LMFDB - Number field 24.0.72340856237421875367936000000000000.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 15*x^22 + 158*x^20 - 789*x^18 + 2798*x^16 - 5124*x^14 + 6639*x^12 - 5271*x^10 + 3030*x^8 - 1062*x^6 + 253*x^4 - 18*x^2 + 1)

gp: K = bnfinit(y^24 - 15*y^22 + 158*y^20 - 789*y^18 + 2798*y^16 - 5124*y^14 + 6639*y^12 - 5271*y^10 + 3030*y^8 - 1062*y^6 + 253*y^4 - 18*y^2 + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 15*x^22 + 158*x^20 - 789*x^18 + 2798*x^16 - 5124*x^14 + 6639*x^12 - 5271*x^10 + 3030*x^8 - 1062*x^6 + 253*x^4 - 18*x^2 + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 15*x^22 + 158*x^20 - 789*x^18 + 2798*x^16 - 5124*x^14 + 6639*x^12 - 5271*x^10 + 3030*x^8 - 1062*x^6 + 253*x^4 - 18*x^2 + 1)

$ x^{24} - 15 x^{22} + 158 x^{20} - 789 x^{18} + 2798 x^{16} - 5124 x^{14} + 6639 x^{12} - 5271 x^{10} + \cdots + 1 $ x^24 - 15*x^22 + 158*x^20 - 789*x^18 + 2798*x^16 - 5124*x^14 + 6639*x^12 - 5271*x^10 + 3030*x^8 - 1062*x^6 + 253*x^4 - 18*x^2 + 1

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$24$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 12]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$72340856237421875367936000000000000$ 72340856237421875367936000000000000 $\medspace = 2^{24}\cdot 3^{12}\cdot 5^{12}\cdot 7^{16}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$28.34$	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:	$2\cdot 3^{1/2}5^{1/2}7^{2/3}\approx 28.344860147201352$
Ramified primes:	$2$, $3$, $5$, $7$ 2, 3, 5, 7	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$24$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$420=2^{2}\cdot 3\cdot 5\cdot 7$
Dirichlet character group:	$\lbrace$$\chi_{420}(1,·)$, $\chi_{420}(331,·)$, $\chi_{420}(389,·)$, $\chi_{420}(71,·)$, $\chi_{420}(11,·)$, $\chi_{420}(79,·)$, $\chi_{420}(401,·)$, $\chi_{420}(211,·)$, $\chi_{420}(149,·)$, $\chi_{420}(151,·)$, $\chi_{420}(281,·)$, $\chi_{420}(239,·)$, $\chi_{420}(29,·)$, $\chi_{420}(289,·)$, $\chi_{420}(359,·)$, $\chi_{420}(169,·)$, $\chi_{420}(109,·)$, $\chi_{420}(221,·)$, $\chi_{420}(319,·)$, $\chi_{420}(179,·)$, $\chi_{420}(361,·)$, $\chi_{420}(121,·)$, $\chi_{420}(379,·)$, $\chi_{420}(191,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{2048}$

