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Properties

Label 23.1.152...439.1
Degree $23$
Signature $[1, 11]$
Discriminant $-1.523\times 10^{33}$
Root discriminant \(27.72\)
Ramified prime $1039$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{23}$ (as 23T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^23 + 4*x^21 - 14*x^20 + 25*x^19 - 6*x^18 - 14*x^17 - 26*x^16 + 31*x^15 + 48*x^14 - 62*x^13 + 19*x^12 + 19*x^11 + 24*x^10 - 17*x^9 + 34*x^8 + 40*x^7 - 228*x^6 + 76*x^5 + 4*x^4 + 28*x^3 + 45*x^2 + 49*x + 1)
 
gp: K = bnfinit(y^23 + 4*y^21 - 14*y^20 + 25*y^19 - 6*y^18 - 14*y^17 - 26*y^16 + 31*y^15 + 48*y^14 - 62*y^13 + 19*y^12 + 19*y^11 + 24*y^10 - 17*y^9 + 34*y^8 + 40*y^7 - 228*y^6 + 76*y^5 + 4*y^4 + 28*y^3 + 45*y^2 + 49*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 + 4*x^21 - 14*x^20 + 25*x^19 - 6*x^18 - 14*x^17 - 26*x^16 + 31*x^15 + 48*x^14 - 62*x^13 + 19*x^12 + 19*x^11 + 24*x^10 - 17*x^9 + 34*x^8 + 40*x^7 - 228*x^6 + 76*x^5 + 4*x^4 + 28*x^3 + 45*x^2 + 49*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 + 4*x^21 - 14*x^20 + 25*x^19 - 6*x^18 - 14*x^17 - 26*x^16 + 31*x^15 + 48*x^14 - 62*x^13 + 19*x^12 + 19*x^11 + 24*x^10 - 17*x^9 + 34*x^8 + 40*x^7 - 228*x^6 + 76*x^5 + 4*x^4 + 28*x^3 + 45*x^2 + 49*x + 1)
 

\( x^{23} + 4 x^{21} - 14 x^{20} + 25 x^{19} - 6 x^{18} - 14 x^{17} - 26 x^{16} + 31 x^{15} + 48 x^{14} + \cdots + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $23$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 11]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-1523249425950439235855871175083439\) \(\medspace = -\,1039^{11}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(27.72\)
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:  $1039^{1/2}\approx 32.2335229225724$
Ramified primes:   \(1039\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-1039}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{6}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{7}$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{8}-\frac{2}{9}$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{9}-\frac{2}{9}a$, $\frac{1}{9}a^{18}+\frac{1}{9}a^{10}-\frac{2}{9}a^{2}$, $\frac{1}{27}a^{19}+\frac{1}{27}a^{17}+\frac{1}{27}a^{16}-\frac{1}{9}a^{15}+\frac{1}{9}a^{14}-\frac{1}{9}a^{12}-\frac{2}{27}a^{11}-\frac{2}{27}a^{9}+\frac{1}{27}a^{8}+\frac{4}{9}a^{7}-\frac{4}{9}a^{6}+\frac{1}{3}a^{5}-\frac{2}{9}a^{4}+\frac{10}{27}a^{3}-\frac{1}{3}a^{2}+\frac{10}{27}a-\frac{11}{27}$, $\frac{1}{1647}a^{20}-\frac{1}{549}a^{19}+\frac{85}{1647}a^{18}-\frac{59}{1647}a^{17}+\frac{14}{549}a^{16}-\frac{65}{549}a^{15}-\frac{7}{61}a^{14}+\frac{41}{549}a^{13}-\frac{209}{1647}a^{12}+\frac{68}{549}a^{11}-\frac{107}{1647}a^{10}-\frac{257}{1647}a^{9}-\frac{14}{549}a^{8}-\frac{145}{549}a^{7}-\frac{83}{183}a^{6}-\frac{233}{549}a^{5}+\frac{28}{1647}a^{4}+\frac{83}{549}a^{3}+\frac{31}{1647}a^{2}-\frac{611}{1647}a+\frac{34}{183}$, $\frac{1}{1684881}a^{21}-\frac{175}{1684881}a^{20}-\frac{4151}{561627}a^{19}+\frac{719}{561627}a^{18}-\frac{36719}{1684881}a^{17}+\frac{56021}{1684881}a^{16}-\frac{16099}{187209}a^{15}+\frac{57115}{561627}a^{14}+\frac{205921}{1684881}a^{13}+\frac{124724}{1684881}a^{12}+\frac{22225}{561627}a^{11}+\frac{13586}{187209}a^{10}+\frac{