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Properties

Label 24.0.575...000.1
Degree $24$
Signature $[0, 12]$
Discriminant $5.758\times 10^{39}$
Root discriminant \(45.36\)
Ramified primes $2,3,5,71$
Class number $44$ (GRH)
Class group [2, 22] (GRH)
Galois group $C_2^2\times D_6$ (as 24T30)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^23 + 5*x^22 - 2*x^21 - 10*x^20 - 14*x^19 - 49*x^18 - 46*x^17 - 229*x^16 - 1464*x^15 + 4190*x^14 - 6920*x^13 + 11088*x^12 + 13840*x^11 + 16760*x^10 + 11712*x^9 - 3664*x^8 + 1472*x^7 - 3136*x^6 + 1792*x^5 - 2560*x^4 + 1024*x^3 + 5120*x^2 + 4096*x + 4096)
 
gp: K = bnfinit(y^24 - 2*y^23 + 5*y^22 - 2*y^21 - 10*y^20 - 14*y^19 - 49*y^18 - 46*y^17 - 229*y^16 - 1464*y^15 + 4190*y^14 - 6920*y^13 + 11088*y^12 + 13840*y^11 + 16760*y^10 + 11712*y^9 - 3664*y^8 + 1472*y^7 - 3136*y^6 + 1792*y^5 - 2560*y^4 + 1024*y^3 + 5120*y^2 + 4096*y + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 2*x^23 + 5*x^22 - 2*x^21 - 10*x^20 - 14*x^19 - 49*x^18 - 46*x^17 - 229*x^16 - 1464*x^15 + 4190*x^14 - 6920*x^13 + 11088*x^12 + 13840*x^11 + 16760*x^10 + 11712*x^9 - 3664*x^8 + 1472*x^7 - 3136*x^6 + 1792*x^5 - 2560*x^4 + 1024*x^3 + 5120*x^2 + 4096*x + 4096);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 2*x^23 + 5*x^22 - 2*x^21 - 10*x^20 - 14*x^19 - 49*x^18 - 46*x^17 - 229*x^16 - 1464*x^15 + 4190*x^14 - 6920*x^13 + 11088*x^12 + 13840*x^11 + 16760*x^10 + 11712*x^9 - 3664*x^8 + 1472*x^7 - 3136*x^6 + 1792*x^5 - 2560*x^4 + 1024*x^3 + 5120*x^2 + 4096*x + 4096)
 

\( x^{24} - 2 x^{23} + 5 x^{22} - 2 x^{21} - 10 x^{20} - 14 x^{19} - 49 x^{18} - 46 x^{17} - 229 x^{16} + \cdots + 4096 \) Copy content Toggle raw display

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:   \(5757603349403224554199842816000000000000\) \(\medspace = 2^{36}\cdot 3^{12}\cdot 5^{12}\cdot 71^{8}\) 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:  \(45.36\)
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:  $2^{3/2}3^{1/2}5^{1/2}71^{1/2}\approx 92.30384607371461$
Ramified primes:   \(2\), \(3\), \(5\), \(71\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{2}a^{7}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{12}-\frac{1}{4}a^{9}-\frac{1}{8}a^{8}-\frac{1}{2}a^{7}+\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{13}-\frac{1}{8}a^{10}+\frac{3}{16}a^{9}-\frac{1}{4}a^{8}-\frac{7}{16}a^{7}-\frac{1}{8}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{32}a^{16}-\frac{1}{32}a^{14}-\frac{1}{8}a^{12}-\frac{1}{16}a^{11}+\frac{3}{32}a^{10}+\frac{1}{8}a^{9}-\frac{7}{32}a^{8}+\frac{7}{16}a^{7}-\frac{3}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{17}-\frac{1}{64}a^{15}+\frac{1}{16}a^{13}+\frac{3}{32}a^{12}-\frac{5}{64}a^{11}+\frac{1}{16}a^{10}-\frac{7}{64}a^{9}-\frac{1}{32}a^{8}+\frac{3}{16}a^{7}+\frac{1}{4}a^{6}-\frac{3}{8}a^{5}+\frac{1}{4}a^{4}-\frac{