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{8}a^{18}-\frac{1}{4}a^{12}+\frac{1}{8}a^{6}-\frac{3}{8}$, $\frac{1}{8}a^{19}-\frac{1}{4}a^{13}+\frac{1}{8}a^{7}-\frac{3}{8}a$, $\frac{1}{183352}a^{20}+\frac{6473}{183352}a^{18}-\frac{9019}{22919}a^{16}-\frac{39485}{91676}a^{14}+\frac{39683}{91676}a^{12}-\frac{5605}{22919}a^{10}+\frac{26393}{183352}a^{8}-\frac{64591}{183352}a^{6}-\frac{88}{22919}a^{4}-\frac{31307}{183352}a^{2}+\frac{39581}{183352}$, $\frac{1}{183352}a^{21}+\frac{6473}{183352}a^{19}-\frac{9019}{22919}a^{17}-\frac{39485}{91676}a^{15}+\frac{39683}{91676}a^{13}-\frac{5605}{22919}a^{11}+\frac{26393}{183352}a^{9}-\frac{64591}{183352}a^{7}-\frac{88}{22919}a^{5}-\frac{31307}{183352}a^{3}+\frac{39581}{183352}a$, $\frac{1}{924825837832}a^{22}-\frac{298179}{924825837832}a^{20}-\frac{18273521245}{924825837832}a^{18}-\frac{14633772693}{462412918916}a^{16}-\frac{143736184233}{462412918916}a^{14}+\frac{4144244749}{35570224532}a^{12}-\frac{98160812223}{924825837832}a^{10}-\frac{322101635347}{924825837832}a^{8}-\frac{105247252133}{924825837832}a^{6}+\frac{46938583917}{924825837832}a^{4}-\frac{29822725259}{71140449064}a^{2}+\frac{272511138511}{924825837832}$, $\frac{1}{924825837832}a^{23}-\frac{298179}{924825837832}a^{21}-\frac{18273521245}{924825837832}a^{19}-\frac{14633772693}{462412918916}a^{17}-\frac{143736184233}{462412918916}a^{15}+\frac{4144244749}{35570224532}a^{13}-\frac{98160812223}{924825837832}a^{11}-\frac{322101635347}{924825837832}a^{9}-\frac{105247252133}{924825837832}a^{7}+\frac{46938583917}{924825837832}a^{5}-\frac{29822725259}{71140449064}a^{3}+\frac{272511138511}{924825837832}a$ 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, a^17, 1/8*a^18 - 1/4*a^12 + 1/8*a^6 - 3/8, 1/8*a^19 - 1/4*a^13 + 1/8*a^7 - 3/8*a, 1/183352*a^20 + 6473/183352*a^18 - 9019/22919*a^16 - 39485/91676*a^14 + 39683/91676*a^12 - 5605/22919*a^10 + 26393/183352*a^8 - 64591/183352*a^6 - 88/22919*a^4 - 31307/183352*a^2 + 39581/183352, 1/183352*a^21 + 6473/183352*a^19 - 9019/22919*a^17 - 39485/91676*a^15 + 39683/91676*a^13 - 5605/22919*a^11 + 26393/183352*a^9 - 64591/183352*a^7 - 88/22919*a^5 - 31307/183352*a^3 + 39581/183352*a, 1/924825837832*a^22 - 298179/924825837832*a^20 - 18273521245/924825837832*a^18 - 14633772693/462412918916*a^16 - 143736184233/462412918916*a^14 + 4144244749/35570224532*a^12 - 98160812223/924825837832*a^10 - 322101635347/924825837832*a^8 - 105247252133/924825837832*a^6 + 46938583917/924825837832*a^4 - 29822725259/71140449064*a^2 + 272511138511/924825837832, 1/924825837832*a^23 - 298179/924825837832*a^21 - 18273521245/924825837832*a^19 - 14633772693/462412918916*a^17 - 143736184233/462412918916*a^15 + 4144244749/35570224532*a^13 - 98160812223/924825837832*a^11 - 322101635347/924825837832*a^9 - 105247252133/924825837832*a^7 + 46938583917/924825837832*a^5 - 29822725259/71140449064*a^3 + 272511138511/924825837832*a