61120}{1684881}a^{9}+\frac{129521}{1684881}a^{8}-\frac{71395}{187209}a^{7}+\frac{91517}{561627}a^{6}-\frac{308513}{1684881}a^{5}-\frac{731626}{1684881}a^{4}+\frac{161272}{561627}a^{3}-\frac{250190}{561627}a^{2}-\frac{388946}{1684881}a-\frac{435895}{1684881}$, $\frac{1}{27918564098931}a^{22}+\frac{784448}{9306188032977}a^{21}-\frac{2349680683}{27918564098931}a^{20}-\frac{23447370737}{3102062677659}a^{19}+\frac{1370586264214}{27918564098931}a^{18}+\frac{331475931283}{9306188032977}a^{17}-\frac{172682317417}{27918564098931}a^{16}-\frac{828516859133}{9306188032977}a^{15}-\frac{3899339361131}{27918564098931}a^{14}-\frac{699736799495}{9306188032977}a^{13}+\frac{5709962494}{34767825777}a^{12}+\frac{156855724351}{9306188032977}a^{11}+\frac{1688941533211}{27918564098931}a^{10}+\frac{75603573679}{3102062677659}a^{9}+\frac{3776769126722}{27918564098931}a^{8}-\frac{661738837027}{9306188032977}a^{7}-\frac{9968983470590}{27918564098931}a^{6}+\frac{1022530562671}{3102062677659}a^{5}+\frac{5743185482693}{27918564098931}a^{4}+\frac{476765817470}{9306188032977}a^{3}+\frac{6400961280025}{27918564098931}a^{2}+\frac{95511075958}{846017093907}a+\frac{3517882628618}{27918564098931}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$

Class group and class number

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

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $\frac{293377085095}{27918564098931}a^{22}-\frac{364133333233}{27918564098931}a^{21}+\frac{5910013252}{282005697969}a^{20}-\frac{176278104253}{846017093907}a^{19}+\frac{884147400773}{2538051281721}a^{18}-\frac{4069011187765}{27918564098931}a^{17}-\frac{1645476101926}{3102062677659}a^{16}-\frac{770103242009}{9306188032977}a^{15}+\frac{24182084610436}{27918564098931}a^{14}+\frac{16515267325412}{27918564098931}a^{13}-\frac{68867213809}{42494009283}a^{12}+\frac{1824885869690}{9306188032977}a^{11}+\frac{27681350169958}{27918564098931}a^{10}-\frac{1072236664289}{2538051281721}a^{9}-\frac{4046614697341}{3102062677659}a^{8}+\frac{1619631878603}{9306188032977}a^{7}+\frac{8679033177004}{27918564098931}a^{6}-\frac{110762712906313}{27918564098931}a^{5}+\frac{582613842115}{344673630851}a^{4}+\frac{9400559743862}{3102062677659}a^{3}+\frac{5892399823000}{27918564098931}a^{2}+\frac{40977770348732}{27918564098931}a+\frac{895056629564}{1034020892553}$, $\frac{44019546343}{27918564098931}a^{22}-\frac{11477141821}{3102062677659}a^{21}+\frac{182578888853}{27918564098931}a^{20}-\frac{325945072676}{9306188032977}a^{19}+\frac{2480032789672}{27918564098931}a^{18}-\frac{1031091422807}{9306188032977}a^{17}-\frac{1176789443851}{27918564098931}a^{16}+\frac{483282737038}{9306188032977}a^{15}+\frac{5198118651496}{27918564098931}a^{14}-\frac{74894675212}{300199613967}a^{13}-\frac{180810114880}{382446083547}a^{12}+\frac{7099108794077}{9306188032977}a^{11}+\frac{7761892006186}{27918564098931}a^{10}-\frac{1893830832505}{3102062677659}a^{9}-\frac{16887841384234}{27918564098931}a^{8}+\frac{5860816923905}{9306188032977}a^{7}+\frac{3917999600947}{27918564098931}a^{6}-\frac{774081795448}{846017093907}a^{5}+\frac{11070956824745}{27918564098931}a^{4}-\frac{2834132792647}{3102062677659}a^{3}-\frac{13293676709597}{27918564098931}a^{2}+\frac{5773387480556}{9306