3}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{128}a^{18}-\frac{1}{128}a^{16}+\frac{1}{32}a^{14}-\frac{5}{64}a^{13}-\frac{5}{128}a^{12}-\frac{3}{32}a^{11}-\frac{7}{128}a^{10}+\frac{15}{64}a^{9}-\frac{5}{32}a^{8}-\frac{1}{4}a^{7}-\frac{3}{16}a^{6}-\frac{1}{4}a^{5}-\frac{7}{16}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{256}a^{19}-\frac{1}{256}a^{17}+\frac{1}{64}a^{15}-\frac{5}{128}a^{14}+\frac{27}{256}a^{13}+\frac{5}{64}a^{12}+\frac{25}{256}a^{11}+\frac{15}{128}a^{10}+\frac{11}{64}a^{9}+\frac{1}{8}a^{8}+\frac{9}{32}a^{7}-\frac{1}{4}a^{6}-\frac{11}{32}a^{5}-\frac{5}{16}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2746880}a^{20}+\frac{2103}{1373440}a^{19}+\frac{171}{2746880}a^{18}-\frac{6191}{1373440}a^{17}-\frac{2811}{343360}a^{16}+\frac{18303}{1373440}a^{15}-\frac{84817}{2746880}a^{14}-\frac{18549}{1373440}a^{13}-\frac{79243}{2746880}a^{12}+\frac{63799}{686720}a^{11}-\frac{13321}{137344}a^{10}-\frac{17291}{85840}a^{9}-\frac{15479}{343360}a^{8}-\frac{13641}{171680}a^{7}-\frac{132079}{343360}a^{6}+\frac{33139}{85840}a^{5}-\frac{7421}{85840}a^{4}+\frac{413}{5365}a^{3}-\frac{5023}{21460}a^{2}+\frac{1159}{5365}a+\frac{2}{5365}$, $\frac{1}{2746880}a^{21}+\frac{701}{549376}a^{19}-\frac{123}{171680}a^{18}-\frac{8243}{1373440}a^{17}-\frac{8733}{1373440}a^{16}+\frac{1143}{94720}a^{15}-\frac{303}{5365}a^{14}-\frac{48333}{549376}a^{13}+\frac{117267}{1373440}a^{12}-\frac{68973}{1373440}a^{11}+\frac{83427}{686720}a^{10}-\frac{8983}{68672}a^{9}+\frac{1109}{5920}a^{8}+\frac{116183}{343360}a^{7}-\frac{15717}{34336}a^{6}+\frac{6715}{34336}a^{5}+\frac{399}{85840}a^{4}-\frac{59}{4292}a^{3}-\frac{1361}{21460}a^{2}-\frac{1299}{10730}a+\frac{2318}{5365}$, $\frac{1}{86\!\cdots\!00}a^{22}+\frac{228139621}{43\!\cdots\!00}a^{21}-\frac{181297147}{17\!\cdots\!20}a^{20}+\frac{351718198121}{43\!\cdots\!00}a^{19}-\frac{15789939922531}{43\!\cdots\!00}a^{18}-\frac{30104532933159}{43\!\cdots\!00}a^{17}+\frac{14651604648899}{17\!\cdots\!20}a^{16}+\frac{8374870687309}{393984064460800}a^{15}+\frac{47450580471}{8077958451200}a^{14}-\frac{125378951877}{5324108979200}a^{13}+\frac{7137388091773}{117130397542400}a^{12}-\frac{37791423954929}{541728088633600}a^{11}+\frac{72047270038107}{10\!\cdots\!00}a^{10}-\frac{2335899453999}{49248008057600}a^{9}+\frac{172268317304827}{10\!\cdots\!00}a^{8}+\frac{8688049349919}{24624004028800}a^{7}+\frac{41351080281713}{108345617726720}a^{6}-\frac{3104903445779}{67716011079200}a^{5}+\frac{6428501158821}{33858005539600}a^{4}+\frac{12238587514667}{33858005539600}a^{3}+\frac{1373907551729}{3385800553960}a^{2}-\frac{593308219732}{2116125346225}a-\frac{3431160183937}{8464501384900}$, $\frac{1}{36\!\cdots\!00}a^{23}+\frac{1}{31\!\cdots\!00}a^{22}+\frac{215557892213}{36\!\cdots\!00}a^{21}+\frac{90632985243}{91\!\cdots\!00}a^{20}-\frac{44964064454289}{34\!