Normalized defining polynomial

\( x^{23} - 299 x^{21} + 38870 x^{19} - 2880267 x^{17} + 134008212 x^{15} - 4064915764 x^{13} + \cdots - 12722733426052 \) x^23 - 299*x^21 + 38870*x^19 - 2880267*x^17 + 134008212*x^15 - 4064915764*x^13 + 80820089896*x^11 - 1031899362065*x^9 + 8048815024107*x^7 - 34878198437797*x^5 + 69756396875594*x^3 - 41219689062851*x - 12722733426052

Invariants

Degree:		$23$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[23, 0]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(281289734904574222488347652007286105552626963997840714384801792\) 281289734904574222488347652007286105552626963997840714384801792 \(\medspace = 2^{22}\cdot 13^{22}\cdot 23^{23}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(519.02\)	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:		$2\cdot 13^{22/23}23^{527/506}\approx 609.2327772055112$
Ramified primes:		\(2\), \(13\), \(23\) 2, 13, 23	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q(\sqrt{23}) \)
$\card{ \Aut(K/\Q) }$:		$1$	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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{87263423403}a^{12}+\frac{2541706348}{6712571031}a^{11}-\frac{4}{2237523677}a^{10}-\frac{985172090}{6712571031}a^{9}+\frac{234}{2237523677}a^{8}-\frac{2471619568}{6712571031}a^{7}-\frac{18928}{6712571031}a^{6}+\frac{2528776924}{6712571031}a^{5}+\frac{76895}{2237523677}a^{4}+\frac{1450906678}{6712571031}a^{3}-\frac{342732}{2237523677}a^{2}+\frac{1597699462}{6712571031}a+\frac{742586}{6712571031}$, $\frac{1}{87263423403}a^{13}-\frac{13}{6712571031}a^{11}-\frac{520672631}{6712571031}a^{10}+\frac{845}{6712571031}a^{9}+\frac{561731720}{6712571031}a^{8}-\frac{8788}{2237523677}a^{7}+\frac{1291490864}{6712571031}a^{6}+\frac{399854}{6712571031}a^{5}-\frac{1929239801}{6712571031}a^{4}-\frac{2599051}{6712571031}a^{3}+\frac{2471202160}{6712571031}a^{2}+\frac{4826809}{6712571031}a+\frac{114978166}{6712571031}$, $\frac{1}{87263423403}a^{14}-\frac{576845803}{6712571031}a^{11}-\frac{1183}{6712571031}a^{10}+\frac{627308095}{2237523677}a^{9}+\frac{30758}{2237523677}a^{8}-\frac{232812206}{6712571031}a^{7}-\frac{2798978}{6712571031}a^{6}+\frac{2542085402}{6712571031}a^{5}+\frac{36386714}{6712571031}a^{4}-\frac{690697405}{6712571031}a^{3}-\frac{168938315}{6712571031}a^{2}+\frac{1623346004}{6712571031}a+\frac{125497034}{6712571031}$, $\frac{1}{87263423403}a^{15}-\frac{455}{2237523677}a^{11}-\frac{280865860}{2237523677}a^{10}+\frac{118300}{6712571031}a^{9}+\frac{1406297668}{6712571031}a^{8}-\frac{1384110}{2237523677}a^{7}-\frac{1129133495}{6712571031}a^{6}+\frac{22391824}{2237523677}a^{5}+\frac{1679178269}{6712571031}a^{4}-\frac{454833925}{6712571031}a^{3}-\frac{500867704}{6712571031}a^{2}+\frac{289608540}{2237523677}a+\frac{788313119}{6712571031}$, $\frac{1}{87263423403}a^{16}-\frac{9403203}{2237523677}a^{11}-\frac{94640}{6712571031}a^{10}-\frac{937474658}{6712571031}a^{9}+\frac{2768220}{2237523677}a^{8}-\frac{79251101}{6712571031}a^{7}-\frac{89567296}{2237523677}a^{6}+\frac{1288352414}{6712571031}a^{5}-\frac{3073899631}{6712571031}a^{4}+\frac{3128229521}{6712571031}a^{3}+\frac{920400231}{2237523677}a^{2}-\frac{1934768635}{6712571031}a-\frac{82651164}{2237523677}$, $\frac{1}{87263423403}a^{17}-\frac{114920}{6712571031}a^{11}+\frac{1374397369}{6712571031}a^{10}+\frac{3734900}{2237523677}a^{9}+\frac{2283941455}{6712571031}a^{8}-\frac{139834656}{2237523677}a^{7}+\frac{717569492}{6712571031}a^{6}+\frac{356847689}{6712571031}a^{5}+\frac{2533003973}{6712571031}a^{4}-\frac{748484861}{2237523677}a^{3}-\frac{1775004004}{6712571031}a^{2}+\frac{676412192}{2237523677}a+\frac{831686241}{2237523677}$, $