188032977}a+\frac{17346357793679}{27918564098931}$, $\frac{473690774699}{27918564098931}a^{22}+\frac{18743478977}{3102062677659}a^{21}+\frac{1696936380124}{27918564098931}a^{20}-\frac{2091376841252}{9306188032977}a^{19}+\frac{8208410773592}{27918564098931}a^{18}+\frac{809636671432}{9306188032977}a^{17}-\frac{10617228874430}{27918564098931}a^{16}-\frac{5710372582960}{9306188032977}a^{15}+\frac{6176284381067}{27918564098931}a^{14}+\frac{1007507844059}{846017093907}a^{13}-\frac{255949971230}{382446083547}a^{12}-\frac{971707529608}{3102062677659}a^{11}+\frac{5024160964169}{27918564098931}a^{10}+\frac{1372512795722}{9306188032977}a^{9}-\frac{2641207327628}{27918564098931}a^{8}-\frac{1513087660061}{9306188032977}a^{7}+\frac{5855882143559}{27918564098931}a^{6}-\frac{44237642417215}{9306188032977}a^{5}-\frac{3331205839018}{2538051281721}a^{4}+\frac{3458344823891}{9306188032977}a^{3}+\frac{6053116949153}{27918564098931}a^{2}+\frac{561366094843}{344673630851}a+\frac{28363542363157}{27918564098931}$, $\frac{1687004474}{27918564098931}a^{22}-\frac{5950435289}{457681378671}a^{21}+\frac{7211194067}{846017093907}a^{20}-\frac{21001925}{447391377}a^{19}+\frac{550207336813}{2538051281721}a^{18}-\frac{11762739199148}{27918564098931}a^{17}+\frac{602755179395}{3102062677659}a^{16}+\frac{2409866554421}{9306188032977}a^{15}+\frac{5895772327301}{27918564098931}a^{14}-\frac{19363576944551}{27918564098931}a^{13}-\frac{82389992057}{127482027849}a^{12}+\frac{13293520930126}{9306188032977}a^{11}-\frac{5834801002864}{27918564098931}a^{10}-\frac{621556252606}{2538051281721}a^{9}-\frac{971425175042}{3102062677659}a^{8}+\frac{4758085226998}{9306188032977}a^{7}+\frac{27396931592}{900598841901}a^{6}-\frac{8460153164255}{27918564098931}a^{5}+\frac{25691344493851}{9306188032977}a^{4}-\frac{9533695317793}{3102062677659}a^{3}-\frac{37459295284492}{27918564098931}a^{2}-\frac{14362658623847}{27918564098931}a-\frac{1559229915013}{3102062677659}$, $\frac{105846277865}{3102062677659}a^{22}-\frac{475475902786}{27918564098931}a^{21}+\frac{3895578708118}{27918564098931}a^{20}-\frac{164637278227}{300199613967}a^{19}+\frac{10216119817280}{9306188032977}a^{18}-\frac{19089749878693}{27918564098931}a^{17}-\frac{737080288729}{2538051281721}a^{16}-\frac{33839374981}{50853486519}a^{15}+\frac{1107574833940}{846017093907}a^{14}+\frac{30400791858683}{27918564098931}a^{13}-\frac{992862431810}{382446083547}a^{12}+\frac{1364342995345}{846017093907}a^{11}-\frac{263212384805}{9306188032977}a^{10}+\frac{19826916165488}{27918564098931}a^{9}-\frac{15366336129497}{27918564098931}a^{8}+\frac{1376270624530}{1034020892553}a^{7}-\frac{26521650371}{300199613967}a^{6}-\frac{222051251430394}{27918564098931}a^{5}+\frac{175216576742506}{27918564098931}a^{4}-\frac{2273428829663}{846017093907}a^{3}+\frac{2274211313609}{1034020892553}a^{2}+\frac{35301528967334}{27918564098931}a+\frac{22302249445744}{27918564098931}$, $\frac{40507386518}{27918564098931}a^{22}+\frac{324158344799}{27918564098931}a^{21}-\frac{154495062025}{27918564098931}a^{20}+\frac{61700004874}{3102062677659}a^{19}-\frac{4902546962320}{27918564098931}a^{18}+\frac{11543926469078}{27918564098931}a^{17}-\frac{8195540256103}{27918564098931}a^{16}-\frac{2186807883418}{9306188032