Normalized defining polynomial

\( x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + \cdots - 74 \) x^17 - x^16 - 4*x^15 + 2*x^14 + 54*x^13 - 6*x^12 - 36*x^11 + 16*x^10 + 714*x^9 + 1238*x^8 + 484*x^7 - 764*x^6 - 1084*x^5 + 520*x^4 + 668*x^3 - 776*x^2 + 382*x - 74

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:		\(861526607800060221948166144\) 861526607800060221948166144 \(\medspace = 2^{30}\cdot 173^{8}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(38.41\)	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:		not computed
Ramified primes:		\(2\), \(173\) 2, 173	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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{51\!\cdots\!19}a^{16}-\frac{29\!\cdots\!61}{51\!\cdots\!19}a^{15}-\frac{14\!\cdots\!39}{51\!\cdots\!19}a^{14}+\frac{87\!\cdots\!85}{51\!\cdots\!19}a^{13}+\frac{90\!\cdots\!97}{51\!\cdots\!19}a^{12}+\frac{21\!\cdots\!08}{51\!\cdots\!19}a^{11}+\frac{72\!\cdots\!54}{51\!\cdots\!19}a^{10}+\frac{31\!\cdots\!96}{51\!\cdots\!19}a^{9}+\frac{25\!\cdots\!45}{51\!\cdots\!19}a^{8}-\frac{49\!\cdots\!91}{51\!\cdots\!19}a^{7}-\frac{27\!\cdots\!52}{51\!\cdots\!19}a^{6}+\frac{22\!\cdots\!06}{51\!\cdots\!19}a^{5}-\frac{63\!\cdots\!14}{51\!\cdots\!19}a^{4}+\frac{14\!\cdots\!17}{51\!\cdots\!19}a^{3}+\frac{30\!\cdots\!20}{51\!\cdots\!19}a^{2}+\frac{22\!\cdots\!38}{51\!\cdots\!19}a+\frac{10\!\cdots\!37}{51\!\cdots\!19}$ 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, 1/518449026921810234250726519*a^16 - 2903744299660867300931161/518449026921810234250726519*a^15 - 145515969622780446567758039/518449026921810234250726519*a^14 + 87449486712529578111145685/518449026921810234250726519*a^13 + 90388261689503535099965497/518449026921810234250726519*a^12 + 213755840339945972595489008/518449026921810234250726519*a^11 + 72953974012171598161596854/518449026921810234250726519*a^10 + 3137750212693082994430496/518449026921810234250726519*a^9 + 256458172663421001979231745/518449026921810234250726519*a^8 - 49611598799627003593660391/518449026921810234250726519*a^7 - 27387611796406411645447552/518449026921810234250726519*a^6 + 222961449395058127219738606/518449026921810234250726519*a^5 - 63208606884273117780803614/518449026921810234250726519*a^4 + 148709660332780900942449817/518449026921810234250726519*a^3 + 30151660364221902163647120/518449026921810234250726519*a^2 + 225255761602774742093792238/518449026921810234250726519*a + 1081393905446205624367037/518449026921810234250726519

Class group and class number

$C_{15}$, which has order $15$ (assuming GRH)