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

$C_{3}$, which has order $3$ (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:	$ -\frac{148787121575}{924825837832} a^{23} + \frac{84389196267}{35570224532} a^{21} - \frac{22945059625745}{924825837832} a^{19} + \frac{55734503392911}{462412918916} a^{17} - \frac{96723380662449}{231206459458} a^{15} + \frac{329284836562241}{462412918916} a^{13} - \frac{61570749574763}{71140449064} a^{11} + \frac{272297101425195}{462412918916} a^{9} - \frac{264636377210105}{924825837832} a^{7} + \frac{54766886055237}{924825837832} a^{5} - \frac{1966812984841}{462412918916} a^{3} - \frac{3161652599205}{924825837832} a $ -(148787121575)/(924825837832)a^(23) + (84389196267)/(35570224532)a^(21) - (22945059625745)/(924825837832)a^(19) + (55734503392911)/(462412918916)a^(17) - (96723380662449)/(231206459458)a^(15) + (329284836562241)/(462412918916)a^(13) - (61570749574763)/(71140449064)a^(11) + (272297101425195)/(462412918916)a^(9) - (264636377210105)/(924825837832)a^(7) + (54766886055237)/(924825837832)a^(5) - (1966812984841)/(462412918916)a^(3) - (3161652599205)/(924825837832)a (order $12$)	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:	$\frac{12740624980}{115603229729}a^{22}-\frac{357550479545}{231206459458}a^{20}+\frac{7317736473999}{462412918916}a^{18}-\frac{8127693029793}{115603229729}a^{16}+\frac{2013933163227}{8892556133}a^{14}-\frac{64601338558191}{231206459458}a^{12}+\frac{27236266780013}{115603229729}a^{10}+\frac{59622353831}{5639181938}a^{8}-\frac{31737293946825}{462412918916}a^{6}+\frac{523791317034}{8892556133}a^{4}-\frac{2916253008573}{231206459458}a^{2}+\frac{427687797371}{462412918916}$, $\frac{9969168111}{71140449064}a^{22}-\frac{144516224125}{71140449064}a^{20}+\frac{1500116894577}{71140449064}a^{18}-\frac{3539247866163}{35570224532}a^{16}+\frac{12001554501441}{35570224532}a^{14}-\frac{18754110637413}{35570224532}a^{12}+\frac{42396917283759}{71140449064}a^{10}-\frac{594736755629}{1735132904}a^{8}+\frac{11036199422865}{71140449064}a^{6}-\frac{2435163891877}{71140449064}a^{4}+\frac{593549542647}{71140449064}a^{2}-\frac{40917319267}{71140449064}$, $\frac{10289446539}{115603229729}a^{22}-\frac{153568851902}{115603229729}a^{20}+\frac{123934994493}{8892556133}a^{18}-\frac{7953534294374}{115603229729}a^{16}+\frac{27738702720228}{115603229729}a^{14}-\frac{48521885753797}{115603229729}a^{12}+\frac{57541479317043}{115603229729}a^{10}-\frac{3012810957294}{8892556133}a^{8}+\frac{16460747389026}{115603229729}a^{6}-\frac{3940266945841}{115603229729}a^{4}+\frac{283014250002}{115603229729}a^{2}-\frac{223867630130}{115603229729}$, $\frac{221565343115}{924825837832}a^{23}-\frac{128789369491}{35570224532}a^{21}+\frac{35375147209613}{924825837832}a^{19}-\frac{89328668918939}{462412918916}a^{17}+\frac{159625910533289}{231206459458}a^{15}-\frac{599812712846541}{462412918916}a^{13}+\frac{121494010259215}{71140449064}a^{11}-\frac{15863692747031}{11278363876}a^{9}+\frac{761788482762693}{924825837832}a^{7}-\frac{280455634159137}{924825837832}a^{5}+\frac{32575214637677}{462412918916}a^{3}-\frac{4641376305263}{924825837832}a-1$, $\frac{6039235439}{21507577624}a^{22}-\frac{481486024351}{115603229729}a^{20}+\frac{20199105185113}{462412918916}a^{18}-\frac{99157830872533}{462412918916}a^{16}+\frac{6685706895855}{8892556133}a^{14}-\frac{305952308810909}{231206459458}a^{12}+\frac{15\!\cdots\!77}{924825837832}a^{10}-\frac{146265584743903}{115603229729}a^{8}+\frac{332043663211493}{462412918916}a^{6}-\frac{16649723263203}{71140449064}a^{4}+\frac{155312389429}{2819590969}a^{2}-\frac{477440242607}{462412918916}$, $\frac{93031693503}{924825837832}a^{22}-\frac{1379319015315}{924825837832}a^{20}+\frac{14454316696335}{924825837832}a^{18}-\frac{35412422416339}{462412918916}a^{16}+\frac{9513034836555}{35570224532}a^{14}-\frac{215615831514075}{462412918916}a^{12}+\frac{535686277034839}{924825837832}a^{10}-\frac{9652980040035}{22556727752}a^{8}+\frac{217466318621295}{924825837832}a^{6}-\frac{5803854655161}{71140449064}a^{4}+\frac{17053