\!\cdots\!78}a^{16}+\frac{40\!\cdots\!43}{22\!\cdots\!06}a^{15}+\frac{22\!\cdots\!59}{14\!\cdots\!89}a^{14}-\frac{11\!\cdots\!85}{14\!\cdots\!89}a^{13}+\frac{47\!\cdots\!65}{29\!\cdots\!78}a^{12}+\frac{21\!\cdots\!35}{14\!\cdots\!89}a^{11}-\frac{65\!\cdots\!70}{14\!\cdots\!89}a^{10}+\frac{10\!\cdots\!11}{29\!\cdots\!78}a^{9}+\frac{10\!\cdots\!83}{29\!\cdots\!78}a^{8}+\frac{26\!\cdots\!67}{14\!\cdots\!89}a^{7}-\frac{56\!\cdots\!59}{29\!\cdots\!78}a^{6}-\frac{65\!\cdots\!63}{14\!\cdots\!89}a^{5}-\frac{78\!\cdots\!11}{29\!\cdots\!78}a^{4}+\frac{97\!\cdots\!75}{22\!\cdots\!06}a^{3}-\frac{98\!\cdots\!35}{29\!\cdots\!78}a^{2}+\frac{13\!\cdots\!91}{14\!\cdots\!89}a+\frac{11\!\cdots\!45}{29\!\cdots\!78}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, 1/2*a^11 - 1/2*a^8 - 1/2*a^7 - 1/2*a^5 - 1/2*a^3 - 1/2*a^2 - 1/2*a - 1/2, 1/2*a^12 - 1/2*a^9 - 1/2*a^8 - 1/2*a^6 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2 - 1/2*a, 1/2*a^13 - 1/2*a^10 - 1/2*a^9 - 1/2*a^7 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2, 1/2*a^14 - 1/2*a^10 - 1/2*a^7 - 1/2*a^6 - 1/2*a^4 - 1/2*a^2 - 1/2*a - 1/2, 1/26*a^15 - 1/26*a^14 - 1/26*a^13 - 5/26*a^12 + 2/13*a^11 + 5/13*a^10 + 3/13*a^9 - 1/2*a^8 - 3/13*a^7 - 2/13*a^6 + 5/26*a^5 - 11/26*a^4 - 6/13*a^3 - 1/26*a^2 - 4/13*a - 2/13, 1/29364921293260777978*a^16 + 4098531100523443/2258840099481598306*a^15 + 2290759332112233059/14682460646630388989*a^14 - 1170527218531311085/14682460646630388989*a^13 + 4745795395826816265/29364921293260777978*a^12 + 2119269114832081635/14682460646630388989*a^11 - 6516938756224068970/14682460646630388989*a^10 + 10822074663067952511/29364921293260777978*a^9 + 10080944291125566983/29364921293260777978*a^8 + 2626619353481275667/14682460646630388989*a^7 - 5697224104791830259/29364921293260777978*a^6 - 6506936825805858263/14682460646630388989*a^5 - 7890559222185381011/29364921293260777978*a^4 + 979884448016261775/2258840099481598306*a^3 - 9844269148539855935/29364921293260777978*a^2 + 1392516107826284591/14682460646630388989*a + 11582301900463138945/29364921293260777978

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$8$	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:		$\frac{43\!\cdots\!63}{29\!\cdots\!78}a^{16}-\frac{32\!\cdots\!57}{14\!\cdots\!89}a^{15}-\frac{61\!\cdots\!01}{29\!\cdots\!78}a^{14}-\frac{14\!\cdots\!13}{29\!\cdots\!78}a^{13}-\frac{82\!\cdots\!69}{29\!\cdots\!78}a^{12}+\frac{51\!\cdots\!81}{29\!\cdots\!78}a^{11}+\frac{53\!\cdots\!15}{14\!\cdots\!89}a^{10}+\frac{74\!\cdots\!49}{14\!\cdots\!89}a^{9}+\frac{19\!\cdots\!42}{14\!\cdots\!89}a^{8}+\frac{16\!\cdots\!03}{29\!\cdots\!78}a^{7}-\frac{62\!\cdots\!04}{14\!\cdots\!89}a^{6}-\frac{11\!\cdots\!25}{14\!\cdots\!89}a^{5}-\frac{84\!\cdots\!79}{29\!\cdots\!78}a^{4}+\frac{86\!\cdots\!27}{29\!\cdots\!78}a^{3}-\frac{17\!\cdots\!19}{14\!\cdots\!89}a^{2}+\frac{40\!\cdots\!90}{14\!\cdots\!89}a-\frac{21\!\cdots\!44}{14\!\cdots\!89}$, $\frac{11\!\cdots\!43}{14\!\cdots\!89}a^{16}-\frac{20\!\cdots\!08}{14\!\cdots\!89}a^{15}-\frac{12\!\cdots\!82}{14\!\cdots\!89}a^{14}-\frac{37\!