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:		$\frac{44\!\cdots\!32}{51\!\cdots\!19}a^{16}-\frac{27\!\cdots\!92}{51\!\cdots\!19}a^{15}-\frac{19\!\cdots\!79}{51\!\cdots\!19}a^{14}+\frac{84\!\cdots\!50}{51\!\cdots\!19}a^{13}+\frac{24\!\cdots\!30}{51\!\cdots\!19}a^{12}+\frac{67\!\cdots\!12}{51\!\cdots\!19}a^{11}-\frac{14\!\cdots\!79}{51\!\cdots\!19}a^{10}-\frac{36\!\cdots\!62}{51\!\cdots\!19}a^{9}+\frac{32\!\cdots\!07}{51\!\cdots\!19}a^{8}+\frac{67\!\cdots\!11}{51\!\cdots\!19}a^{7}+\frac{45\!\cdots\!38}{51\!\cdots\!19}a^{6}-\frac{26\!\cdots\!08}{51\!\cdots\!19}a^{5}-\frac{70\!\cdots\!44}{51\!\cdots\!19}a^{4}-\frac{96\!\cdots\!88}{51\!\cdots\!19}a^{3}+\frac{28\!\cdots\!28}{51\!\cdots\!19}a^{2}-\frac{17\!\cdots\!58}{51\!\cdots\!19}a+\frac{75\!\cdots\!83}{51\!\cdots\!19}$, $\frac{21\!\cdots\!47}{51\!\cdots\!19}a^{16}-\frac{12\!\cdots\!41}{51\!\cdots\!19}a^{15}-\frac{93\!\cdots\!11}{51\!\cdots\!19}a^{14}+\frac{43\!\cdots\!83}{51\!\cdots\!19}a^{13}+\frac{11\!\cdots\!10}{51\!\cdots\!19}a^{12}+\frac{37\!\cdots\!85}{51\!\cdots\!19}a^{11}-\frac{69\!\cdots\!45}{51\!\cdots\!19}a^{10}+\frac{39\!\cdots\!07}{51\!\cdots\!19}a^{9}+\frac{15\!\cdots\!48}{51\!\cdots\!19}a^{8}+\frac{33\!\cdots\!96}{51\!\cdots\!19}a^{7}+\frac{24\!\cdots\!55}{51\!\cdots\!19}a^{6}-\frac{82\!\cdots\!39}{51\!\cdots\!19}a^{5}-\frac{29\!\cdots\!47}{51\!\cdots\!19}a^{4}-\frac{20\!\cdots\!35}{51\!\cdots\!19}a^{3}+\frac{14\!\cdots\!31}{51\!\cdots\!19}a^{2}-\frac{11\!\cdots\!92}{51\!\cdots\!19}a+\frac{34\!\cdots\!07}{51\!\cdots\!19}$, $\frac{68\!\cdots\!28}{51\!\cdots\!19}a^{16}+\frac{16\!\cdots\!86}{51\!\cdots\!19}a^{15}-\frac{28\!\cdots\!39}{51\!\cdots\!19}a^{14}-\frac{21\!\cdots\!35}{51\!\cdots\!19}a^{13}+\frac{35\!\cdots\!80}{51\!\cdots\!19}a^{12}+\frac{40\!\cdots\!41}{51\!\cdots\!19}a^{11}+\frac{11\!\cdots\!07}{51\!\cdots\!19}a^{10}+\frac{96\!\cdots\!91}{51\!\cdots\!19}a^{9}+\frac{49\!\cdots\!40}{51\!\cdots\!19}a^{8}+\frac{14\!\cdots\!69}{51\!\cdots\!19}a^{7}+\frac{19\!\cdots\!93}{51\!\cdots\!19}a^{6}+\frac{12\!\cdots\!64}{51\!\cdots\!19}a^{5}+\frac{14\!\cdots\!20}{51\!\cdots\!19}a^{4}-\frac{27\!\cdots\!16}{51\!\cdots\!19}a^{3}+\frac{36\!\cdots\!57}{51\!\cdots\!19}a^{2}-\frac{12\!\cdots\!34}{51\!\cdots\!19}a+\frac{23\!\cdots\!51}{51\!\cdots\!19}$, $\frac{32\!\cdots\!72}{51\!\cdots\!19}a^{16}-\frac{23\!\cdots\!33}{51\!\cdots\!19}a^{15}-\frac{20\!\cdots\!47}{51\!\cdots\!19}a^{14}+\frac{89\!\cdots\!57}{51\!\cdots\!19}a^{13}+\frac{25\!\cdots\!11}{51\!\cdots\!19}a^{12}-\frac{10\!\cdots\!39}{51\!\cdots\!19}a^{11}-\frac{14\!\cdots\!20}{51\!\cdots\!19}a^{10}-\frac{27\!\cdots\!98}{51\!\cdots\!19}a^{9}+\frac{23\!\cdots\!53}{51\!\cdots\!19}a^{8}-\frac{11\!\cdots\!88}{51\!\cdots\!19}a^{7}-\frac{42\!\cdots\!38}{51\!\cdots\!19}a^{6}-\frac{62\!\cdots\!55}{51\!\cdots\!19}a^{5}-\frac{45\!\cdots\!78}{51\!\cdots\!19}a^{4}-\frac{95\!\cdots\!35}{51\!\cdots\!19}a^{3}-\frac{65\!\cdots\!16}{51\!\cdots\!19}a^{2}-\frac{73\!\cdots\!26}{51\!\cdots\!19}a+\frac{16\!\cdots\!27}{51\!\cdots\!19}$, $\frac{15\!\cdots\!79}{51\!\cdots\!19}a^{16}-\frac{83\!\cdots\!62}{51\!\cdots\!19}a^{15}-\frac{67\!\cdots\!22}{51\!\cdots\!19}a^{14}+\frac{19\!\cdots\!51}{51\!\cdots\!19}a^{13}+\frac{84\!\cdots\!23}{51\!\cdots\!19}a^{12}+\frac{29\!\cdots\!81}{51\!\cdots\!19}a^{11}-\frac{50\!\cdots\!52}{51\!\cdots\!19}a^{10}+\frac{46\!\cdots\!34}{51\!\cdots\!19}a^{9}+\frac{11\!\cdots\!52}{51\!\cdots\!19}a^{8}+\frac{24\!\cdots\!11}{51\!\cdots\!19}a^{7}+\frac{17\!