\cdots\!00}a^{19}-\frac{208777040750053}{18\!\cdots\!00}a^{18}+\frac{20\!\cdots\!83}{36\!\cdots\!00}a^{17}+\frac{18\!\cdots\!57}{91\!\cdots\!00}a^{16}-\frac{19\!\cdots\!93}{73\!\cdots\!40}a^{15}-\frac{51\!\cdots\!47}{18\!\cdots\!00}a^{14}+\frac{76\!\cdots\!49}{18\!\cdots\!00}a^{13}-\frac{10\!\cdots\!89}{83\!\cdots\!00}a^{12}-\frac{914669836772253}{18\!\cdots\!36}a^{11}-\frac{13\!\cdots\!91}{22\!\cdots\!20}a^{10}+\frac{17\!\cdots\!19}{18\!\cdots\!36}a^{9}+\frac{10\!\cdots\!41}{61\!\cdots\!00}a^{8}+\frac{83\!\cdots\!37}{22\!\cdots\!00}a^{7}-\frac{24\!\cdots\!31}{11\!\cdots\!00}a^{6}-\frac{54\!\cdots\!17}{14\!\cdots\!00}a^{5}+\frac{174501249077191}{12\!\cdots\!00}a^{4}-\frac{12\!\cdots\!83}{35\!\cdots\!00}a^{3}-\frac{6296859530521}{67396595932600}a^{2}+\frac{13\!\cdots\!91}{35\!\cdots\!00}a+\frac{570787501600971}{17\!\cdots\!00}$ 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:  $2$

Class group and class number

$C_{2}\times C_{22}$, which has order $44$ (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:   \( -\frac{1803253231}{28172714598400} a^{23} + \frac{1122405231}{14086357299200} a^{22} - \frac{12686318967}{28172714598400} a^{21} + \frac{10533622779}{14086357299200} a^{20} - \frac{4155596799}{2817271459840} a^{19} + \frac{12382668561}{2817271459840} a^{18} + \frac{110446663143}{28172714598400} a^{17} + \frac{73223856941}{14086357299200} a^{16} + \frac{680605089719}{28172714598400} a^{15} + \frac{176812457893}{1760794662400} a^{14} - \frac{2148437245723}{14086357299200} a^{13} + \frac{1728501716419}{3521589324800} a^{12} - \frac{6556061017449}{3521589324800} a^{11} + \frac{3935237239413}{1760794662400} a^{10} - \frac{28767063860629}{3521589324800} a^{9} + \frac{181387900551}{110049666400} a^{8} - \frac{3143264584423}{1760794662400} a^{7} + \frac{852503602563}{440198665600} a^{6} - \frac{18422767413}{13756208300} a^{5} - \frac{178833552803}{110049666400} a^{4} + \frac{85789822399}{55024833200} a^{3} - \frac{8438871203}{6878104150} a^{2} + \frac{6338801579}{5502483320} a - \frac{12757778407}{6878104150} \)  (order $6$) 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:   $\frac{144765796479031}{36\!\cdots\!00}a^{23}-\frac{90073047676651}{18\!\cdots\!00}a^{22}+\frac{10\!\cdots\!07}{36\!\cdots\!00}a^{21}-\frac{844896575302039}{18\!\cdots\!00}a^{20}+\frac{30317059876109}{33\!\cdots\!20}a^{19}-\frac{993503916508561}{36\!\cdots\!20}a^{18}-\frac{88\!\cdots\!03}{36\!\cdots\!00}a^{17}-\frac{58\!\cdots\!21}{18\!\cdots\!00}a^{16}-\frac{54\!\cdots\!19}{36\!\cdots\!00}a^{15}-\frac{28\!\cdots\!61}{45\!\cdots\!00}a^{14}+\frac{17\!\cdots\!23}{18\!\cdots\!00}a^{13}-\frac{13\!\cdots\!49}{45\!\cdots\!00}a^{12}+\frac{47\!\cdots\!99}{41\!\cdots\!00}a^{11}-\frac{31\!\cdots\!93}{22\!\cdots\!00}a^{10}+\frac{23\!\cdots\!29}{45\!\cdots\!00}a^{9}-\frac{58\!\cdots\!99}{57\!\cdots\!00}a^{8}+\frac{25\!