\frac{1}{87263423403}a^{18}+\frac{511271061}{2237523677}a^{11}-\frac{2240940}{2237523677}a^{10}-\frac{124935618}{2237523677}a^{9}+\frac{209751984}{2237523677}a^{8}-\frac{663836697}{2237523677}a^{7}-\frac{356847689}{2237523677}a^{6}-\frac{1283003066}{6712571031}a^{5}+\frac{15861812}{2237523677}a^{4}-\frac{1821385520}{6712571031}a^{3}+\frac{1041435505}{2237523677}a^{2}-\frac{1411891954}{6712571031}a+\frac{1819560445}{6712571031}$, $\frac{1}{87263423403}a^{19}-\frac{2838524}{2237523677}a^{11}-\frac{917502474}{2237523677}a^{10}+\frac{295206496}{2237523677}a^{9}+\frac{950850839}{2237523677}a^{8}-\frac{325434959}{2237523677}a^{7}+\frac{617840671}{6712571031}a^{6}+\frac{419984043}{2237523677}a^{5}+\frac{603758491}{6712571031}a^{4}-\frac{153958152}{2237523677}a^{3}+\frac{3313701983}{6712571031}a^{2}+\frac{403517320}{6712571031}a-\frac{764722372}{2237523677}$, $\frac{1}{87263423403}a^{20}-\frac{112465016}{2237523677}a^{11}-\frac{147603248}{2237523677}a^{10}-\frac{55798128}{2237523677}a^{9}+\frac{966174618}{2237523677}a^{8}+\frac{3265945810}{6712571031}a^{7}+\frac{68801731}{2237523677}a^{6}-\frac{905636900}{6712571031}a^{5}+\frac{769790760}{2237523677}a^{4}-\frac{2808196765}{6712571031}a^{3}+\frac{1668604531}{6712571031}a^{2}+\frac{513380591}{2237523677}a-\frac{926092387}{2237523677}$, $\frac{1}{87263423403}a^{21}-\frac{193729263}{2237523677}a^{11}+\frac{299848792}{2237523677}a^{10}+\frac{849623732}{2237523677}a^{9}+\frac{1263050629}{6712571031}a^{8}-\frac{583123228}{2237523677}a^{7}-\frac{602216564}{6712571031}a^{6}-\frac{620963081}{2237523677}a^{5}+\frac{206438921}{6712571031}a^{4}+\frac{3012422929}{6712571031}a^{3}+\frac{573469888}{2237523677}a^{2}-\frac{300716057}{2237523677}a+\frac{826750271}{2237523677}$, $\frac{1}{87263423403}a^{22}+\frac{424325214}{2237523677}a^{11}-\frac{284333495}{2237523677}a^{10}+\frac{1563233839}{6712571031}a^{9}-\frac{253573920}{2237523677}a^{8}-\frac{2786601422}{6712571031}a^{7}+\frac{23604572}{2237523677}a^{6}-\frac{443088292}{6712571031}a^{5}-\frac{2514547238}{6712571031}a^{4}+\frac{26741345}{2237523677}a^{3}+\frac{1070352950}{2237523677}a^{2}-\frac{884288665}{2237523677}a-\frac{1114523376}{2237523677}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, 1/87263423403*a^12 + 2541706348/6712571031*a^11 - 4/2237523677*a^10 - 985172090/6712571031*a^9 + 234/2237523677*a^8 - 2471619568/6712571031*a^7 - 18928/6712571031*a^6 + 2528776924/6712571031*a^5 + 76895/2237523677*a^4 + 1450906678/6712571031*a^3 - 342732/2237523677*a^2 + 1597699462/6712571031*a + 742586/6712571031, 1/87263423403*a^13 - 13/6712571031*a^11 - 520672631/6712571031*a^10 + 845/6712571031*a^9 + 561731720/6712571031*a^8 - 8788/2237523677*a^7 + 1291490864/6712571031*a^6 + 399854/6712571031*a^5 - 1929239801/6712571031*a^4 - 2599051/6712571031*a^3 + 2471202160/6712571031*a^2 + 4826809/6712571031*a + 114978166/6712571031, 1/87263423403*a^14 - 576845803/6712571031*a^11 - 1183/6712571031*a^10 + 627308095/2237523677*a^9 + 30758/2237523677*a^8 - 232812206/6712571031*a^7 - 2798978/6712571031*a^6 + 2542085402/6712571031*a^5 + 36386714/6712571031*a^4 - 690697405/6712571031*a^3 - 168938315/6712571031*a^2 + 1623346004/6712571031*a + 125497034/6712571031, 1/87263423403*a^15 - 455/2237523677*a^11 - 280865860/2237523677*a^10 + 118300/6712571031*a^9 + 1406297668/6712571031*a^8 - 1384110/2237523677*a^7 - 1129133495/6712571031*a^6 + 22391824/2237523677*a^5 + 1679178269/6712571031*a^4 - 454833925/6712571031*a^3 - 500867704/6712571031*a^2 + 289608540/2237523677*a + 788313119/6712571031, 1/87263423403*a^16 - 9403203/2237523677*a^11 - 94640/6712571031*a^10 - 937474658/6712571031*a^9 + 2768220/2237523677*a^8 - 79251101/6712571031*a^7 - 89567296/2237523677*a^6 + 1288352414/6712571031*a^5 - 3073899631/6712571031*a^4 + 3128229521/6712571031*a^3 + 920400231/2237523677*a^2 - 1934768635/6712571031*a - 82651164/2237523677, 1/87263423403*a^17 - 114920/6712571031*a^11 + 1374397369/6712571031*a^10 + 3734900/2237523677*a^9 + 2283941455/6712571031*a^8 - 139834656/2237523677*a^7 + 717569492/6712571031*a^6 + 356847689/6712571031*a^5 + 2533003973/6712571031*a^4 - 748484861/2237523677*a^3 - 1775004004/6712571031*a^2 + 676412192/2237523677*a + 831686241/2237523677, 1/87263423403*a^18 + 511271061/2237523677*a^11 - 2240940/2237523677*a^10 - 124935618/2237523677*a^9 + 209751984/2237523677*a^8 - 663836697/2237523677*a^7 - 356847689/2237523677*a^6 - 1283003066/6712571031*a^5 + 15861812/2237523677*a^4 - 1821385520/6712571031*a^3 + 1041435505/2237523677*a^2 - 1411891954/6712571031*a + 1819560445/6712571031, 1/87263423403*a^19 - 2838524/2237523677*a^11 - 917502474/2237523677*a^10 + 295206496/2237523677*a^9 + 950850839/2237523677*a^8 - 325434959/2237523677*a^7 + 617840671/6712571031*a^6 + 419984043/2237523677*a^5 + 603758491/6712571031*a^4 - 153958152/2237523677*a^3 + 3313701983/6712571031*a^2 + 403517320/6712571031*a - 764722372/2237523677, 1/87263423403*a^20 - 112465016/2237523677*a^11 - 147603248/2237523677*a^10 - 55798128/2237523677*a^9 + 966174618/2237523677*a^8 + 3265945810/6712571031*a^7 + 68801731/2237523677*a^6 - 905636900/6712571031*a^5 + 769790760/2237523677*a^4 - 2808196765/6712571031*a^3 + 1668604531/6712571031*a^2 + 513380591/2237523677*a - 926092387/2237523677, 1/87263423403*a^21 - 193729263/2237523677*a^11 + 299848792/2237523677*a^10 + 849623732/2237523677*a^9 + 1263050629/6712571031*a^8 - 583123228/2237523677*a^7 - 602216564/6712571031*a^6 - 620963081/2237523677*a^5 + 206438921/6712571031*a^4 + 3012422929/6712571031*a^3 + 573469888/2237523677*a^2 - 300716057/2237523677*a + 826750271/2237523677, 1/87263423403*a^22 + 424325214/2237523677*a^11 - 284333495/2237523677*a^10 + 1563233839/6712571031*a^9 - 253573920/2237523677*a^8 - 2786601422/6712571031*a^7 + 23604572/2237523677*a^6 - 443088292/6712571031*a^5 - 2514547238/6712571031*a^4 + 26741345/2237523677*a^3 + 1070352950/2237523677*a^2 - 884288665/2237523677*a - 1114523376/2237523677

Class group and class number

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

Unit group

Rank:		$22$	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:		$\frac{2}{87263423403}a^{22}+\frac{2}{87263423403}a^{21}-\frac{44}{6712571031}a^{20}-\frac{638}{87263423403}a^{19}+\frac{70523}{87263423403}a^{18}+\frac{6662}{6712571031}a^{17}-\frac{4898152}{87263423403}a^{16}-\frac{6539830}{87263423403}a^{15}+\frac{16159595}{6712571031}a^{14}+\frac{300900794}{87263423403}a^{13}-\frac{1910896005}{29087807801}a^{12}-\frac{666958376}{6712571031}a^{11}+\frac{7602802495}{6712571031}a^{10}+\frac{11879997142}{6712571031}a^{9}-\frac{79484679437}{6712571031}a^{8}-\frac{123738307424}{6712571031}a^{7}+\frac{154576732986}{2237523677}a^{6}+\frac{666760247030}{6712571031}a^{5}-\frac{1320013565867}{6712571031}a^{4}-\frac{1438400447198}{6712571031}a^{3}+\frac{525148258598}{2237523677}a^{2}+\frac{256694816888}{2237523677}a-\frac{49937222789}{2237523677}$, $\frac{1}{87263423403}a^{22}-\frac{16}{87263423403}a^{21}-\frac{233}{87263423403}a^{20}+\frac{4321}{87263423403}a^{19}+\frac{21544}{87263423403}a^{18}-\frac{498728}{87263423403}a^{17}-\frac{956912}{87263423403}a^{16}+\frac{32111816}{87263423403}a^{15}+\frac{426802}{2237523677}a^{14}-\frac{421022704}{29087807801}a^{13}+\frac{220284928}{87263423403}a^{12}+\frac{2403696389}{6712571031}a^{11}-\frac{1137584188}{6712