\cdots\!71}{51\!\cdots\!19}a^{6}-\frac{53\!\cdots\!23}{51\!\cdots\!19}a^{5}-\frac{19\!\cdots\!94}{51\!\cdots\!19}a^{4}+\frac{34\!\cdots\!29}{51\!\cdots\!19}a^{3}+\frac{12\!\cdots\!49}{51\!\cdots\!19}a^{2}-\frac{74\!\cdots\!71}{51\!\cdots\!19}a+\frac{20\!\cdots\!81}{51\!\cdots\!19}$, $\frac{12\!\cdots\!34}{51\!\cdots\!19}a^{16}-\frac{55\!\cdots\!88}{51\!\cdots\!19}a^{15}-\frac{55\!\cdots\!10}{51\!\cdots\!19}a^{14}-\frac{49\!\cdots\!28}{51\!\cdots\!19}a^{13}+\frac{69\!\cdots\!96}{51\!\cdots\!19}a^{12}+\frac{31\!\cdots\!77}{51\!\cdots\!19}a^{11}-\frac{31\!\cdots\!48}{51\!\cdots\!19}a^{10}+\frac{35\!\cdots\!03}{51\!\cdots\!19}a^{9}+\frac{91\!\cdots\!12}{51\!\cdots\!19}a^{8}+\frac{21\!\cdots\!91}{51\!\cdots\!19}a^{7}+\frac{17\!\cdots\!64}{51\!\cdots\!19}a^{6}-\frac{46\!\cdots\!04}{51\!\cdots\!19}a^{5}-\frac{14\!\cdots\!99}{51\!\cdots\!19}a^{4}-\frac{14\!\cdots\!03}{51\!\cdots\!19}a^{3}+\frac{80\!\cdots\!99}{51\!\cdots\!19}a^{2}-\frac{59\!\cdots\!78}{51\!\cdots\!19}a+\frac{17\!\cdots\!29}{51\!\cdots\!19}$, $\frac{33\!\cdots\!74}{51\!\cdots\!19}a^{16}-\frac{88\!\cdots\!23}{51\!\cdots\!19}a^{15}-\frac{15\!\cdots\!99}{51\!\cdots\!19}a^{14}-\frac{19\!\cdots\!30}{51\!\cdots\!19}a^{13}+\frac{18\!\cdots\!25}{51\!\cdots\!19}a^{12}+\frac{10\!\cdots\!04}{51\!\cdots\!19}a^{11}-\frac{12\!\cdots\!24}{51\!\cdots\!19}a^{10}+\frac{25\!\cdots\!97}{51\!\cdots\!19}a^{9}+\frac{23\!\cdots\!80}{51\!\cdots\!19}a^{8}+\frac{59\!\cdots\!89}{51\!\cdots\!19}a^{7}+\frac{48\!\cdots\!06}{51\!\cdots\!19}a^{6}-\frac{62\!\cdots\!87}{51\!\cdots\!19}a^{5}-\frac{48\!\cdots\!83}{51\!\cdots\!19}a^{4}-\frac{12\!\cdots\!77}{51\!\cdots\!19}a^{3}+\frac{37\!\cdots\!60}{51\!\cdots\!19}a^{2}-\frac{19\!\cdots\!86}{51\!\cdots\!19}a+\frac{73\!\cdots\!79}{51\!\cdots\!19}$, $\frac{36\!\cdots\!06}{51\!\cdots\!19}a^{16}-\frac{78\!\cdots\!91}{51\!\cdots\!19}a^{15}-\frac{62\!\cdots\!61}{51\!\cdots\!19}a^{14}+\frac{78\!\cdots\!32}{51\!\cdots\!19}a^{13}+\frac{18\!\cdots\!98}{51\!\cdots\!19}a^{12}-\frac{13\!\cdots\!02}{51\!\cdots\!19}a^{11}+\frac{37\!\cdots\!44}{51\!\cdots\!19}a^{10}-\frac{86\!\cdots\!77}{51\!\cdots\!19}a^{9}+\frac{26\!\cdots\!17}{51\!\cdots\!19}a^{8}+\frac{31\!\cdots\!15}{51\!\cdots\!19}a^{7}-\frac{30\!\cdots\!55}{51\!\cdots\!19}a^{6}-\frac{11\!\cdots\!90}{51\!\cdots\!19}a^{5}-\frac{20\!\cdots\!47}{51\!\cdots\!19}a^{4}-\frac{58\!\cdots\!75}{51\!\cdots\!19}a^{3}+\frac{39\!\cdots\!91}{51\!\cdots\!19}a^{2}-\frac{30\!\cdots\!87}{51\!\cdots\!19}a+\frac{22\!\cdots\!57}{51\!\cdots\!19}$ 4464450691457888219127032/518449026921810234250726519*a^16 - 2778077651056340703627492/518449026921810234250726519*a^15 - 19099343160650306364113179/518449026921810234250726519*a^14 + 840000348123006134290950/518449026921810234250726519*a^13 + 243626097347702830887753430/518449026921810234250726519*a^12 + 67759552090794232799747212/518449026921810234250726519*a^11 - 147594529874799854508116079/518449026921810234250726519*a^10 - 36133204068351849659180062/518449026921810234250726519*a^9 + 