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Properties

Label 24.0.134...625.1
Degree $24$
Signature $[0, 12]$
Discriminant $1.349\times 10^{36}$
Root discriminant \(32.02\)
Ramified primes $3,5,13$
Class number $16$ (GRH)
Class group [4, 4] (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 4*x^22 + 15*x^21 - 16*x^20 + 62*x^19 + 163*x^18 - 809*x^17 + 670*x^16 + 3344*x^15 + 2796*x^14 - 2771*x^13 - 4594*x^12 - 11446*x^11 - 11444*x^10 + 8739*x^9 + 34835*x^8 + 8891*x^7 - 162*x^6 - 698*x^5 - 196*x^4 - 50*x^3 + x^2 + 4*x + 1)
 
gp: K = bnfinit(y^24 - y^23 - 4*y^22 + 15*y^21 - 16*y^20 + 62*y^19 + 163*y^18 - 809*y^17 + 670*y^16 + 3344*y^15 + 2796*y^14 - 2771*y^13 - 4594*y^12 - 11446*y^11 - 11444*y^10 + 8739*y^9 + 34835*y^8 + 8891*y^7 - 162*y^6 - 698*y^5 - 196*y^4 - 50*y^3 + y^2 + 4*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - x^23 - 4*x^22 + 15*x^21 - 16*x^20 + 62*x^19 + 163*x^18 - 809*x^17 + 670*x^16 + 3344*x^15 + 2796*x^14 - 2771*x^13 - 4594*x^12 - 11446*x^11 - 11444*x^10 + 8739*x^9 + 34835*x^8 + 8891*x^7 - 162*x^6 - 698*x^5 - 196*x^4 - 50*x^3 + x^2 + 4*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - x^23 - 4*x^22 + 15*x^21 - 16*x^20 + 62*x^19 + 163*x^18 - 809*x^17 + 670*x^16 + 3344*x^15 + 2796*x^14 - 2771*x^13 - 4594*x^12 - 11446*x^11 - 11444*x^10 + 8739*x^9 + 34835*x^8 + 8891*x^7 - 162*x^6 - 698*x^5 - 196*x^4 - 50*x^3 + x^2 + 4*x + 1)
 

\( x^{24} - x^{23} - 4 x^{22} + 15 x^{21} - 16 x^{20} + 62 x^{19} + 163 x^{18} - 809 x^{17} + 670 x^{16} + \cdots + 1 \) 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:   \(1348990128329918967746280670166015625\) \(\medspace = 3^{12}\cdot 5^{18}\cdot 13^{16}\) 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:  \(32.02\)
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:  $3^{1/2}5^{3/4}13^{2/3}\approx 32.01968330451422$
Ramified primes:   \(3\), \(5\), \(13\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $24$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(195=3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{195}(1,·)$, $\chi_{195}(131,·)$, $\chi_{195}(68,·)$, $\chi_{195}(133,·)$, $\chi_{195}(74,·)$, $\chi_{195}(139,·)$, $\chi_{195}(14,·)$, $\chi_{195}(79,·)$, $\chi_{195}(16,·)$, $\chi_{195}(146,·)$, $\chi_{195}(22,·)$, $\chi_{195}(152,·)$, $\chi_{195}(92,·)$, $\chi_{195}(29,·)$, $\chi_{195}(94,·)$, $\chi_{195}(107,·)$, $\chi_{195}(172,·)$, $\chi_{195}(157,·)$, $\chi_{195}(113,·)$, $\chi_{195}(178,·)$, $\chi_{195}(53,·)$, $\chi_{195}(118,·)$, $\chi_{195}(61,·)$, $\chi_{195}(191,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{6781}a^{18}+\frac{675}{6781}a^{17}-\frac{1401}{6781}a^{15}-\frac{3116}{6781}a^{14}-\frac{532}{6781}a^{13}+\frac{3385}{6781}a^{12}-\frac{1448}{6781}a^{11}-\frac{578}{6781}a^{10}-\frac{2466}{6781}a^{9}+\frac{2556}{6781}a^{8}+\frac{3162}{6781}a^{7}+\frac{3337}{6781}a^{6}-\frac{588}{6781}a^{5}-\frac{1969}{6781}a^{4}+\frac{1}{6781}a^{3}-\frac{1298}{6781}a-\frac{1401}{6781}$, $\frac{1}{6781}a^{19}-\frac{1298}{6781}a^{17}-\frac