571031}a^{10}-\frac{37106989216}{6712571031}a^{9}+\frac{19118625200}{6712571031}a^{8}+\frac{341810278451}{6712571031}a^{7}-\frac{44322094945}{2237523677}a^{6}-\frac{566164708969}{2237523677}a^{5}+\frac{323784360746}{6712571031}a^{4}+\frac{1235902319798}{2237523677}a^{3}-\frac{61026262575}{2237523677}a^{2}-\frac{801275320958}{2237523677}a-\frac{742979592287}{6712571031}$, $\frac{4}{87263423403}a^{22}-\frac{29}{87263423403}a^{21}-\frac{1006}{87263423403}a^{20}+\frac{7753}{87263423403}a^{19}+\frac{34735}{29087807801}a^{18}-\frac{294184}{29087807801}a^{17}-\frac{5618072}{87263423403}a^{16}+\frac{55734650}{87263423403}a^{15}+\frac{158015785}{87263423403}a^{14}-\frac{710497830}{29087807801}a^{13}-\frac{1538990036}{87263423403}a^{12}+\frac{1295734166}{2237523677}a^{11}-\frac{889628440}{2237523677}a^{10}-\frac{56051281465}{6712571031}a^{9}+\frac{31951039707}{2237523677}a^{8}+\frac{458874080614}{6712571031}a^{7}-\frac{1174061358923}{6712571031}a^{6}-\frac{598770257489}{2237523677}a^{5}+\frac{2148490041846}{2237523677}a^{4}+\frac{1665450351500}{6712571031}a^{3}-\frac{4623604442107}{2237523677}a^{2}+\frac{1574723183858}{2237523677}a+\frac{4596597462161}{6712571031}$, $\frac{181}{87263423403}a^{22}+\frac{904}{87263423403}a^{21}-\frac{1274}{2237523677}a^{20}-\frac{82611}{29087807801}a^{19}+\frac{1939778}{29087807801}a^{18}+\frac{28984075}{87263423403}a^{17}-\frac{126338984}{29087807801}a^{16}-\frac{1884458140}{87263423403}a^{15}+\frac{14998433951}{87263423403}a^{14}+\frac{74405683997}{87263423403}a^{13}-\frac{123376742068}{29087807801}a^{12}-\frac{46933394072}{2237523677}a^{11}+\frac{433314297919}{6712571031}a^{10}+\frac{2131990692793}{6712571031}a^{9}-\frac{3890994635470}{6712571031}a^{8}-\frac{6320283364206}{2237523677}a^{7}+\frac{18917295730013}{6712571031}a^{6}+\frac{30060264818581}{2237523677}a^{5}-\frac{14142625254247}{2237523677}a^{4}-\frac{189819451264718}{6712571031}a^{3}+\frac{33670671465313}{6712571031}a^{2}+\frac{39513802913469}{2237523677}a+\frac{32330925743659}{6712571031}$, $\frac{10}{2237523677}a^{22}+\frac{551}{87263423403}a^{21}-\frac{38606}{29087807801}a^{20}-\frac{4195}{2237523677}a^{19}+\frac{4974800}{29087807801}a^{18}+\frac{21078820}{87263423403}a^{17}-\frac{364358946}{29087807801}a^{16}-\frac{1543567435}{87263423403}a^{15}+\frac{16684225588}{29087807801}a^{14}+\frac{70668691186}{87263423403}a^{13}-\frac{494753590592}{29087807801}a^{12}-\frac{53724802271}{2237523677}a^{11}+\frac{2194484029769}{6712571031}a^{10}+\frac{3097458202255}{6712571031}a^{9}-\frac{26541876195536}{6712571031}a^{8}-\frac{12486998292404}{2237523677}a^{7}+\frac{188159322961952}{6712571031}a^{6}+\frac{265586513866033}{6712571031}a^{5}-\frac{669089489731630}{6712571031}a^{4}-\frac{944715853302805}{6712571031}a^{3}+\frac{753162913544065}{6712571031}a^{2}+\frac{10\!\cdots\!99}{6712571031}a+\frac{269718255930269}{6712571031}$, $\frac{34}{87263423403}a^{22}+\frac{107}{87263423403}a^{21}-\frac{9940}{87263423403}a^{20}-\frac{10538}{29087807801}a^{19}+\frac{416720}{29087807801}a^{18}+\frac{4018618}{87263423403}a^{17}-\frac{88285618}{87263423403}a^{16}-\frac{286824991}{87263423403}a^{15}+\frac{3830044357}{87263423403}a^{14}+\frac{4191876709}{29087807801}a^{13}-\frac{34944181612}{29087807801}a^{12}-\frac{8920330201}{2237523677}a^{11}+\frac{137429175167}{6712571031}a^{10}+\frac{461185199075}{6712571031}a^{9}-\frac{1381081308472}{6712571031}a^{8}-\frac{1565368993465}{2237523677}a^{7}+\frac{7279710104005}{6712571031}a^{6}+\frac{8419648559682}{2237523677}a^{5}-\frac{15903339295282}{6712571031}a^{4}-\frac{57904650668219}{6712571031}a^{3}+\frac{3099172275279}{2237523677}a^{2}+\frac{13568868853462}