410106425}{924825837832}a^{2}-\frac{28113220695}{21507577624}$, $\frac{31204365473}{115603229729}a^{23}+\frac{163570440861}{924825837832}a^{22}-\frac{295501389701}{71140449064}a^{21}-\frac{584649418051}{231206459458}a^{20}+\frac{40923657105971}{924825837832}a^{19}+\frac{464698912227}{17785112266}a^{18}-\frac{26579760000835}{115603229729}a^{17}-\frac{1360641666237}{11278363876}a^{16}+\frac{389257927597393}{462412918916}a^{15}+\frac{46767967996527}{115603229729}a^{14}-\frac{782209939192343}{462412918916}a^{13}-\frac{69118738410596}{115603229729}a^{12}+\frac{21103690355455}{8892556133}a^{11}+\frac{626411592329253}{924825837832}a^{10}-\frac{19\!\cdots\!17}{924825837832}a^{9}-\frac{6548540636343}{17785112266}a^{8}+\frac{12\!\cdots\!79}{924825837832}a^{7}+\frac{46531036757115}{231206459458}a^{6}-\frac{1474528039927}{2819590969}a^{5}-\frac{47952344599631}{924825837832}a^{4}+\frac{115890145338923}{924825837832}a^{3}+\frac{4686133545897}{231206459458}a^{2}-\frac{6422928291529}{924825837832}a-\frac{465446728157}{231206459458}$, $\frac{71245923029}{231206459458}a^{23}+\frac{211914757755}{924825837832}a^{22}-\frac{84893489619}{17785112266}a^{21}-\frac{3107261728841}{924825837832}a^{20}+\frac{47070911759269}{924825837832}a^{19}+\frac{2491596850521}{71140449064}a^{18}-\frac{30765543257593}{115603229729}a^{17}-\frac{77824359437615}{462412918916}a^{16}+\frac{112388961139206}{115603229729}a^{15}+\frac{266975019399645}{462412918916}a^{14}-\frac{903666431649013}{462412918916}a^{13}-\frac{437890981301557}{462412918916}a^{12}+\frac{47130626800037}{17785112266}a^{11}+\frac{10\!\cdots\!11}{924825837832}a^{10}-\frac{532002952358039}{231206459458}a^{9}-\frac{48486694639141}{71140449064}a^{8}+\frac{12\!\cdots\!17}{924825837832}a^{7}+\frac{275156571609453}{924825837832}a^{6}-\frac{108487766286615}{231206459458}a^{5}-\frac{63179266161129}{924825837832}a^{4}+\frac{21852524696115}{231206459458}a^{3}+\frac{9980258054427}{924825837832}a^{2}-\frac{3456130860375}{924825837832}a-\frac{1398040353679}{924825837832}$, $\frac{76897871}{229144162}a^{23}+\frac{35044656896}{115603229729}a^{22}-\frac{42890471}{8813237}a^{21}-\frac{521505407000}{115603229729}a^{20}+\frac{11593239755}{229144162}a^{19}+\frac{841511140571}{17785112266}a^{18}-\frac{27444262420}{114572081}a^{17}-\frac{26924107963371}{115603229729}a^{16}+\frac{93880387062}{114572081}a^{15}+\frac{94066055491767}{115603229729}a^{14}-\frac{150386351373}{114572081}a^{13}-\frac{164679971027751}{115603229729}a^{12}+\frac{27910765877}{17626474}a^{11}+\frac{200566502256359}{115603229729}a^{10}-\frac{117913124156}{114572081}a^{9}-\frac{10895719599920}{8892556133}a^{8}+\frac{132049066505}{229144162}a^{7}+\frac{136543642653271}{231206459458}a^{6}-\frac{34930198995}{229144162}a^{5}-\frac{18192737052150}{115603229729}a^{4}+\frac{4679076214}{114572081}a^{3}+\frac{2849628939356}{115603229729}a^{2}+\frac{689511311}{229144162}a-\frac{7164754721}{231206459458}$, $\frac{25731524041}{231206459458}a^{23}-\frac{348409494455}{924825837832}a^{22}-\frac{1808356473919}{924825837832}a^{21}+\frac{5004893235041}{924825837832}a^{20}+\frac{10033841764289}{462412918916}a^{19}-\frac{6469293440946}{115603229729}a^{18}-\frac{15073855796150}{115603229729}a^{17}+\frac{120177162819059}{462412918916}a^{16}+\frac{235730421017251}{462412918916}a^{15}-\frac{402432908402193}{462412918916}a^{14}-\frac{285903747222957}{231206459458}a^{13}+\frac{149120197281759}{115603229729}a^{12}+\frac{401869593713347}{231206459458}a^{11}-\frac{12\!\cdots\!07}{924825837832}a^{10}-\frac{15\!\cdots\!19}{924825837832}a^{9}+\frac{14840192576497}{22556727752}a^{8}+\frac{406027093108109}{462412918916}a^{7}-\frac{27484470720921}{115603229729}a^{6}-\frac{74966614419261}{231206459458}a^{5}+\frac{3171041755853}{924825837832}a^{4}+\frac{36795323118509}{924825837832}a^{3}-\frac{4069323014259}{924825837832}a^{2}-\frac{181953355999}{35570224532}a-\frac{86556838408}{115603229729}$, $\frac{315392784371}{924825837832}a^{23}-\frac{25358136913}{924825837832}a^{22}-\frac{4724267300833}{924825837832}a^{21}+\frac{378213096051}{924825837832}a^{20}+\frac{49715969514597}{924825837832}a^{19}-\frac{1982957348071}{462412918916}a^{18}-\frac{123783373246047}{462412918916}a^{17}+\frac{9778569530185}{462412918916}a^{16}+\frac{437438660889413}{462412918916}a^{15}-\frac{34077498250075}{462412918916}a^{14}-\frac{793603314636473}{462412918916}a^{13}+\frac{29896416089841}{231206459458}a^{12}+\frac{20\!