\cdots\!03}{22\!\cdots\!00}a^{7}-\frac{68\!\cdots\!53}{57\!\cdots\!00}a^{6}+\frac{59\!\cdots\!37}{71\!\cdots\!00}a^{5}+\frac{14\!\cdots\!63}{14\!\cdots\!00}a^{4}-\frac{68\!\cdots\!99}{71\!\cdots\!00}a^{3}+\frac{13\!\cdots\!21}{17\!\cdots\!00}a^{2}-\frac{508655518247907}{714403916885560}a+\frac{10\!\cdots\!17}{893004896106950}$, $\frac{5529874640427}{86\!\cdots\!00}a^{23}-\frac{8563156069221}{86\!\cdots\!00}a^{22}+\frac{3284508608633}{17\!\cdots\!20}a^{21}+\frac{522083733711}{298884462694400}a^{20}-\frac{2799354200857}{270864044316800}a^{19}-\frac{204127518159}{18680278918400}a^{18}-\frac{44448094508157}{17\!\cdots\!20}a^{17}-\frac{28581426650089}{787968128921600}a^{16}-\frac{11\!\cdots\!49}{86\!\cdots\!00}a^{15}-\frac{769741320499347}{787968128921600}a^{14}+\frac{10\!\cdots\!77}{43\!\cdots\!00}a^{13}-\frac{95\!\cdots\!19}{43\!\cdots\!00}a^{12}+\frac{19\!\cdots\!09}{10\!\cdots\!00}a^{11}+\frac{16\!\cdots\!39}{98496016115200}a^{10}+\frac{89\!\cdots\!79}{10\!\cdots\!00}a^{9}-\frac{26933268598983}{98496016115200}a^{8}-\frac{357495558254903}{108345617726720}a^{7}+\frac{600877386213941}{541728088633600}a^{6}+\frac{104356020970457}{33858005539600}a^{5}-\frac{98520252689181}{33858005539600}a^{4}-\frac{10580766208769}{6771601107920}a^{3}+\frac{61825105697}{583758716200}a^{2}+\frac{46928614417741}{8464501384900}a+\frac{173357995}{75626548}$, $\frac{523086305059}{17\!\cdots\!20}a^{23}-\frac{1046178763477}{17\!\cdots\!20}a^{22}+\frac{523091420465}{346705976725504}a^{21}-\frac{1046967912317}{17\!\cdots\!20}a^{20}-\frac{326936314923}{108345617726720}a^{19}-\frac{57228216459}{13543202215840}a^{18}-\frac{5126511625013}{346705976725504}a^{17}-\frac{2187219165273}{157593625784320}a^{16}-\frac{119773634078333}{17\!\cdots\!20}a^{15}-\frac{69612931755539}{157593625784320}a^{14}+\frac{10\!\cdots\!09}{866764941813760}a^{13}-\frac{18\!\cdots\!63}{866764941813760}a^{12}+\frac{725128602979153}{216691235453440}a^{11}+\frac{82281482946863}{19699203223040}a^{10}+\frac{10\!\cdots\!03}{216691235453440}a^{9}+\frac{69623320298809}{19699203223040}a^{8}-\frac{23959191269919}{21669123545344}a^{7}+\frac{48123795306677}{108345617726720}a^{6}-\frac{400512101716}{423225069245}a^{5}+\frac{3661677529623}{6771601107920}a^{4}-\frac{1046223335395}{1354320221584}a^{3}+\frac{1046617219701}{3385800553960}a^{2}-\frac{123349933023}{1692900276980}a+\frac{17858535}{75626548}$, $\frac{8538340264537}{63\!\cdots\!00}a^{23}+\frac{5576802665051}{63\!\cdots\!00}a^{22}-\frac{56652501348541}{63\!\cdots\!00}a^{21}+\frac{237635777157259}{63\!\cdots\!00}a^{20}-\frac{23611415567771}{31\!\cdots\!20}a^{19}-\frac{88414581173}{85222461660160}a^{18}-\frac{294865831379111}{63\!\cdots\!00}a^{17}-\frac{27689609353847}{17\!\cdots\!00}a^{16}-\frac{925524326199163}{63\!\cdots\!00}a^{15}-\frac{16\!\cdots\!01}{63\!\cdots\!00}a^{14}+\frac{68\!