{2237523677}a+\frac{4035390043335}{2237523677}$, $\frac{32}{87263423403}a^{22}+\frac{73}{87263423403}a^{21}-\frac{9094}{87263423403}a^{20}-\frac{21751}{87263423403}a^{19}+\frac{1112458}{87263423403}a^{18}+\frac{2782877}{87263423403}a^{17}-\frac{76496006}{87263423403}a^{16}-\frac{199581953}{87263423403}a^{15}+\frac{3237085448}{87263423403}a^{14}+\frac{2926884158}{29087807801}a^{13}-\frac{28884726515}{29087807801}a^{12}-\frac{6243377045}{2237523677}a^{11}+\frac{111524079907}{6712571031}a^{10}+\frac{323302869124}{6712571031}a^{9}-\frac{1106673368837}{6712571031}a^{8}-\frac{3294537957416}{6712571031}a^{7}+\frac{5820627428302}{6712571031}a^{6}+\frac{17703549328919}{6712571031}a^{5}-\frac{4343322544259}{2237523677}a^{4}-\frac{40346008127017}{6712571031}a^{3}+\frac{8643459128624}{6712571031}a^{2}+\frac{27961147422841}{6712571031}a+\frac{7938053281837}{6712571031}$, $\frac{1}{87263423403}a^{22}+\frac{2}{6712571031}a^{21}-\frac{93}{29087807801}a^{20}-\frac{2547}{29087807801}a^{19}+\frac{11325}{29087807801}a^{18}+\frac{321330}{29087807801}a^{17}-\frac{2368216}{87263423403}a^{16}-\frac{5240630}{6712571031}a^{15}+\frac{103781293}{87263423403}a^{14}+\frac{983523914}{29087807801}a^{13}-\frac{2939433307}{87263423403}a^{12}-\frac{6187584065}{6712571031}a^{11}+\frac{4062911590}{6712571031}a^{10}+\frac{34962624023}{2237523677}a^{9}-\frac{14348669167}{2237523677}a^{8}-\frac{1050464494324}{6712571031}a^{7}+\frac{227106841883}{6712571031}a^{6}+\frac{5583525095527}{6712571031}a^{5}-\frac{420633838721}{6712571031}a^{4}-\frac{4242203312967}{2237523677}a^{3}+\frac{155337036780}{2237523677}a^{2}+\frac{7961591232890}{6712571031}a+\frac{2427971244059}{6712571031}$, $\frac{7}{29087807801}a^{22}-\frac{38}{29087807801}a^{21}-\frac{5591}{87263423403}a^{20}+\frac{30175}{87263423403}a^{19}+\frac{211509}{29087807801}a^{18}-\frac{3396676}{87263423403}a^{17}-\frac{40074793}{87263423403}a^{16}+\frac{212020135}{87263423403}a^{15}+\frac{514655014}{29087807801}a^{14}-\frac{8029602257}{87263423403}a^{13}-\frac{37473905800}{87263423403}a^{12}+\frac{4871992316}{2237523677}a^{11}+\frac{44033938955}{6712571031}a^{10}-\frac{215088001559}{6712571031}a^{9}-\frac{410344843477}{6712571031}a^{8}+\frac{1901204990188}{6712571031}a^{7}+\frac{716402777351}{2237523677}a^{6}-\frac{9312411408026}{6712571031}a^{5}-\frac{1706903195725}{2237523677}a^{4}+\frac{6956176092550}{2237523677}a^{3}+\frac{2788479048010}{6712571031}a^{2}-\frac{11783746413091}{6712571031}a-\frac{3891851435345}{6712571031}$, $\frac{6}{29087807801}a^{22}+\frac{24}{29087807801}a^{21}-\frac{5126}{87263423403}a^{20}-\frac{6896}{29087807801}a^{19}+\frac{208669}{29087807801}a^{18}+\frac{2560582}{87263423403}a^{17}-\frac{42738845}{87263423403}a^{16}-\frac{178280228}{87263423403}a^{15}+\frac{1780490194}{87263423403}a^{14}+\frac{2548354037}{29087807801}a^{13}-\frac{46256226538}{87263423403}a^{12}-\frac{15969932437}{6712571031}a^{11}+\frac{56218054405}{6712571031}a^{10}+\frac{271554725938}{6712571031}a^{9}-\frac{494209881058}{6712571031}a^{8}-\frac{2749942831037}{6712571031}a^{7}+\frac{1854922822382}{6712571031}a^{6}+\frac{4976577961806}{2237523677}a^{5}+\frac{606951132266}{6712571031}a^{4}-\frac{35895428055911}{6712571031}a^{3}-\frac{4298227390106}{2237523677}a^{2}+\frac{10021190722969}{2237523677}a+\frac{15027643853375}{6712571031}$, $\frac{8}{29087807801}a^{22}+\frac{29}{29087807801}a^{21}-\frac{6967}{87263423403}a^{20}-\frac{8410}{29087807801}a^{19}+\frac{871360}{87263423403}a^{18}+\frac{1050268}{29087807801}a^{17}-\frac{61320797}{87263423403}a^{16}-\frac{73755023}{29087807801}a^{15}+\frac{2658674725}{87263423403}a^{14}+\frac{3188507542}{29087807801}a^{13}-