\cdots\!26}{14\!\cdots\!89}a^{13}-\frac{13\!\cdots\!95}{29\!\cdots\!78}a^{12}+\frac{14\!\cdots\!75}{29\!\cdots\!78}a^{11}+\frac{41\!\cdots\!34}{14\!\cdots\!89}a^{10}+\frac{12\!\cdots\!75}{22\!\cdots\!06}a^{9}+\frac{81\!\cdots\!57}{14\!\cdots\!89}a^{8}+\frac{13\!\cdots\!25}{29\!\cdots\!78}a^{7}+\frac{73\!\cdots\!05}{29\!\cdots\!78}a^{6}+\frac{98\!\cdots\!67}{29\!\cdots\!78}a^{5}+\frac{86\!\cdots\!11}{29\!\cdots\!78}a^{4}+\frac{48\!\cdots\!54}{14\!\cdots\!89}a^{3}+\frac{24\!\cdots\!33}{14\!\cdots\!89}a^{2}+\frac{88\!\cdots\!06}{14\!\cdots\!89}a+\frac{40\!\cdots\!77}{22\!\cdots\!06}$, $\frac{12\!\cdots\!95}{29\!\cdots\!78}a^{16}-\frac{66\!\cdots\!64}{14\!\cdots\!89}a^{15}-\frac{21\!\cdots\!38}{14\!\cdots\!89}a^{14}-\frac{15\!\cdots\!52}{11\!\cdots\!53}a^{13}-\frac{19\!\cdots\!68}{14\!\cdots\!89}a^{12}+\frac{62\!\cdots\!84}{14\!\cdots\!89}a^{11}+\frac{16\!\cdots\!18}{14\!\cdots\!89}a^{10}+\frac{26\!\cdots\!33}{14\!\cdots\!89}a^{9}+\frac{18\!\cdots\!67}{14\!\cdots\!89}a^{8}+\frac{14\!\cdots\!96}{11\!\cdots\!53}a^{7}+\frac{83\!\cdots\!53}{14\!\cdots\!89}a^{6}+\frac{22\!\cdots\!15}{14\!\cdots\!89}a^{5}+\frac{71\!\cdots\!51}{14\!\cdots\!89}a^{4}+\frac{16\!\cdots\!96}{14\!\cdots\!89}a^{3}+\frac{17\!\cdots\!86}{14\!\cdots\!89}a^{2}+\frac{88\!\cdots\!33}{29\!\cdots\!78}a+\frac{55\!\cdots\!12}{14\!\cdots\!89}$, $\frac{50\!\cdots\!87}{29\!\cdots\!78}a^{16}+\frac{15\!\cdots\!65}{14\!\cdots\!89}a^{15}-\frac{54\!\cdots\!91}{29\!\cdots\!78}a^{14}-\frac{66\!\cdots\!89}{11\!\cdots\!53}a^{13}-\frac{22\!\cdots\!62}{14\!\cdots\!89}a^{12}+\frac{26\!\cdots\!37}{14\!\cdots\!89}a^{11}+\frac{20\!\cdots\!73}{29\!\cdots\!78}a^{10}+\frac{26\!\cdots\!52}{14\!\cdots\!89}a^{9}+\frac{35\!\cdots\!98}{14\!\cdots\!89}a^{8}+\frac{46\!\cdots\!79}{22\!\cdots\!06}a^{7}+\frac{34\!\cdots\!53}{29\!\cdots\!78}a^{6}+\frac{12\!\cdots\!15}{14\!\cdots\!89}a^{5}+\frac{33\!\cdots\!69}{29\!\cdots\!78}a^{4}+\frac{18\!\cdots\!36}{14\!\cdots\!89}a^{3}+\frac{26\!\cdots\!41}{29\!\cdots\!78}a^{2}+\frac{38\!\cdots\!50}{14\!\cdots\!89}a+\frac{11\!\cdots\!87}{29\!\cdots\!78}$, $\frac{21\!\cdots\!85}{29\!\cdots\!78}a^{16}+\frac{26\!\cdots\!59}{29\!\cdots\!78}a^{15}-\frac{36\!\cdots\!01}{29\!\cdots\!78}a^{14}-\frac{70\!\cdots\!83}{29\!\cdots\!78}a^{13}-\frac{89\!\cdots\!15}{11\!\cdots\!53}a^{12}-\frac{12\!\cdots\!04}{14\!\cdots\!89}a^{11}+\frac{46\!\cdots\!02}{14\!\cdots\!89}a^{10}+\frac{24\!\cdots\!27}{29\!\cdots\!78}a^{9}+\frac{18\!\cdots\!09}{14\!\cdots\!89}a^{8}+\frac{18\!\cdots\!94}{14\!\cdots\!89}a^{7}+\frac{30\!\cdots\!67}{29\!\cdots\!78}a^{6}+\frac{21\!\cdots\!67}{29\!\cdots\!78}a^{5}+\frac{92\!\cdots\!03}{11\!\cdots\!53}a^{4}+\frac{18\!\cdots\!95}{29\!\cdots\!78}a^{3}+\frac{10\!\cdots\!28}{14\!\cdots\!89}a^{2}+\frac{33\!\cdots\!65}{11\!\cdots\!53}a+\frac{31\!\cdots\!03}{14\!\cdots\!89}$, $\frac{32\!\cdots\!89}{29\!\cdots\!78}a^{16}-\frac{85\!\cdots\!13}{29\!\cdots\!78}a^{15}-\frac{13\!\cdots\!34}{11\!\cdots\!53}a^{14}-\frac{53\!\cdots\!03}{14\!\cdots\!89}a^{13}-\frac{96\!\cdots\!22}{14\!\cdots\!89}a^{12}+\frac{11\!\cdots\!11}{14\!