3202203210414220049184157507/518449026921810234250726519*a^8 + 6713751819624370883477867011/518449026921810234250726519*a^7 + 4572565643288289463236353338/518449026921810234250726519*a^6 - 2601143819586072033657002008/518449026921810234250726519*a^5 - 7031843738159010148818909944/518449026921810234250726519*a^4 - 962539560041004528037778688/518449026921810234250726519*a^3 + 2818164637943871543542195128/518449026921810234250726519*a^2 - 1701556418368547617748555358/518449026921810234250726519*a + 753870258585790541092256583/518449026921810234250726519, 21807114262001114193418947/518449026921810234250726519*a^16 - 12478962613728058427905741/518449026921810234250726519*a^15 - 93871049867252055582147211/518449026921810234250726519*a^14 + 4378359612658816967470483/518449026921810234250726519*a^13 + 1183818512001398316884742110/518449026921810234250726519*a^12 + 375488680681567271487083285/518449026921810234250726519*a^11 - 691419555286801464088251645/518449026921810234250726519*a^10 + 39793502062838552064680907/518449026921810234250726519*a^9 + 15573498264929675222952215248/518449026921810234250726519*a^8 + 33641701402804420490291569596/518449026921810234250726519*a^7 + 24003479781306459178168174355/518449026921810234250726519*a^6 - 8282768568802960438555996039/518449026921810234250726519*a^5 - 29128376247862596322639233247/518449026921810234250726519*a^4 - 2093340239503375426799419635/518449026921810234250726519*a^3 + 14043454754952792951881820431/518449026921810234250726519*a^2 - 11668980007201171541164764692/518449026921810234250726519*a + 3493001323678648651803439307/518449026921810234250726519, 6807316263347736197711928/518449026921810234250726519*a^16 + 1644718135120494275109286/518449026921810234250726519*a^15 - 28022360360928334904201139/518449026921810234250726519*a^14 - 21610416755385761438662235/518449026921810234250726519*a^13 + 353196750820526296316835980/518449026921810234250726519*a^12 + 401862204840515833247385841/518449026921810234250726519*a^11 + 110727974486214994430068007/518449026921810234250726519*a^10 + 96864175059792416980901791/518449026921810234250726519*a^9 + 4968771816827912292952664340/518449026921810234250726519*a^8 + 14417964725199345546031521469/518449026921810234250726519*a^7 + 19353976898566384896726636093/518449026921810234250726519*a^6 + 12828063805413987612388650464/518449026921810234250726519*a^5 + 1436926428934288367847636120/518449026921810234250726519*a^4 - 275376696257824771268076216/518449026921810234250726519*a^3 + 3698081061343389499541397257/518449026921810234250726519*a^2 - 1235252271638767828511873434/518449026921810234250726519*a + 231286364568997557179504551/518449026921810234250726519, 325772978888328922753772/518449026921810234250726519*a^16 - 2332028059572478195157833/518449026921810234250726519*a^15 - 2076517759100208201644247/518449026921810234250726519*a^14 + 8960758668820467182745357/518449026921810234250726519*a^13 + 25879770227137916038356811/518449026921810234250726519*a^12 - 