\frac{73013553916}{87263423403}a^{12}-\frac{20128992065}{6712571031}a^{11}+\frac{32181157489}{2237523677}a^{10}+\frac{114764261315}{2237523677}a^{9}-\frac{328560442042}{2237523677}a^{8}-\frac{3494354062483}{6712571031}a^{7}+\frac{5335177493048}{6712571031}a^{6}+\frac{6267762300053}{2237523677}a^{5}-\frac{12221299864694}{6712571031}a^{4}-\frac{42947279084500}{6712571031}a^{3}+\frac{2833421072161}{2237523677}a^{2}+\frac{29394992487986}{6712571031}a+\frac{8334613228945}{6712571031}$, $\frac{7}{87263423403}a^{22}+\frac{18}{29087807801}a^{21}-\frac{1745}{87263423403}a^{20}-\frac{354}{2237523677}a^{19}+\frac{183049}{87263423403}a^{18}+\frac{500018}{29087807801}a^{17}-\frac{10474931}{87263423403}a^{16}-\frac{90311743}{87263423403}a^{15}+\frac{353631416}{87263423403}a^{14}+\frac{3293947366}{87263423403}a^{13}-\frac{7065974665}{87263423403}a^{12}-\frac{5750127175}{6712571031}a^{11}+\frac{1989370182}{2237523677}a^{10}+\frac{80245132775}{6712571031}a^{9}-\frac{26292630668}{6712571031}a^{8}-\frac{218309324292}{2237523677}a^{7}-\frac{71976131414}{6712571031}a^{6}+\frac{2800614325646}{6712571031}a^{5}+\frac{304394245053}{2237523677}a^{4}-\frac{4883158025122}{6712571031}a^{3}-\frac{1411126832146}{6712571031}a^{2}+\frac{1826707306235}{6712571031}a+\frac{670097721569}{6712571031}$, $\frac{7}{29087807801}a^{22}-\frac{77}{87263423403}a^{21}-\frac{5986}{87263423403}a^{20}+\frac{6743}{29087807801}a^{19}+\frac{247583}{29087807801}a^{18}-\frac{2274893}{87263423403}a^{17}-\frac{17552055}{29087807801}a^{16}+\frac{143213360}{87263423403}a^{15}+\frac{784069049}{29087807801}a^{14}-\frac{5543319635}{87263423403}a^{13}-\frac{68811150695}{87263423403}a^{12}+\frac{10501253011}{6712571031}a^{11}+\frac{101972373227}{6712571031}a^{10}-\frac{164857164730}{6712571031}a^{9}-\frac{422035993585}{2237523677}a^{8}+\frac{1612892430670}{6712571031}a^{7}+\frac{3201704055519}{2237523677}a^{6}-\frac{9376739848646}{6712571031}a^{5}-\frac{13374978245959}{2237523677}a^{4}+\frac{9923152606001}{2237523677}a^{3}+\frac{24740835602726}{2237523677}a^{2}-\frac{43309578916096}{6712571031}a-\frac{32332760874989}{6712571031}$, $\frac{7}{87263423403}a^{22}-\frac{58}{87263423403}a^{21}-\frac{1922}{87263423403}a^{20}+\frac{15605}{87263423403}a^{19}+\frac{76676}{29087807801}a^{18}-\frac{601372}{29087807801}a^{17}-\frac{5268299}{29087807801}a^{16}+\frac{117231385}{87263423403}a^{15}+\frac{230296078}{29087807801}a^{14}-\frac{1566722188}{29087807801}a^{13}-\frac{20087529382}{87263423403}a^{12}+\frac{9235101650}{6712571031}a^{11}+\frac{10073710317}{2237523677}a^{10}-\frac{149641638008}{6712571031}a^{9}-\frac{130102583256}{2237523677}a^{8}+\frac{493621018908}{2237523677}a^{7}+\frac{1048103535828}{2237523677}a^{6}-\frac{8240195773522}{6712571031}a^{5}-\frac{4672476931786}{2237523677}a^{4}+\frac{22125445357586}{6712571031}a^{3}+\frac{27214596535033}{6712571031}a^{2}-\frac{22545056052647}{6712571031}a-\frac{5352238964463}{2237523677}$, $\frac{398}{87263423403}a^{22}-\frac{329}{29087807801}a^{21}-\frac{38847}{29087807801}a^{20}+\frac{288923}{87263423403}a^{19}+\frac{14750534}{87263423403}a^{18}-\frac{36556982}{87263423403}a^{17}-\frac{1055373017}{87263423403}a^{16}+\frac{871561690}{29087807801}a^{15}+\frac{46835593681}{87263423403}a^{14}-\frac{115994008894}{87263423403}a^{13}-\frac{1329822656131}{87263423403}a^{12}+\frac{84418413174}{2237523677}a^{11}+\frac{615412732120}{2237523677}a^{10}-\frac{1523133879360}{2237523677}a^{9}-\frac{20272574863711}{6712571031}a^{8}+\frac{50162767339775}{6712571031}a^{7}+\frac{122292146834446}{6712571031}a^{6}-\frac{100863538537567}{2237523677}a^{5}-\frac{106614412748429}{2237523677}a^{4}+\frac{263925