\cdots\!95}{924825837832}a^{11}-\frac{144474548539289}{924825837832}a^{10}-\frac{15\!\cdots\!05}{924825837832}a^{9}+\frac{2719671411555}{22556727752}a^{8}+\frac{874232001304853}{924825837832}a^{7}-\frac{717239867673}{10753788812}a^{6}-\frac{295725372941641}{924825837832}a^{5}+\frac{30284040782947}{924825837832}a^{4}+\frac{56613171917515}{924825837832}a^{3}-\frac{4884573718633}{924825837832}a^{2}-\frac{45647294779}{71140449064}a-\frac{289184898755}{462412918916}$ 12740624980/115603229729a^22 - 357550479545/231206459458a^20 + 7317736473999/462412918916a^18 - 8127693029793/115603229729a^16 + 2013933163227/8892556133a^14 - 64601338558191/231206459458a^12 + 27236266780013/115603229729a^10 + 59622353831/5639181938a^8 - 31737293946825/462412918916a^6 + 523791317034/8892556133a^4 - 2916253008573/231206459458a^2 + 427687797371/462412918916, 9969168111/71140449064a^22 - 144516224125/71140449064a^20 + 1500116894577/71140449064a^18 - 3539247866163/35570224532a^16 + 12001554501441/35570224532a^14 - 18754110637413/35570224532a^12 + 42396917283759/71140449064a^10 - 594736755629/1735132904a^8 + 11036199422865/71140449064a^6 - 2435163891877/71140449064a^4 + 593549542647/71140449064a^2 - 40917319267/71140449064, 10289446539/115603229729a^22 - 153568851902/115603229729a^20 + 123934994493/8892556133a^18 - 7953534294374/115603229729a^16 + 27738702720228/115603229729a^14 - 48521885753797/115603229729a^12 + 57541479317043/115603229729a^10 - 3012810957294/8892556133a^8 + 16460747389026/115603229729a^6 - 3940266945841/115603229729a^4 + 283014250002/115603229729a^2 - 223867630130/115603229729, 221565343115/924825837832a^23 - 128789369491/35570224532a^21 + 35375147209613/924825837832a^19 - 89328668918939/462412918916a^17 + 159625910533289/231206459458a^15 - 599812712846541/462412918916a^13 + 121494010259215/71140449064a^11 - 15863692747031/11278363876a^9 + 761788482762693/924825837832a^7 - 280455634159137/924825837832a^5 + 32575214637677/462412918916a^3 - 4641376305263/924825837832a - 1, 6039235439/21507577624a^22 - 481486024351/115603229729a^20 + 20199105185113/462412918916a^18 - 99157830872533/462412918916a^16 + 6685706895855/8892556133a^14 - 305952308810909/231206459458a^12 + 1552721641603677/924825837832a^10 - 146265584743903/115603229729a^8 + 332043663211493/462412918916a^6 - 16649723263203/71140449064a^4 + 155312389429/2819590969a^2 - 477440242607/462412918916, 93031693503/924825837832a^22 - 1379319015315/924825837832a^20 + 14454316696335/924825837832a^18 - 35412422416339/462412918916a^16 + 9513034836555/35570224532a^14 - 215615831514075/462412918916a^12 + 535686277034839/924825837832a^10 - 9652980040035/22556727752a^8 + 217466318621295/924825837832a^6 - 5803854655161/71140449064a^4 + 17053410106425/924825837832a^2 - 28113220695/21507577624, 31204365473/115603229729a^23 + 163570440861/924825837832a^22 - 295501389701/71140449064a^21 - 584649418051/231206459458a^20 + 40923657105971/924825837832a^19 + 464698912227/17785112266a^18 - 26579760000835/115603229729a^17 - 1360641666237/11278363876a^16 + 389257927597393/462412918916a^15 + 46767967996527/115603229729a^14 - 782209939192343/462412918916a^13 - 69118738410596/115603229729a^12 + 21103690355455/8892556133a^11 + 626411592329253/924825837832a^10 - 1980355937989217/924825837832a^9 - 6548540636343/17785112266a^8 + 1254351869626579/924825837832a^7 + 46531036757115/231206459458a^6 - 1474528039927/2819590969a^5 - 47952344599631/924825837832a^4 + 115890145338923/924825837832a^3 + 4686133545897/231206459458a^2 - 6422928291529/924825837832a - 465446728157/231206459458, 71245923029/231206459458a^23 + 211914757755/924825837832a^22 - 84893489619/17785112266a^21 - 3107261728841/924825837832a^20 + 47070911759269/924825837832a^19 + 2491596850521/71140449064a^18 - 30765543257593/115603229729a^17 - 77824359437615/462412918916a^16 + 112388961139206/115603229729a^15 + 266975019399645/462412918916a^14 - 903666431649013/462412918916a^13 - 437890981301557/462412918916a^12 + 47130626800037/17785112266a^11 + 1011491759225211/924825837832a^10 - 532002952358039/231206459458a^9 - 48486694639141/71140449064a^8 + 1223286042741917/924825837832a^7 + 275156571609453/924825837832a^6 - 108487766286615/231206459458a^5 - 63179266161129/924825837832a^4 + 21852524696115/231206459458a^3 + 9980258054427/924825837832a^2 - 3456130860375/924825837832a - 1398040353679/924825837832, 76897871/229144162a^23 + 35044656896/115603229729a^22 - 42890471/8813237a^21 - 521505407000/115603229729a^20 + 11593239755/229144162a^19 + 841511140571/17785112266a^18 - 27444262420/114572081a^17 - 26924107963371/115603229729a^16 + 93880387062/114572081a^15 + 94066055491767/115603229729a^14 - 150386351373/114572081a^13 - 164679971027751/115603229729a^12 + 27910765877/17626474a^11 + 200566502256359/115603229729a^10 - 117913124156/114572081a^9 - 10895719599920/8892556133a^8 + 132049066505/229144162a^7 + 136543642653271/231206459458a^6 - 34930198995/229144162a^5 - 18192737052150/115603229729a^4 + 4679076214/114572081a^3 + 2849628939356/115603229729a^2 + 689511311/229144162a - 7164754721/231206459458, 25731524041/231206459458a^23 - 348409494455/924825837832a^22 - 1808356473919/924825837832a^21 + 5004893235041/924825837832a^20 + 10033841764289/462412918916a^19 - 6469293440946/115603229729a^18 - 15073855796150/115603229729a^17 + 120177162819059/462412918916a^16 + 235730421017251/462412918916a^15 - 402432908402193/462412918916a^14 - 285903747222957/231206459458a^13 + 149120197281759/115603229729a^12 + 401869593713347/231206459458a^11 - 1282446268352607/924825837832a^10 - 1550358064349319/924825837832a^9 + 14840192576497/22556727752a^8 + 406027093108109/462412918916a^7 - 27484470720921/115603229729a^6 - 74966614419261/231206459458a^5 + 3171041755853/924825837832a^4 + 36795323118509/924825837832a^3 - 4069323014259/924825837832a^2 - 181953355999/35570224532a - 86556838408/115603229729, 315392784371/924825837832a^23 - 25358136913/924825837832a^22 - 4724267300833/924825837832a^21 + 378213096051/924825837832a^20 + 49715969514597/924825837832a^19 - 1982957348071/462412918916a^18 - 123783373246047/462412918916a^17 + 9778569530185/462412918916a^16 + 437438660889413/462412918916a^15 - 34077498250075/462412918916a^14 - 793603314636473/462412918916a^13 + 29896416089841/231206459458a^12 + 2026393864611395/924825837832a^11 - 144474548539289/924825837832a^10 - 1576914774175105/924825837832a^9 + 2719671411555/22556727752a^8 + 874232001304853/924825837832a^7 - 717239867673/10753788812a^6 - 295725372941641/924825837832a^5 + 30284040782947/924825837832a^4 + 56613171917515/924825837832a^3 - 4884573718633/924825837832a^2 - 45647294779/71140449064a - 289184898755/462412918916 (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:	$ 39799331.07802784 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 39799331.07802784 \cdot 3}{12\cdot\sqrt{72340856237421875367936000000000000}}\cr\approx \mathstrut & 0.140049763360777 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^24 - 15*x^22 + 158*x^20 - 789*x^18 + 2798*x^16 - 5124*x^14 + 6639*x^12 - 5271*x^10 + 3030*x^8 - 1062*x^6 + 253*x^4 - 18*x^2 + 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^24 - 15*x^22 + 158*x^20 - 789*x^18 + 2798*x^16 - 5124*x^14 + 6639*x^12 - 5271*x^10 + 3030*x^8 - 1062*x^6 + 253*x^4 - 18*x^2 + 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^24 - 15*x^22 + 158*x^20 - 789*x^18 + 2798*x^16 - 5124*x^14 + 6639*x^12 - 5271*x^10 + 3030*x^8 - 1062*x^6 + 253*x^4 - 18*x^2 + 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^24 - 15*x^22 + 158*x^20 - 789*x^18 + 2798*x^16 - 5124*x^14 + 6639*x^12 - 5271*x^10 + 3030*x^8 - 1062*x^6 + 253*x^4 - 18*x^2 + 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