\cdots\!89}a^{11}+\frac{56\!\cdots\!45}{14\!\cdots\!89}a^{10}+\frac{11\!\cdots\!35}{14\!\cdots\!89}a^{9}+\frac{12\!\cdots\!55}{14\!\cdots\!89}a^{8}+\frac{10\!\cdots\!38}{14\!\cdots\!89}a^{7}+\frac{61\!\cdots\!82}{14\!\cdots\!89}a^{6}+\frac{68\!\cdots\!03}{14\!\cdots\!89}a^{5}+\frac{66\!\cdots\!66}{14\!\cdots\!89}a^{4}+\frac{67\!\cdots\!15}{14\!\cdots\!89}a^{3}+\frac{44\!\cdots\!96}{14\!\cdots\!89}a^{2}+\frac{31\!\cdots\!35}{29\!\cdots\!78}a+\frac{97\!\cdots\!13}{29\!\cdots\!78}$, $\frac{34\!\cdots\!29}{22\!\cdots\!06}a^{16}-\frac{52\!\cdots\!57}{14\!\cdots\!89}a^{15}+\frac{11\!\cdots\!29}{29\!\cdots\!78}a^{14}-\frac{14\!\cdots\!17}{29\!\cdots\!78}a^{13}+\frac{20\!\cdots\!74}{14\!\cdots\!89}a^{12}+\frac{73\!\cdots\!07}{29\!\cdots\!78}a^{11}+\frac{41\!\cdots\!10}{14\!\cdots\!89}a^{10}+\frac{14\!\cdots\!41}{29\!\cdots\!78}a^{9}-\frac{17\!\cdots\!73}{22\!\cdots\!06}a^{8}-\frac{21\!\cdots\!87}{29\!\cdots\!78}a^{7}-\frac{25\!\cdots\!01}{29\!\cdots\!78}a^{6}+\frac{10\!\cdots\!36}{14\!\cdots\!89}a^{5}-\frac{82\!\cdots\!43}{14\!\cdots\!89}a^{4}-\frac{13\!\cdots\!19}{14\!\cdots\!89}a^{3}-\frac{21\!\cdots\!61}{29\!\cdots\!78}a^{2}-\frac{45\!\cdots\!03}{29\!\cdots\!78}a-\frac{18\!\cdots\!66}{14\!\cdots\!89}$, $\frac{14\!\cdots\!43}{29\!\cdots\!78}a^{16}-\frac{28\!\cdots\!66}{14\!\cdots\!89}a^{15}+\frac{65\!\cdots\!17}{22\!\cdots\!06}a^{14}-\frac{52\!\cdots\!07}{29\!\cdots\!78}a^{13}+\frac{82\!\cdots\!55}{29\!\cdots\!78}a^{12}+\frac{63\!\cdots\!66}{14\!\cdots\!89}a^{11}-\frac{27\!\cdots\!94}{14\!\cdots\!89}a^{10}+\frac{12\!\cdots\!12}{14\!\cdots\!89}a^{9}-\frac{12\!\cdots\!93}{29\!\cdots\!78}a^{8}-\frac{18\!\cdots\!85}{14\!\cdots\!89}a^{7}-\frac{14\!\cdots\!11}{14\!\cdots\!89}a^{6}-\frac{30\!\cdots\!87}{29\!\cdots\!78}a^{5}-\frac{52\!\cdots\!03}{29\!\cdots\!78}a^{4}-\frac{65\!\cdots\!47}{14\!\cdots\!89}a^{3}-\frac{26\!\cdots\!03}{29\!\cdots\!78}a^{2}+\frac{43\!\cdots\!29}{29\!\cdots\!78}a-\frac{94\!\cdots\!19}{29\!\cdots\!78}$ 437521165359222863/29364921293260777978*a^16 - 325093634480706857/14682460646630388989*a^15 - 6109123389535901/29364921293260777978*a^14 - 14142519287366914513/29364921293260777978*a^13 - 8299844152909776369/29364921293260777978*a^12 + 51374052937125360381/29364921293260777978*a^11 + 53481188610464529815/14682460646630388989*a^10 + 74498523097528508349/14682460646630388989*a^9 + 19072705992113553142/14682460646630388989*a^8 + 16140164162915225903/29364921293260777978*a^7 - 62688453213824719404/14682460646630388989*a^6 - 1104818661529248025/14682460646630388989*a^5 - 84322170506356319779/29364921293260777978*a^4 + 86578579359844712727/29364921293260777978*a^3 - 17382800976680055419/14682460646630388989*a^2 + 4024684022148519590/14682460646630388989*a - 211670739792748544/14682460646630388989, 1123081425176787443/14682460646630388989*a^16 - 209191916685823308/14682460646630388989*a^15 - 1253177957200433982/14682460646630388989*a^14 - 37202993087223126126/14682460646630388989*a^13 - 136400329881542192195/29364921293260777978*a^12 + 147802490061000761475/29364921293260777978*a^11 + 411118464026491399534/14682460646630388989*a^10 + 123655896408988076275/2258840099481598306*a^9 + 815574408596599009457/14682460646630388989*a^8 + 1333782320794211810225/29364921293260777978*a^7 + 731578974382086178105/29364921293260777978*a^6 + 986412564338914517167/29364921293260777978*a^5 + 866718218130048581511/29364921293260777978*a^4 + 481437466428572391254/14682460646630388989*a^3 + 248857314006459783133/14682460646630388989*a^2 + 88867936812048129906/14682460646630388989*a + 4092171653405328377/2258840099481598306, 1219625553590893795/29364921293260777978*a^16 - 667780967489262664/14682460646630388989*a^15 - 218293265714180238/14682460646630388989*a^14 - 1511489233548390652/1129420049740799153*a^13 - 19319835829649467568/14682460646630388989*a^12 + 62292693341077414084/14682460646630388989*a^11 + 164893705098648344318/14682460646630388989*a^10 + 264936609459171331133/14682460646630388989*a^9 + 181669464443836792867/14682460646630388989*a^8 + 14395926240151305396/1129420049740799153*a^7 + 83628691134794967053/14682460646630388989*a^6 + 220022352347044099215/14682460646630388989*a^5 + 71661572758204132551/14682460646630388989*a^4 + 162551489361209534796/14682460646630388989*a^3 + 17387429398774257486/14682460646630388989*a^2 + 88600626710112966633/29364921293260777978*a + 5574662919873687512/14682460646630388989, 502295389779462187/29364921293260777978*a^16 + 159628234054998465/14682460646630388989*a^15 - 547816154016094591/29364921293260777978*a^14 - 660405440680538989/1129420049740799153*a^13 - 22284232259732513962/14682460646630388989*a^12 + 2674862270827637337/14682460646630388989*a^11 + 207853318400073261973/29364921293260777978*a^10 + 264581282585248360552/14682460646630388989*a^9 + 350273752686643153098/14682460646630388989*a^8 + 46476331913728970379/2258840099481598306*a^7 + 340464406479229606653/29364921293260777978*a^6 + 129231982576719398615/14682460646630388989*a^5 + 334904360869375106169/29364921293260777978*a^4 + 188287910936457530136/14682460646630388989*a^3 + 267421493781114464041/29364921293260777978*a^2 + 38822332934355355250/14682460646630388989*a + 11438178803722651987/29364921293260777978, 2102683008354330485/29364921293260777978*a^16 + 2698107624333760859/29364921293260777978*a^15 - 3688134512415306001/29364921293260777978*a^14 - 70454540672514275683/29364921293260777978*a^13 - 8911341388657211615/1129420049740799153*a^12 - 12847458937833592104/14682460646630388989*a^11 + 469741702336211539802/14682460646630388989*a^10 + 2498758616589590866327/29364921293260777978*a^9 + 1864089635586121643009/14682460646630388989*a^8 + 1833764908117013817794/14682460646630388989*a^7 + 3055497056318846060567/29364921293260777978*a^6 + 2135236867914369456267/29364921293260777978*a^5 + 92408304282952486803/1129420049740799153*a^4 + 1891760137904587074695/29364921293260777978*a^3 + 1090663802695497965328/14682460646630388989*a^2 + 33416304512393171165/1129420049740799153*a + 318921430388171609003/14682460646630388989, 3273577680286173589/29364921293260777978*a^16 - 