\cdots\!71}{31\!\cdots\!00}a^{13}+\frac{53\!\cdots\!73}{31\!\cdots\!00}a^{12}-\frac{41\!\cdots\!77}{78\!\cdots\!00}a^{11}+\frac{11\!\cdots\!73}{78\!\cdots\!00}a^{10}-\frac{48\!\cdots\!67}{78\!\cdots\!00}a^{9}+\frac{17\!\cdots\!11}{78\!\cdots\!00}a^{8}-\frac{21\!\cdots\!29}{39\!\cdots\!00}a^{7}+\frac{12\!\cdots\!21}{39\!\cdots\!00}a^{6}-\frac{24742013969967}{246346178236400}a^{5}-\frac{531613692259919}{246346178236400}a^{4}+\frac{413517966537579}{246346178236400}a^{3}-\frac{211086149433327}{123173089118200}a^{2}+\frac{52426228332147}{12317308911820}a+\frac{16411987600731}{61586544559100}$, $\frac{703935398456869}{73\!\cdots\!40}a^{23}-\frac{36\!\cdots\!07}{18\!\cdots\!00}a^{22}+\frac{13\!\cdots\!27}{33\!\cdots\!00}a^{21}-\frac{12093514004467}{73\!\cdots\!44}a^{20}-\frac{26\!\cdots\!19}{18\!\cdots\!00}a^{19}-\frac{19\!\cdots\!31}{18\!\cdots\!00}a^{18}-\frac{12\!\cdots\!33}{36\!\cdots\!00}a^{17}-\frac{11\!\cdots\!97}{36\!\cdots\!20}a^{16}-\frac{64\!\cdots\!57}{36\!\cdots\!00}a^{15}-\frac{12\!\cdots\!33}{91\!\cdots\!00}a^{14}+\frac{79\!\cdots\!77}{18\!\cdots\!00}a^{13}-\frac{32\!\cdots\!17}{57\!\cdots\!00}a^{12}+\frac{31\!\cdots\!07}{45\!\cdots\!00}a^{11}+\frac{12\!\cdots\!23}{61\!\cdots\!00}a^{10}+\frac{20\!\cdots\!47}{45\!\cdots\!00}a^{9}-\frac{53\!\cdots\!97}{11\!\cdots\!00}a^{8}-\frac{71\!\cdots\!11}{61\!\cdots\!00}a^{7}-\frac{87\!\cdots\!73}{28\!\cdots\!40}a^{6}-\frac{11\!\cdots\!31}{71\!\cdots\!00}a^{5}-\frac{30\!\cdots\!89}{14\!\cdots\!00}a^{4}-\frac{19\!\cdots\!49}{71\!\cdots\!00}a^{3}+\frac{125645383160897}{71440391688556}a^{2}+\frac{28\!\cdots\!47}{35\!\cdots\!00}a+\frac{19\!\cdots\!16}{446502448053475}$, $\frac{53506064258633}{36\!\cdots\!20}a^{23}-\frac{901394778898533}{18\!\cdots\!00}a^{22}+\frac{26\!\cdots\!19}{18\!\cdots\!00}a^{21}-\frac{139737502502725}{73\!\cdots\!44}a^{20}+\frac{324557823239291}{45\!\cdots\!00}a^{19}-\frac{572435710557461}{45\!\cdots\!00}a^{18}-\frac{10\!\cdots\!11}{18\!\cdots\!00}a^{17}-\frac{271792784765899}{73\!\cdots\!44}a^{16}-\frac{14\!\cdots\!03}{34\!\cdots\!00}a^{15}-\frac{33\!\cdots\!89}{18\!\cdots\!00}a^{14}+\frac{75\!\cdots\!19}{91\!\cdots\!00}a^{13}-\frac{21\!\cdots\!23}{91\!\cdots\!00}a^{12}+\frac{97\!\cdots\!59}{22\!\cdots\!00}a^{11}-\frac{58\!\cdots\!71}{22\!\cdots\!00}a^{10}+\frac{34\!\cdots\!21}{78\!\cdots\!00}a^{9}+\frac{10\!\cdots\!59}{22\!\cdots\!00}a^{8}+\frac{85\!\cdots\!11}{11\!\cdots\!00}a^{7}+\frac{97\!\cdots\!29}{22\!\cdots\!20}a^{6}+\frac{199054702690759}{17\!\cdots\!00}a^{5}-\frac{52\!\cdots\!07}{17\!\cdots\!00}a^{4}-\frac{28\!\cdots\!31}{71\!\cdots\!00}a^{3}+\frac{70701742224227}{64945810625960}a^{2}+\frac{20\!\cdots\!49}{17\!\cdots\!00}a+\frac{37\!\cdots\!31}{17\!\cdots\!00}$, $\frac{24\!\cdots\!03}{18\!\cdots\!00}a^{23}-\frac{678716507845073}{18\!\cdots\!60}a^{22}+\frac{16\!\cdots\!43}{18\!\cdots\!00}a^{21}-\frac{80\!