377271374}{2237523677}a^{3}+\frac{60129276888444}{2237523677}a^{2}-\frac{447760946431300}{6712571031}a-\frac{157864008742775}{6712571031}$, $\frac{3}{29087807801}a^{22}-\frac{680}{87263423403}a^{21}-\frac{397}{6712571031}a^{20}+\frac{184040}{87263423403}a^{19}+\frac{980071}{87263423403}a^{18}-\frac{21355294}{87263423403}a^{17}-\frac{93724988}{87263423403}a^{16}+\frac{1389299804}{87263423403}a^{15}+\frac{5200790845}{87263423403}a^{14}-\frac{55634562977}{87263423403}a^{13}-\frac{58874400047}{29087807801}a^{12}+\frac{36331630936}{2237523677}a^{11}+\frac{94733166213}{2237523677}a^{10}-\frac{1759778779483}{6712571031}a^{9}-\frac{3541708678823}{6712571031}a^{8}+\frac{17338298667772}{6712571031}a^{7}+\frac{8036763089288}{2237523677}a^{6}-\frac{95120602492748}{6712571031}a^{5}-\frac{74163837056677}{6712571031}a^{4}+\frac{235159300962700}{6712571031}a^{3}+\frac{69760419798754}{6712571031}a^{2}-\frac{50070018745118}{2237523677}a-\frac{62943875491201}{6712571031}$, $\frac{10}{29087807801}a^{22}+\frac{28}{87263423403}a^{21}-\frac{8821}{87263423403}a^{20}-\frac{2891}{29087807801}a^{19}+\frac{1124687}{87263423403}a^{18}+\frac{387369}{29087807801}a^{17}-\frac{27146892}{29087807801}a^{16}-\frac{88176470}{87263423403}a^{15}+\frac{3683733572}{87263423403}a^{14}+\frac{1387794692}{29087807801}a^{13}-\frac{107796850052}{87263423403}a^{12}-\frac{9731899822}{6712571031}a^{11}+\frac{157104382498}{6712571031}a^{10}+\frac{190054367951}{6712571031}a^{9}-\frac{1871102925592}{6712571031}a^{8}-\frac{769389140621}{2237523677}a^{7}+\frac{4353042215780}{2237523677}a^{6}+\frac{5386560904186}{2237523677}a^{5}-\frac{45941391559039}{6712571031}a^{4}-\frac{55321698946219}{6712571031}a^{3}+\frac{17867750351752}{2237523677}a^{2}+\frac{18637060790044}{2237523677}a+\frac{9910912140301}{6712571031}$, $\frac{59}{87263423403}a^{22}+\frac{80}{29087807801}a^{21}-\frac{5636}{29087807801}a^{20}-\frac{67931}{87263423403}a^{19}+\frac{2085284}{87263423403}a^{18}+\frac{8267047}{87263423403}a^{17}-\frac{144522823}{87263423403}a^{16}-\frac{188299742}{29087807801}a^{15}+\frac{6161655671}{87263423403}a^{14}+\frac{7910473468}{29087807801}a^{13}-\frac{55365441334}{29087807801}a^{12}-\frac{48480054532}{6712571031}a^{11}+\frac{215127568084}{6712571031}a^{10}+\frac{804634174717}{6712571031}a^{9}-\frac{2146807367411}{6712571031}a^{8}-\frac{2642722448884}{2237523677}a^{7}+\frac{11348404055375}{6712571031}a^{6}+\frac{41522816244190}{6712571031}a^{5}-\frac{25553910777935}{6712571031}a^{4}-\frac{93202739207501}{6712571031}a^{3}+\frac{5792806729680}{2237523677}a^{2}+\frac{63006137932243}{6712571031}a+\frac{5976305030307}{2237523677}$, $\frac{107}{87263423403}a^{22}-\frac{10}{29087807801}a^{21}-\frac{2408}{6712571031}a^{20}+\frac{6574}{87263423403}a^{19}+\frac{1321045}{29087807801}a^{18}-\frac{38464}{6712571031}a^{17}-\frac{284218726}{87263423403}a^{16}+\frac{7218038}{87263423403}a^{15}+\frac{12691596703}{87263423403}a^{14}+\frac{1225743523}{87263423403}a^{13}-\frac{365246995583}{87263423403}a^{12}-\frac{6871852160}{6712571031}a^{11}+\frac{521259319837}{6712571031}a^{10}+\frac{73501781121}{2237523677}a^{9}-\frac{2017896950280}{2237523677}a^{8}-\frac{3831667059613}{6712571031}a^{7}+\frac{13687949419978}{2237523677}a^{6}+\frac{11849009854779}{2237523677}a^{5}-\frac{46551591022710}{2237523677}a^{4}-\frac{51058800789705}{2237523677}a^{3}+\frac{148227784733128}{6712571031}a^{2}+\frac{65790711770814}{2237523677}a+\frac{17225026863745}{2237523677}$, $\frac{80}{87263423403}a^{22}-\frac{22}{29087807801}a^{21}-\frac{23134}{87263423403}a^{20}+\frac{17846}{87263423403}a^{19}+\frac{962840}{29087807801}a^{18}-\frac{674962}{29087807801}a^{17}-\