852407943473214513/29364921293260777978*a^15 - 132801554221080034/1129420049740799153*a^14 - 53422717849839157503/14682460646630388989*a^13 - 96734021216976729022/14682460646630388989*a^12 + 112462614212558611611/14682460646630388989*a^11 + 566001785027582940145/14682460646630388989*a^10 + 1126942641367682202935/14682460646630388989*a^9 + 1227866602197275027755/14682460646630388989*a^8 + 1058659699805318313538/14682460646630388989*a^7 + 613082440035524621882/14682460646630388989*a^6 + 683442688814080938503/14682460646630388989*a^5 + 661727075467906106566/14682460646630388989*a^4 + 671995974415283069015/14682460646630388989*a^3 + 444126356144572490596/14682460646630388989*a^2 + 310193644003981936435/29364921293260777978*a + 97432499151696865913/29364921293260777978, 344879784141823429/2258840099481598306*a^16 - 5203426435423758357/14682460646630388989*a^15 + 1136065530203004629/29364921293260777978*a^14 - 141294627701154438517/29364921293260777978*a^13 + 20361702289373196074/14682460646630388989*a^12 + 737604502266715530107/29364921293260777978*a^11 + 414748260980735795910/14682460646630388989*a^10 + 143743680254097969341/29364921293260777978*a^9 - 175072191725610538873/2258840099481598306*a^8 - 2158847795571950934487/29364921293260777978*a^7 - 2508646297804749618501/29364921293260777978*a^6 + 100216371201492256336/14682460646630388989*a^5 - 823247119164456224843/14682460646630388989*a^4 - 138885984790930770819/14682460646630388989*a^3 - 2100789241661229747361/29364921293260777978*a^2 - 455740676347958632003/29364921293260777978*a - 186844030230650931566/14682460646630388989, 142391629105643143/29364921293260777978*a^16 - 280213920157560966/14682460646630388989*a^15 + 65445654367432217/2258840099481598306*a^14 - 5287618819320119107/29364921293260777978*a^13 + 8284098954796318855/29364921293260777978*a^12 + 6329930276058577866/14682460646630388989*a^11 - 2770291342103857894/14682460646630388989*a^10 + 12436217027325757612/14682460646630388989*a^9 - 12910088893010157593/29364921293260777978*a^8 - 180935197416601285/14682460646630388989*a^7 - 148959717905539070111/14682460646630388989*a^6 - 303543277091417241387/29364921293260777978*a^5 - 529194670737517016003/29364921293260777978*a^4 - 65677894329565153147/14682460646630388989*a^3 - 261140999762126916103/29364921293260777978*a^2 + 43354742748941319729/29364921293260777978*a - 94258445579451309019/29364921293260777978	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:		\( 4992205.48619 \)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 4992205.48619 \cdot 1}{2\cdot\sqrt{92267897090016343666010049}}\cr\approx \mathstrut & 1.26242646052 \end{aligned}\]

Galois group

$C_{17}:C_8$ (as 17T4):

Intermediate 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
\(3\) 3	$\Q_{3}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	3.8.4.2	$x^{8} - 6 x^{6} + 162$	$2$	$4$	$4$	$C_8$	$[\ ]_{2}^{4}$
	3.8.4.2	$x^{8} - 6 x^{6} + 162$	$2$	$4$	$4$	$C_8$	$[\ ]_{2}^{4}$
\(17\) 17	17.17.18.4	$x^{17} + 136 x^{2} + 17$	$17$	$1$	$18$	$C_{17}:C_{8}$	$[9/8]_{8}$