\cdots\!97}{91\!\cdots\!00}a^{20}-\frac{720782393014119}{83\!\cdots\!00}a^{19}-\frac{10\!\cdots\!81}{91\!\cdots\!00}a^{18}-\frac{98\!\cdots\!47}{18\!\cdots\!00}a^{17}-\frac{20\!\cdots\!43}{91\!\cdots\!00}a^{16}-\frac{51\!\cdots\!19}{18\!\cdots\!00}a^{15}-\frac{80\!\cdots\!89}{45\!\cdots\!00}a^{14}+\frac{12\!\cdots\!47}{17\!\cdots\!00}a^{13}-\frac{32\!\cdots\!83}{22\!\cdots\!00}a^{12}+\frac{50\!\cdots\!19}{20\!\cdots\!00}a^{11}+\frac{40\!\cdots\!37}{11\!\cdots\!00}a^{10}+\frac{37\!\cdots\!69}{22\!\cdots\!00}a^{9}-\frac{29\!\cdots\!21}{57\!\cdots\!00}a^{8}-\frac{26\!\cdots\!33}{11\!\cdots\!00}a^{7}+\frac{13\!\cdots\!81}{28\!\cdots\!00}a^{6}-\frac{307643246446963}{14\!\cdots\!20}a^{5}+\frac{22\!\cdots\!07}{14\!\cdots\!20}a^{4}-\frac{26\!\cdots\!07}{446502448053475}a^{3}+\frac{64\!\cdots\!93}{893004896106950}a^{2}+\frac{36408853606897}{17\!\cdots\!00}a+\frac{27\!\cdots\!73}{446502448053475}$, $\frac{913348065276201}{36\!\cdots\!00}a^{23}-\frac{17\!\cdots\!11}{18\!\cdots\!00}a^{22}+\frac{10\!\cdots\!97}{36\!\cdots\!00}a^{21}-\frac{86\!\cdots\!79}{18\!\cdots\!00}a^{20}+\frac{117387267923111}{33\!\cdots\!20}a^{19}-\frac{339385693330583}{73\!\cdots\!44}a^{18}-\frac{26\!\cdots\!93}{36\!\cdots\!00}a^{17}+\frac{31\!\cdots\!59}{18\!\cdots\!00}a^{16}-\frac{23\!\cdots\!89}{36\!\cdots\!00}a^{15}-\frac{66\!\cdots\!21}{24\!\cdots\!00}a^{14}+\frac{28\!\cdots\!53}{18\!\cdots\!00}a^{13}-\frac{52\!\cdots\!31}{11\!\cdots\!00}a^{12}+\frac{39\!\cdots\!09}{41\!\cdots\!00}a^{11}-\frac{21\!\cdots\!53}{22\!\cdots\!00}a^{10}+\frac{55\!\cdots\!79}{45\!\cdots\!00}a^{9}-\frac{94\!\cdots\!73}{11\!\cdots\!00}a^{8}+\frac{23\!\cdots\!33}{22\!\cdots\!00}a^{7}-\frac{20\!\cdots\!91}{71\!\cdots\!00}a^{6}-\frac{72\!\cdots\!21}{14\!\cdots\!00}a^{5}+\frac{52\!\cdots\!13}{14\!\cdots\!00}a^{4}-\frac{32\!\cdots\!69}{71\!\cdots\!00}a^{3}+\frac{75\!\cdots\!41}{17\!\cdots\!00}a^{2}-\frac{23\!\cdots\!69}{714403916885560}a+\frac{597387236201201}{446502448053475}$, $\frac{25\!\cdots\!99}{18\!\cdots\!00}a^{23}-\frac{247012899788657}{73\!\cdots\!44}a^{22}+\frac{408871344995757}{49\!\cdots\!00}a^{21}-\frac{11\!\cdots\!97}{18\!\cdots\!00}a^{20}-\frac{23\!\cdots\!23}{22\!\cdots\!00}a^{19}-\frac{336415028898427}{20\!\cdots\!00}a^{18}-\frac{10\!\cdots\!41}{18\!\cdots\!00}a^{17}-\frac{64\!\cdots\!63}{18\!\cdots\!00}a^{16}-\frac{53\!\cdots\!57}{18\!\cdots\!00}a^{15}-\frac{33\!\cdots\!73}{18\!\cdots\!00}a^{14}+\frac{60\!\cdots\!13}{91\!\cdots\!00}a^{13}-\frac{11\!\cdots\!91}{91\!\cdots\!00}a^{12}+\frac{46\!\cdots\!17}{22\!\cdots\!00}a^{11}+\frac{20\!\cdots\!57}{22\!\cdots\!00}a^{10}+\frac{47\!\cdots\!87}{22\!\cdots\!00}a^{9}-\frac{54\!\cdots\!17}{22\!\cdots\!00}a^{8}-\frac{29\!\cdots\!59}{11\!\cdots\!00}a^{7}+\frac{95\!\cdots\!57}{11\!\cdots\!00}a^{6}-\frac{709659723281057}{14\!\cdots\!20}a^{5}+\frac{165293606824619}{714403916885560}a^{4}-\frac{26\!