{1401}{6781}a^{16}+\frac{658}{6781}a^{14}+\frac{3092}{6781}a^{13}-\frac{1126}{6781}a^{12}+\frac{358}{6781}a^{11}+\frac{1167}{6781}a^{10}-\frac{1020}{6781}a^{9}+\frac{236}{6781}a^{8}-\frac{1779}{6781}a^{7}-\frac{1771}{6781}a^{6}+\frac{1633}{6781}a^{5}-\frac{675}{6781}a^{3}-\frac{1298}{6781}a^{2}+\frac{3116}{6781}$, $\frac{1}{535699}a^{20}-\frac{5}{535699}a^{19}+\frac{243825}{535699}a^{17}+\frac{115501}{535699}a^{16}-\frac{136152}{535699}a^{15}-\frac{240625}{535699}a^{14}+\frac{99817}{535699}a^{13}+\frac{5630}{535699}a^{12}-\frac{164534}{535699}a^{11}+\frac{110869}{535699}a^{10}-\frac{137301}{535699}a^{9}+\frac{222593}{535699}a^{8}-\frac{4667}{535699}a^{7}+\frac{185161}{535699}a^{6}+\frac{83017}{535699}a^{5}-\frac{33}{79}a^{4}-\frac{125464}{535699}a^{3}+\frac{683}{6781}a^{2}-\frac{30}{79}a-\frac{247324}{535699}$, $\frac{1}{47184545236369}a^{21}-\frac{17080124}{47184545236369}a^{20}+\frac{1500080850}{47184545236369}a^{19}-\frac{1414680224}{47184545236369}a^{18}-\frac{1877983318866}{47184545236369}a^{17}-\frac{7007454641359}{47184545236369}a^{16}-\frac{3379018699049}{47184545236369}a^{15}+\frac{7040323998634}{47184545236369}a^{14}-\frac{4027367370299}{47184545236369}a^{13}+\frac{3887941698954}{47184545236369}a^{12}+\frac{13286602800625}{47184545236369}a^{11}-\frac{23009996697831}{47184545236369}a^{10}-\frac{22749799957235}{47184545236369}a^{9}-\frac{5503154423107}{47184545236369}a^{8}+\frac{8677897803958}{47184545236369}a^{7}-\frac{19535957558520}{47184545236369}a^{6}-\frac{1983064517430}{47184545236369}a^{5}+\frac{10595512088738}{47184545236369}a^{4}-\frac{11404493465426}{47184545236369}a^{3}-\frac{10846588671388}{47184545236369}a^{2}-\frac{11873330592484}{47184545236369}a-\frac{2387836934179}{47184545236369}$, $\frac{1}{47184545236369}a^{22}-\frac{33076046}{47184545236369}a^{20}+\frac{372334935}{47184545236369}a^{19}-\frac{2619680}{597272724511}a^{18}+\frac{11004207107753}{47184545236369}a^{17}+\frac{7332756595023}{47184545236369}a^{16}-\frac{22941253632919}{47184545236369}a^{15}+\frac{760943789928}{47184545236369}a^{14}-\frac{7122442825305}{47184545236369}a^{13}-\frac{8561061457225}{47184545236369}a^{12}+\frac{7808099354305}{47184545236369}a^{11}-\frac{11281529368793}{47184545236369}a^{10}-\frac{12651888130908}{47184545236369}a^{9}-\frac{20849388313915}{47184545236369}a^{8}+\frac{3928612349415}{47184545236369}a^{7}+\frac{15741425842458}{47184545236369}a^{6}-\frac{5418135874254}{47184545236369}a^{5}-\frac{1681538238348}{47184545236369}a^{4}-\frac{6454077701611}{47184545236369}a^{3}-\frac{270554649143}{597272724511}a^{2}+\frac{19384762548293}{47184545236369}a+\frac{7060940013033}{47184545236369}$, $\frac{1}{47184545236369}a^{23}-\frac{19998312}{47184545236369}a^{20}-\frac{22417316}{597272724511}a^{19}-\frac{172364210}{47184545236369}a^{18}+\frac{23018114605162}{47184545236369}a^{17}+\frac{174855771762}{597272724511}a^{16}+\frac{21648219442167}{47184545236369}a^{15}+\frac{22172224036193}{47184545236369}a^{14}-\frac{6893582052213}{47184545236