977}a^{15}-\frac{93462183154}{900598841901}a^{14}+\frac{18910124303219}{27918564098931}a^{13}+\frac{172251754244}{382446083547}a^{12}-\frac{11642106646231}{9306188032977}a^{11}+\frac{13759015050860}{27918564098931}a^{10}-\frac{4418769286201}{27918564098931}a^{9}+\frac{1037901095201}{27918564098931}a^{8}+\frac{1997517331924}{9306188032977}a^{7}+\frac{22290207072050}{27918564098931}a^{6}-\frac{349430616011}{2538051281721}a^{5}-\frac{105975995441599}{27918564098931}a^{4}+\frac{24873725424736}{9306188032977}a^{3}-\frac{10927631019067}{27918564098931}a^{2}+\frac{3598173506657}{27918564098931}a+\frac{30125781048458}{27918564098931}$, $\frac{50114129}{11589275259}a^{22}-\frac{225005978}{127482027849}a^{21}+\frac{2913163265}{127482027849}a^{20}-\frac{8545777835}{127482027849}a^{19}+\frac{19921351319}{127482027849}a^{18}-\frac{6101659480}{42494009283}a^{17}+\frac{10497802610}{127482027849}a^{16}-\frac{5433105806}{42494009283}a^{15}+\frac{8790521419}{127482027849}a^{14}+\frac{3269043955}{127482027849}a^{13}-\frac{29142274672}{127482027849}a^{12}+\frac{41824881454}{127482027849}a^{11}-\frac{38742564148}{127482027849}a^{10}+\frac{2655568496}{14164669761}a^{9}+\frac{52850776700}{127482027849}a^{8}+\frac{20368427735}{42494009283}a^{7}-\frac{11408859317}{127482027849}a^{6}-\frac{171597375935}{127482027849}a^{5}-\frac{7645184398}{127482027849}a^{4}-\frac{194364921869}{127482027849}a^{3}+\frac{71309156120}{127482027849}a^{2}+\frac{45495363772}{42494009283}a+\frac{80331419588}{127482027849}$, $\frac{8435549215}{382446083547}a^{22}-\frac{4598980492}{382446083547}a^{21}+\frac{11633922617}{127482027849}a^{20}-\frac{44422197070}{127482027849}a^{19}+\frac{278464897210}{382446083547}a^{18}-\frac{173611019800}{382446083547}a^{17}-\frac{12651183334}{42494009283}a^{16}-\frac{2338394533}{11589275259}a^{15}+\frac{193750957}{202245417}a^{14}+\frac{100001888543}{382446083547}a^{13}-\frac{277980512941}{127482027849}a^{12}+\frac{87635594449}{42494009283}a^{11}+\frac{21170301671}{34767825777}a^{10}-\frac{365250650305}{382446083547}a^{9}-\frac{58605261632}{42494009283}a^{8}+\frac{28507940974}{11589275259}a^{7}+\frac{546855783118}{382446083547}a^{6}-\frac{2760843618133}{382446083547}a^{5}+\frac{410631313139}{127482027849}a^{4}-\frac{54112638908}{127482027849}a^{3}+\frac{134678165}{34767825777}a^{2}+\frac{135811215437}{382446083547}a+\frac{40219910512}{42494009283}$, $\frac{7239087247}{382446083547}a^{22}+\frac{13917425}{12336970437}a^{21}+\frac{10129654637}{127482027849}a^{20}-\frac{1222066917}{4721556587}a^{19}+\frac{181407260032}{382446083547}a^{18}-\frac{48599676397}{382446083547}a^{17}-\frac{24264685919}{127482027849}a^{16}-\frac{64340758760}{127482027849}a^{15}+\frac{191619753124}{382446083547}a^{14}+\frac{319387460318}{382446083547}a^{13}-\frac{118283100418}{127482027849}a^{12}+\frac{20635535230}{127482027849}a^{11}+\frac{171716692957}{382446083547}a^{10}+\frac{20670815003}{382446083547}a^{9}+\frac{103433682937}{127482027849}a^{8}-\frac{40155599239}{127482027849}a^{7}+\frac{628348048498}{382446083547}a^{6}-\frac{1951193662543}{382446083547}a^{5}+\frac{360626391428}{127482027849}a^{4}-\frac{221972742820}{127482027849}a^{3}+\frac{420028577482}{382446083547}a^{2}+\frac{28717233332}{382446083547