108143820900599749795679839/518449026921810234250726519*a^11 - 146027666973307301970534320/518449026921810234250726519*a^10 - 27158525472672144104636298/518449026921810234250726519*a^9 + 231090400024750115865464653/518449026921810234250726519*a^8 - 1173326604813166143095850288/518449026921810234250726519*a^7 - 4247275270316821458870022538/518449026921810234250726519*a^6 - 6266294556557209603037776755/518449026921810234250726519*a^5 - 4595940222004033160610255178/518449026921810234250726519*a^4 - 952082239258945208945595835/518449026921810234250726519*a^3 - 65446657689732667907716016/518449026921810234250726519*a^2 - 732925405642264225906652726/518449026921810234250726519*a + 164209407965228746273363327/518449026921810234250726519, 15442180745624312526534179/518449026921810234250726519*a^16 - 8302649535143152334241462/518449026921810234250726519*a^15 - 67258135352961026094748622/518449026921810234250726519*a^14 + 1991699725933660549700851/518449026921810234250726519*a^13 + 840121564762739215573554423/518449026921810234250726519*a^12 + 292572981690985096541108881/518449026921810234250726519*a^11 - 508993745325468478256603152/518449026921810234250726519*a^10 + 46168377789770301005546534/518449026921810234250726519*a^9 + 11072745140146991838489672952/518449026921810234250726519*a^8 + 24269720052298977113461344211/518449026921810234250726519*a^7 + 17456832247686245979476065671/518449026921810234250726519*a^6 - 5357384579887323470115410023/518449026921810234250726519*a^5 - 19752787877096831280172763994/518449026921810234250726519*a^4 + 340471438365957011547248229/518449026921810234250726519*a^3 + 12148836636963571290185933449/518449026921810234250726519*a^2 - 7482621978539213829960316671/518449026921810234250726519*a + 2046079679417802978730096681/518449026921810234250726519, 12816263711183996506298934/518449026921810234250726519*a^16 - 5500712006779831271717188/518449026921810234250726519*a^15 - 55128804009581710942417810/518449026921810234250726519*a^14 - 4983307045611374061073228/518449026921810234250726519*a^13 + 691825566766543100779574696/518449026921810234250726519*a^12 + 315991277606627538908773377/518449026921810234250726519*a^11 - 318807140157446402379278548/518449026921810234250726519*a^10 + 35155130510720349834856803/518449026921810234250726519*a^9 + 9184756822494719070763372012/518449026921810234250726519*a^8 + 21079952562020856461420890791/518449026921810234250726519*a^7 + 17732077566812578278385186664/518449026921810234250726519*a^6 - 465596996761661216876943104/518449026921810234250726519*a^5 - 14536631465834807606342921399/518449026921810234250726519*a^4 - 1438774751754740286691115203/518449026921810234250726519*a^3 + 8088294769967018109853381199/518449026921810234250726519*a^2 - 5903975854379506921537345578/518449026921810234250726519*a + 1765516403022442175378704029/518449026921810234250726519, 3354074379222128372277674/518449026921810234250726519*a^16 - 889858522483991251557523/518449026921810234250726519*a^15 - 