\cdots\!11}{71\!\cdots\!00}a^{3}+\frac{13\!\cdots\!01}{35\!\cdots\!00}a^{2}+\frac{79\!\cdots\!31}{17\!\cdots\!00}a+\frac{97\!\cdots\!51}{17\!\cdots\!00}$, $\frac{597329347939}{446502448053475}a^{23}-\frac{41\!\cdots\!83}{18\!\cdots\!00}a^{22}+\frac{43\!\cdots\!89}{91\!\cdots\!00}a^{21}+\frac{27\!\cdots\!13}{18\!\cdots\!00}a^{20}-\frac{689127797312583}{36\!\cdots\!72}a^{19}-\frac{844242445712243}{36\!\cdots\!72}a^{18}-\frac{47\!\cdots\!61}{83\!\cdots\!00}a^{17}-\frac{12\!\cdots\!93}{18\!\cdots\!00}a^{16}-\frac{25\!\cdots\!33}{91\!\cdots\!00}a^{15}-\frac{36\!\cdots\!17}{18\!\cdots\!00}a^{14}+\frac{23\!\cdots\!21}{45\!\cdots\!00}a^{13}-\frac{52\!\cdots\!19}{91\!\cdots\!00}a^{12}+\frac{84\!\cdots\!63}{11\!\cdots\!00}a^{11}+\frac{67\!\cdots\!71}{22\!\cdots\!00}a^{10}+\frac{22\!\cdots\!43}{11\!\cdots\!00}a^{9}+\frac{19\!\cdots\!67}{22\!\cdots\!00}a^{8}-\frac{89\!\cdots\!99}{57\!\cdots\!00}a^{7}-\frac{78\!\cdots\!83}{11\!\cdots\!00}a^{6}+\frac{38\!\cdots\!77}{14\!\cdots\!00}a^{5}+\frac{37\!\cdots\!47}{71\!\cdots\!00}a^{4}+\frac{15\!\cdots\!63}{71\!\cdots\!00}a^{3}+\frac{26\!\cdots\!01}{35\!\cdots\!00}a^{2}+\frac{44671760231559}{8118226328245}a+\frac{53\!\cdots\!07}{17\!\cdots\!00}$, $\frac{39\!\cdots\!29}{36\!\cdots\!00}a^{23}-\frac{56\!\cdots\!39}{18\!\cdots\!00}a^{22}+\frac{29\!\cdots\!73}{36\!\cdots\!00}a^{21}-\frac{16\!\cdots\!91}{18\!\cdots\!00}a^{20}-\frac{97\!\cdots\!19}{36\!\cdots\!20}a^{19}-\frac{49\!\cdots\!59}{36\!\cdots\!20}a^{18}-\frac{15\!\cdots\!17}{36\!\cdots\!00}a^{17}-\frac{27\!\cdots\!09}{18\!\cdots\!00}a^{16}-\frac{87\!\cdots\!21}{36\!\cdots\!00}a^{15}-\frac{12\!\cdots\!23}{91\!\cdots\!00}a^{14}+\frac{10\!\cdots\!57}{18\!\cdots\!00}a^{13}-\frac{19\!\cdots\!41}{15\!\cdots\!00}a^{12}+\frac{10\!\cdots\!11}{45\!\cdots\!00}a^{11}-\frac{10\!\cdots\!57}{22\!\cdots\!00}a^{10}+\frac{10\!\cdots\!11}{45\!\cdots\!00}a^{9}-\frac{78\!\cdots\!67}{11\!\cdots\!00}a^{8}+\frac{51\!\cdots\!97}{22\!\cdots\!00}a^{7}-\frac{11\!\cdots\!53}{14\!\cdots\!00}a^{6}-\frac{40\!\cdots\!79}{14\!\cdots\!00}a^{5}+\frac{68\!\cdots\!57}{14\!\cdots\!00}a^{4}-\frac{50\!\cdots\!41}{71\!\cdots\!00}a^{3}+\frac{196939081941673}{2799388389050}a^{2}-\frac{813069716619613}{142880783377112}a+\frac{24\!\cdots\!34}{446502448053475}$ 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:  \( 2670961305.4082894 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 2670961305.4082894 \cdot 44}{6\cdot\sqrt{5757603349403224554199842816000000000000}}\cr\approx \mathstrut & 0.977252008341264 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^23 + 5*x^22 - 2*x^21 - 10*x^20 - 14*x^19 - 49*x^18 - 46*x^17 - 229*x^16 - 1464*x^15 + 4190*x^14 - 6920*x^13 + 11088*x^12 + 13840*x^11 + 16760*x^10 + 11712*x^9 - 3664*x^8 + 1472*x^7 - 3136*x^6 + 1792*x^5 - 2560*x^4 + 1024*x^3 + 5120*x^2 + 4096*x + 4096)
 