$C_2^2\times C_6$ (as 24T3):

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)

An abelian group of order 24

The 24 conjugacy class representatives for $C_2^2\times C_6$

Character table for $C_2^2\times C_6$

Intermediate fields

$\Q(\sqrt{-1}) $, $\Q(\sqrt{3}) $, $\Q(\sqrt{-3}) $, $\Q(\sqrt{-5}) $, $\Q(\sqrt{5}) $, $\Q(\sqrt{-15}) $, $\Q(\sqrt{15}) $, $\Q(\zeta_{7})^+$, $\Q(\zeta_{12})$, $\Q(i, \sqrt{5})$, $\Q(i, \sqrt{15})$, $\Q(\sqrt{3}, \sqrt{-5})$, $\Q(\sqrt{3}, \sqrt{5})$, $\Q(\sqrt{-3}, \sqrt{-5})$, $\Q(\sqrt{-3}, \sqrt{5})$, 6.0.153664.1, 6.6.4148928.1, 6.0.64827.1, 6.0.19208000.1, 6.6.300125.1, 6.0.8103375.1, 6.6.518616000.1, 8.0.12960000.1, 12.0.17213603549184.1, 12.0.368947264000000.1, 12.0.268962555456000000.3, 12.0.268962555456000000.1, 12.12.268962555456000000.1, 12.0.268962555456000000.2, 12.0.65664686390625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

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]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	R	R	${\href{/padicField/11.6.0.1}{6} }^{4}$	${\href{/padicField/13.2.0.1}{2} }^{12}$	${\href{/padicField/17.6.0.1}{6} }^{4}$	${\href{/padicField/19.6.0.1}{6} }^{4}$	${\href{/padicField/23.6.0.1}{6} }^{4}$	${\href{/padicField/29.2.0.1}{2} }^{12}$	${\href{/padicField/31.6.0.1}{6} }^{4}$	${\href{/padicField/37.6.0.1}{6} }^{4}$	${\href{/padicField/41.2.0.1}{2} }^{12}$	${\href{/padicField/43.2.0.1}{2} }^{12}$	${\href{/padicField/47.6.0.1}{6} }^{4}$	${\href{/padicField/53.6.0.1}{6} }^{4}$	${\href{/padicField/59.6.0.1}{6} }^{4}$

In the table, R denotes a ramified prime. 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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.12.12.26	$x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$	$2$	$6$	$12$	$C_6\times C_2$	$[2]^{6}$
$2$ 2	2.12.12.26	$x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$	$2$	$6$	$12$	$C_6\times C_2$	$[2]^{6}$
$3$ 3	3.12.6.2	$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
$3$ 3	3.12.6.2	$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
$5$ 5	5.12.6.1	$x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
$5$ 5	5.12.6.1	$x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
$7$ 7	7.6.4.3	$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$	$3$	$2$	$4$	$C_6$	$[\ ]_{3}^{2}$
	7.6.4.3	$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$	$3$	$2$	$4$	$C_6$	$[\ ]_{3}^{2}$
	7.6.4.3	$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$	$3$	$2$	$4$	$C_6$	$[\ ]_{3}^{2}$
	7.6.4.3	$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$	$3$	$2$	$4$	$C_6$	$[\ ]_{3}^{2}$

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