}a+\frac{98868545002}{127482027849}$, $\frac{125327129}{17096487507}a^{22}-\frac{255437993}{17096487507}a^{21}+\frac{149066161}{5698829169}a^{20}-\frac{956583223}{5698829169}a^{19}+\frac{6428641610}{17096487507}a^{18}-\frac{6891242915}{17096487507}a^{17}-\frac{231980483}{5698829169}a^{16}-\frac{493147009}{5698829169}a^{15}+\frac{11374643462}{17096487507}a^{14}-\frac{380744891}{17096487507}a^{13}-\frac{93474512}{78066153}a^{12}+\frac{5054555788}{5698829169}a^{11}-\frac{1428894295}{17096487507}a^{10}+\frac{4525855039}{17096487507}a^{9}-\frac{448817458}{518075379}a^{8}+\frac{1134640147}{5698829169}a^{7}-\frac{5465262943}{17096487507}a^{6}-\frac{35410692770}{17096487507}a^{5}+\frac{20393367271}{5698829169}a^{4}-\frac{2806096019}{1899609723}a^{3}+\frac{6493359254}{17096487507}a^{2}-\frac{5825789063}{17096487507}a+\frac{30193717}{183833199}$, $a$ Copy content Toggle raw display (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:  \( 55921894.4803 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 55921894.4803 \cdot 1}{2\cdot\sqrt{1523249425950439235855871175083439}}\cr\approx \mathstrut & 0.863326190258 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^23 + 4*x^21 - 14*x^20 + 25*x^19 - 6*x^18 - 14*x^17 - 26*x^16 + 31*x^15 + 48*x^14 - 62*x^13 + 19*x^12 + 19*x^11 + 24*x^10 - 17*x^9 + 34*x^8 + 40*x^7 - 228*x^6 + 76*x^5 + 4*x^4 + 28*x^3 + 45*x^2 + 49*x + 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^23 + 4*x^21 - 14*x^20 + 25*x^19 - 6*x^18 - 14*x^17 - 26*x^16 + 31*x^15 + 48*x^14 - 62*x^13 + 19*x^12 + 19*x^11 + 24*x^10 - 17*x^9 + 34*x^8 + 40*x^7 - 228*x^6 + 76*x^5 + 4*x^4 + 28*x^3 + 45*x^2 + 49*x + 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^23 + 4*x^21 - 14*x^20 + 25*x^19 - 6*x^18 - 14*x^17 - 26*x^16 + 31*x^15 + 48*x^14 - 62*x^13 + 19*x^12 + 19*x^11 + 24*x^10 - 17*x^9 + 34*x^8 + 40*x^7 - 228*x^6 + 76*x^5 + 4*x^4 + 28*x^3 + 45*x^2 + 49*x + 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 + 4*x^21 - 14*x^20 + 25*x^19 - 6*x^18 - 14*x^17 - 26*x^16 + 31*x^15 + 48*x^14 - 62*x^13 + 19*x^12 + 19*x^11 + 24*x^10 - 17*x^9 + 34*x^8 + 40*x^7 - 228*x^6 + 76*x^5 + 4*x^4 + 28*x^3 + 45*x^2 + 49*x + 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$D_{23}$ (as 23T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 46
The 13 conjugacy class representatives for $D_{23}$
Character table for $D_{23}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Galois closure: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $23$ ${\href{/padicField/3.2.0.1}{2} }^{11}{,}\,{\href{/padicField/3.1.0.1}{1} }$ $23$ $23$ ${\href{/padicField/11.2.0.1}{2} }^{11}{,}\,{\href{/padicField/11.1.0.1}{1} }$ $23$ $23$ $23$ ${\href{/padicField/23.2.0.1}{2} }^{11}{,}\,{\href{/padicField/23.1.0.1}{1} }$ $23$ ${\href{/padicField/31.2.0.1}{2} }^{11}{,}\,{\href{/padicField/31.1.0.1}{1} }$ $23$ ${\href{/padicField/41.2.0.1}{2} }^{11}{,}\,{\href{/padicField/41.1.0.1}{1} }$ $23$ $23$ ${\href{/padicField/53.2.0.1}{2} }^{11}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.2.0.1}{2} }^{11}{,}\,{\href{/padicField/59.1.0.1}{1} }$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(1039\) Copy content Toggle raw display $\Q_{1039}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$