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 - 2*x^23 + 5*x^22 - 2*x^21 - 10*x^20 - 14*x^19 - 49*x^18 - 46*x^17 - 229*x^16 - 1464*x^15 + 4190*x^14 - 6920*x^13 + 11088*x^12 + 13840*x^11 + 16760*x^10 + 11712*x^9 - 3664*x^8 + 1472*x^7 - 3136*x^6 + 1792*x^5 - 2560*x^4 + 1024*x^3 + 5120*x^2 + 4096*x + 4096, 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 - 2*x^23 + 5*x^22 - 2*x^21 - 10*x^20 - 14*x^19 - 49*x^18 - 46*x^17 - 229*x^16 - 1464*x^15 + 4190*x^14 - 6920*x^13 + 11088*x^12 + 13840*x^11 + 16760*x^10 + 11712*x^9 - 3664*x^8 + 1472*x^7 - 3136*x^6 + 1792*x^5 - 2560*x^4 + 1024*x^3 + 5120*x^2 + 4096*x + 4096);
 
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 - 2*x^23 + 5*x^22 - 2*x^21 - 10*x^20 - 14*x^19 - 49*x^18 - 46*x^17 - 229*x^16 - 1464*x^15 + 4190*x^14 - 6920*x^13 + 11088*x^12 + 13840*x^11 + 16760*x^10 + 11712*x^9 - 3664*x^8 + 1472*x^7 - 3136*x^6 + 1792*x^5 - 2560*x^4 + 1024*x^3 + 5120*x^2 + 4096*x + 4096);
 
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

$C_2^2\times D_6$ (as 24T30):

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 48
The 24 conjugacy class representatives for $C_2^2\times D_6$
Character table for $C_2^2\times D_6$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{6}) \), 3.3.568.1, \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-15})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{6}, \sqrt{-10})\), 6.0.2580992.1, 6.0.322624000.1, 6.0.1088856000.6, 6.0.8710848.1, 6.6.40328000.1, 6.6.8710848000.1, 6.6.69686784.1, 8.0.207360000.2, 12.12.75878872879104000000.1, 12.0.104086245376000000.1, 12.0.1185607388736000000.1, 12.0.75878872879104000000.1, 12.0.75878872879104000000.2, 12.0.4856247864262656.1, 12.0.75878872879104000000.3

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]
 

Sibling fields

Degree 24 siblings: 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 R R R ${\href{/padicField/7.6.0.1}{6} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{12}$ ${\href{/padicField/13.6.0.1}{6} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{12}$ ${\href{/padicField/19.3.0.1}{3} }^{8}$ ${\href{/padicField/23.6.0.1}{6} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{12}$ ${\href{/padicField/31.6.0.1}{6} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{12}$ ${\href{/padicField/41.2.0.1}{2} }^{12}$ ${\href{/padicField/43.6.0.1}{6} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{12}$ ${\href{/padicField/59.2.0.1}{2} }^{12}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.4.6.2$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$$2$$2$$6$$C_2^2$$[3]^{2}$
\(3\) Copy content Toggle raw display 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$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(71\) Copy content Toggle raw display 71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$