15815512437409584669141999/518449026921810234250726519*a^14 - 1908437195839031835926930/518449026921810234250726519*a^13 + 183119562528060292472499225/518449026921810234250726519*a^12 + 109727503285562280989946804/518449026921810234250726519*a^11 - 122366979299948848019838924/518449026921810234250726519*a^10 + 25027928726451208800162997/518449026921810234250726519*a^9 + 2346575274392929302995968880/518449026921810234250726519*a^8 + 5920752582048878595877574389/518449026921810234250726519*a^7 + 4896773393207362191807341206/518449026921810234250726519*a^6 - 624380622029945679773348087/518449026921810234250726519*a^5 - 4836004685627735175399066483/518449026921810234250726519*a^4 - 1247640344802977173168098777/518449026921810234250726519*a^3 + 3798198255632122682306544560/518449026921810234250726519*a^2 - 1999437758969872094709529686/518449026921810234250726519*a + 734264881612328059726664879/518449026921810234250726519, 368877631866614265508606/518449026921810234250726519*a^16 - 783403095225277713119991/518449026921810234250726519*a^15 - 629538891732819858439261/518449026921810234250726519*a^14 + 78054757550634795674832/518449026921810234250726519*a^13 + 18869413147164132311323498/518449026921810234250726519*a^12 - 13795080337073087994226802/518449026921810234250726519*a^11 + 3744194309540716388985944/518449026921810234250726519*a^10 - 86819335162187412427143077/518449026921810234250726519*a^9 + 260035635654685895319186117/518449026921810234250726519*a^8 + 311673458180276975210090015/518449026921810234250726519*a^7 - 306990998524667548611984655/518449026921810234250726519*a^6 - 1195270954512322766268836690/518449026921810234250726519*a^5 - 2017172352018197375497096447/518449026921810234250726519*a^4 - 581090801736428039525125375/518449026921810234250726519*a^3 + 395484407223711547007707191/518449026921810234250726519*a^2 - 305212630429169614825795687/518449026921810234250726519*a + 222309746828792703832780857/518449026921810234250726519 (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:		\( 781146.759847 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 781146.759847 \cdot 15}{2\cdot\sqrt{861526607800060221948166144}}\cr\approx \mathstrut & 0.969680124809 \end{aligned}\] (assuming GRH)

Galois group

$\SL(2,16)$ (as 17T6):

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
\(2\) 2	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $16$	$16$	$1$	$30$
\(173\) 173	$\Q_{173}$	$x + 171$	$1$	$1$	$0$	Trivial	$[\ ]$
	173.4.2.1	$x^{4} + 55698 x^{3} + 780245071 x^{2} + 130285141230 x + 2355971094$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	173.4.2.1	$x^{4} + 55698 x^{3} + 780245071 x^{2} + 130285141230 x + 2355971094$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	173.4.2.1	$x^{4} + 55698 x^{3} + 780245071 x^{2} + 130285141230 x + 2355971094$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	173.4.2.1	$x^{4} + 55698 x^{3} + 780245071 x^{2} + 130285141230 x + 2355971094$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

Artin representations