frac{67911640}{29087807801}a^{16}+\frac{41304792}{29087807801}a^{15}+\frac{2971944448}{29087807801}a^{14}-\frac{4329666812}{87263423403}a^{13}-\frac{250312550360}{87263423403}a^{12}+\frac{2077832711}{2237523677}a^{11}+\frac{346471747042}{6712571031}a^{10}-\frac{38709019126}{6712571031}a^{9}-\frac{3874441770061}{6712571031}a^{8}-\frac{205207523057}{2237523677}a^{7}+\frac{8363583638528}{2237523677}a^{6}+\frac{4036835864397}{2237523677}a^{5}-\frac{80556826401160}{6712571031}a^{4}-\frac{66082693262642}{6712571031}a^{3}+\frac{77619478111816}{6712571031}a^{2}+\frac{87181568700898}{6712571031}a+\frac{6906321426549}{2237523677}$, $\frac{73}{87263423403}a^{22}+\frac{244}{87263423403}a^{21}-\frac{21035}{87263423403}a^{20}-\frac{23472}{29087807801}a^{19}+\frac{2607541}{87263423403}a^{18}+\frac{2912311}{29087807801}a^{17}-\frac{60528036}{29087807801}a^{16}-\frac{608619224}{87263423403}a^{15}+\frac{2592481691}{29087807801}a^{14}+\frac{26061492551}{87263423403}a^{13}-\frac{70215703510}{29087807801}a^{12}-\frac{54257414291}{6712571031}a^{11}+\frac{274319344814}{6712571031}a^{10}+\frac{305871747434}{2237523677}a^{9}-\frac{918926170288}{2237523677}a^{8}-\frac{9213427175242}{6712571031}a^{7}+\frac{4901378274264}{2237523677}a^{6}+\frac{49154963569331}{6712571031}a^{5}-\frac{33282416213531}{6712571031}a^{4}-\frac{111727589256848}{6712571031}a^{3}+\frac{22490988072479}{6712571031}a^{2}+\frac{76816749138304}{6712571031}a+\frac{21882790625927}{6712571031}$, $\frac{124}{87263423403}a^{22}+\frac{433}{87263423403}a^{21}-\frac{34438}{87263423403}a^{20}-\frac{119008}{87263423403}a^{19}+\frac{4095896}{87263423403}a^{18}+\frac{13933150}{87263423403}a^{17}-\frac{90764320}{29087807801}a^{16}-\frac{904856306}{87263423403}a^{15}+\frac{11078923922}{87263423403}a^{14}+\frac{11853278809}{29087807801}a^{13}-\frac{94711319421}{29087807801}a^{12}-\frac{66587187124}{6712571031}a^{11}+\frac{351300892310}{6712571031}a^{10}+\frac{987995594747}{6712571031}a^{9}-\frac{1138346636658}{2237523677}a^{8}-\frac{8467996987186}{6712571031}a^{7}+\frac{18680188066799}{6712571031}a^{6}+\frac{12589275044859}{2237523677}a^{5}-\frac{50156087352218}{6712571031}a^{4}-\frac{71545422347402}{6712571031}a^{3}+\frac{17647016809060}{2237523677}a^{2}+\frac{39386516112838}{6712571031}a-\frac{785248269757}{6712571031}$ 2/87263423403*a^22 + 2/87263423403*a^21 - 44/6712571031*a^20 - 638/87263423403*a^19 + 70523/87263423403*a^18 + 6662/6712571031*a^17 - 4898152/87263423403*a^16 - 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11853278809/29087807801*a^13 - 94711319421/29087807801*a^12 - 66587187124/6712571031*a^11 + 351300892310/6712571031*a^10 + 987995594747/6712571031*a^9 - 1138346636658/2237523677*a^8 - 8467996987186/6712571031*a^7 + 18680188066799/6712571031*a^6 + 12589275044859/2237523677*a^5 - 50156087352218/6712571031*a^4 - 71545422347402/6712571031*a^3 + 17647016809060/2237523677*a^2 + 39386516112838/6712571031*a - 785248269757/6712571031 (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:		\( 3930853533530000000000000 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{23}\cdot(2\pi)^{0}\cdot 3930853533530000000000000 \cdot 1}{2\cdot\sqrt{281289734904574222488347652007286105552626963997840714384801792}}\cr\approx \mathstrut & 0.983036910370839 \end{aligned}\] (assuming GRH)

Galois group

$F_{23}$ (as 23T4):

Intermediate fields

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	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $22$	$2$	$11$	$22$
\(13\) 13	13.23.22.1	$x^{23} + 13$	$23$	$1$	$22$	$C_{23}:C_{11}$	$[\ ]_{23}^{11}$
\(23\) 23	23.23.23.11	$x^{23} + 506 x + 23$	$23$	$1$	$23$	$F_{23}$	$[23/22]_{22}$