369}a^{13}-\frac{7596374724848}{47184545236369}a^{12}-\frac{4217535432354}{47184545236369}a^{11}+\frac{19943358953226}{47184545236369}a^{10}-\frac{20112121712830}{47184545236369}a^{9}+\frac{7114695148}{597272724511}a^{8}+\frac{14206707948609}{47184545236369}a^{7}+\frac{19539348120385}{47184545236369}a^{6}+\frac{154812702830}{597272724511}a^{5}+\frac{4932422499134}{47184545236369}a^{4}-\frac{280990663123}{597272724511}a^{3}+\frac{103429228327}{597272724511}a^{2}+\frac{17904962031941}{47184545236369}a+\frac{30858710583}{597272724511}$ Copy content Toggle raw display

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

$C_{4}\times C_{4}$, which has order $16$ (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:   \( -\frac{766315375}{1522082104399} a^{23} - \frac{75711959050}{47184545236369} a^{22} + \frac{313852124985}{47184545236369} a^{21} - \frac{376230196510}{47184545236369} a^{20} - \frac{1211416582770}{47184545236369} a^{19} + \frac{3003189954625}{47184545236369} a^{18} - \frac{16723423039085}{47184545236369} a^{17} + \frac{16453220237860}{47184545236369} a^{16} + \frac{63936327861140}{47184545236369} a^{15} - \frac{286037666260465}{47184545236369} a^{14} - \frac{65923812765125}{47184545236369} a^{13} - \frac{80688717621400}{47184545236369} a^{12} - \frac{209958458230150}{47184545236369} a^{11} - \frac{183454674670400}{47184545236369} a^{10} + \frac{1844357238745210}{47184545236369} a^{9} + \frac{659239690934615}{47184545236369} a^{8} + \frac{42332763293135}{47184545236369} a^{7} - \frac{37561837031460}{47184545236369} a^{6} - \frac{13260014722850}{47184545236369} a^{5} - \frac{8388487554590514}{47184545236369} a^{4} - \frac{237680376710}{47184545236369} a^{3} + \frac{213005021635}{47184545236369} a^{2} + \frac{75711959050}{47184545236369} a + \frac{4444629175}{47184545236369} \)  (order $30$) 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:   $\frac{184075041600}{597272724511}a^{23}-\frac{233161005570}{597272724511}a^{22}-\frac{674941819200}{597272724511}a^{21}+\frac{2945200665600}{597272724511}a^{20}-\frac{3730587509760}{597272724511}a^{19}+\frac{12382114464960}{597272724511}a^{18}+\frac{26764604662765}{597272724511}a^{17}-\frac{156156993624000}{597272724511}a^{16}+\frac{165041682298560}{597272724511}a^{15}+\frac{572731084434240}{597272724511}a^{14}+\frac{358749984408960}{597272724511}a^{13}-\frac{606269564398364}{597272724511}a^{12}-\frac{675309969283200}{597272724511}a^{11}-\frac{61787855630400}{19266862081}a^{10}-\frac{16\!\cdots\!00}{597272724511}a^{9}+\frac{20\!\cdots\!00}{597272724511}a^{8}+\frac{58\!\cdots\!75}{597272724511}a^{7}+\frac{33918894332160}{597272724511}a^{6}-\frac{38766203760960}{597272724511}a^{5}-\frac{11424924248640}{597272724511}a^{4}-\frac{122716694400}{19266862081}a^{3}-\frac{8148392184745}{597272724511}a^{2}+\frac{233161719360}{597272724511}a+\frac{670902741151}{597272724511}$, $\frac{416884680676}{1522082104399}a^{23}-\frac{12923305038076}{47184545236369}a^{22}-\frac{51693964542160}{47184545236369}a^{21}+\frac{193851400526916}{47184545236369}a^{20}-\frac{206772756925392}{47184545236369}a^{19}+\frac{801246953429672}{47184545236369}a^{18}+\frac{21\!\cdots\!24}{47184545236369}a^{17}-\frac{10\!\cdots\!40}{47184545236369}a^{16}+\frac{86\!\cdots\!56}{47184545236369}a^{15}+\frac{43\!\cdots\!09}{47184545236369}a^{14}+\frac{36\!\cdots\!76}{47184545236369}a^{13}-\frac{35\!\cdots\!36}{47184545236369}a^{12}-\frac{59\!\cdots\!24}{47184545236369}a^{11}-\frac{14\!\cdots\!36}{47184545236369}a^{10}-\frac{14\!\cdots\!84}{47184545236369}a^{9}+\frac{11\!\cdots\!60}{47184545236369}a^{8}+\frac{45\!\cdots\!84}{47184545236369}a^{7}+\frac{11\!\cdots\!92}{47184545236369}a^{6}-\frac{20\!\cdots\!12}{47184545236369}a^{5}-\frac{90\!\cdots\!68}{47184545236369}a^{4}-\frac{25\!\cdots\!80}{47184545236369}a^{3}-\frac{646170078431576}{47184545236369}a^{2}+\frac{12923305038076}{47184545236369}a+\frac{4509035104575}{47184545236369}$, $\frac{14158349264670}{47184545236369}a^{23}-\frac{17697986156955}{47184545236369}a^{22}-\frac{53093958470865}{47184545236369}a^{21}+\frac{226534222809024}{47184545236369}a^{20}-\frac{3539597231391}{597272724511}a^{19}+\frac{934448467529499}{47184545236369}a^{18}+\frac{20\!\cdots\!90}{47184545236369}a^{17}-\frac{12\!\cdots\!09}{47184545236369}a^{16}+\frac{12\!\cdots\!99}{47184545236369}a^{15}+\frac{44\!\cdots\!46}{47184545236369}a^{14}+\frac{27\!\cdots\!44}{47184545236369}a^{13}-\frac{49\!\cdots\!80}{47184545236369}a^{12}-\frac{55\!\cdots\!55}{47184545236369}a^{11}-\frac{14\!\cdots\!90}{47184545236369}a^{10}-\frac{12\!\cdots\!30}{47184545236369}a^{9}+\frac{16\!\cdots\!25}{47184545236369}a^{8}+\frac{46\!\cdots\!91}{47184545236369}a^{7}+\frac{25\!\cdots\!39}{47184545236369}a^{6}-\frac{33\!\cdots\!49}{47184545236369}a^{5}-\frac{93\!\cdots\!30}{47184545236369}a^{4}-\frac{304405157614545}{47184545236369}a^{3}-\frac{14158388925564}{47184545236369}a^{2}+\frac{191138250495114}{47184545236369}a+\frac{5909413234496}{47184545236369}$, $\frac{575191115712}{1522082104399}a^{23}-\frac{17693748031882}{47184545236369}a^{22}-\frac{71625486769706}{47184545236369}a^{21}+\frac{267491304117118}{47184545236369}a^{20}-\frac{283100362155784}{47184545236369}a^{19}+\frac{10\!\cdots\!14}{47184545236369}a^{18}+\frac{29\!\cdots\!84}{47184545236369}a^{17}-\frac{14\!\cdots\!66}{47184545236369}a^{16}+\frac{11\!\cdots\!08}{47184545236369}a^{15}+\frac{59\!\cdots\!98}{47184545236369}a^{14}+\frac{49\!\cdots\!62}{47184545236369}a^{13}-\frac{49\!\cdots\!42}{47184545236369}a^{12}-\frac{81\!\cdots\!48}{47184545236369}a^{11}-\frac{20\!\cdots\!92}{47184545236369}a^{10}-\frac{20\!\cdots\!37}{47184545236369}a^{9}+\frac{15\!\cdots\!96}{47184545236369}a^{8}+\frac{61\!\cdots\!32}{47184545236369}a^{7}+\frac{15\!\cdots\!50}{47184545236369}a^{6}-\frac{28\!\cdots\!84}{47184545236369}a^{5}-\frac{870450109107334}{47184545236369}a^{4}-\frac{34\!\cdots\!64}{47184545236369}a^{3}-\frac{890201899112738}{47184545236369}a^{2}+\frac{17693748031882}{47184545236369}a+\frac{71186521793098}{47184545236369}$, $\frac{293892525}{1522082104399}a^{23}+\frac{29036581470}{47184545236369}a^{22}-\frac{120366622539}{47184545236369}a^{21}+\frac{144289474074}{47184545236369}a^{20}+\frac{464618072607}{47184545236369}a^{19}-\frac{1151764805475}{47184545236369}a^{18}+\frac{6413663595879}{47184545236369}a^{17}-\frac{6310037091564}{47184545236369}a^{16}-\frac{24520464351036}{47184545236369}a^{15}+\frac{109699393650291}{47184545236369}a^{14}+\frac{25282692248175}{47184545236369}a^{13}+\frac{30945237084360}{47184545236369}a^{12}+\frac{80521967126610}{47184545236369}a^{11}+\frac{70357400256960}{47184545236369}a^{10}-\frac{707336461697454}{47184545236369}a^{9}-\frac{252827522022501}{47184545236369}a^{8}-\frac{16235199110349}{47184545236369}a^{7}+\frac{14405483028204}{47184545236369}a^{6}+\frac{5085398695590}{47184545236369}a^{5}+\frac{31\!\cdots\!03}{47184545236369}a^{4}+\frac{91153705554}{47184545236369}a^{3}-\frac{81690366249}{47184545236369}a^{2}-\frac{29036581470}{47184545236369}a-\frac{1704576645}{47184545236369}$, $\frac{21378677356537}{47184545236369}a^{23}-\frac{21206195540853}{47184545236369}a^{22}-\frac{85849917356435}{47184545236369}a^{21}+\frac{320605226475597}{47184545236369}a^{20}-\frac{339301825317114}{47184545236369}a^{19}+\frac{13\!\cdots\!79}{47184545236369}a^{18}+\frac{35\!\cdots\!68}{47184545236369}a^{17}-\frac{17\!\cdots\!95}{47184545236369}a^{16}+\frac{14\!\cdots\!53}{47184545236369}a^{15}+\frac{71\!\cdots\!03}{47184545236369}a^{14}+\frac{59\!\cdots\!22}{47184545236369}a^{13}-\frac{58\!\cdots\!87}{47184545236369}a^{12}-\frac{97\!\cdots\!98}{47184545236369}a^{11}-\frac{24\!\cdots\!22}{47184545236369}a^{10}-\frac{24\!\cdots\!22}{47184545236369}a^{9}+\frac{18\!\cdots\!35}{47184545236369}a^{8}+\frac{94\!\cdots\!82}{597272724511}a^{7}+\frac{18\!\cdots\!45}{47184545236369}a^{6}-\frac{34\!\cdots\!94}{47184545236369}a^{5}-\frac{10\!\cdots\!33}{47184545236369}a^{4}-\frac{12\!\cdots\!79}{47184545236369}a^{3}-\frac{10\!\cdots\!93}{47184545236369}a^{2}+\frac{21140648257947}{47184545236369}a+\frac{25215743178838}{47184545236369}$, $\frac{34566346739885}{47184545236369}a^{23}-\frac{43800599451775}{47184545236369}a^{22}-\frac{126791478229009}{47184545236369}a^{21}+\frac{553270371023104}{47184545236369}a^{20}-\frac{700806715952248}{47184545236369}a^{19}+\frac{23\!\cdots\!13}{47184545236369}a^{18}+\frac{50\!\cdots\!56}{47184545236369}a^{17}-\frac{29\!\cdots\!59}{47184545236369}a^{16}+\frac{31\!\cdots\!08}{47184545236369}a^{15}+\frac{10\!\cdots\!77}{47184545236369}a^{14}+\frac{67\!\cdots\!98}{47184545236369}a^{13}-\frac{11\!\cdots\!98}{47184545236369}a^{12}-\frac{12\!\cdots\!70}{47184545236369}a^{11}-\frac{11\!\cdots\!65}{1522082104399}a^{10}-\frac{30\!\cdots\!14}{47184545236369}a^{9}+\frac{38\!\cdots\!49}{47184545236369}a^{8}+\frac{10\!\cdots\!26}{47184545236369}a^{7}+\frac{63\!\cdots\!28}{47184545236369}a^{6}-\frac{73\!\cdots\!13}{47184545236369}a^{5}-\frac{21\!\cdots\!21}{47184545236369}a^{4}-\frac{61\!\cdots\!29}{47184545236369}a^{3}-\frac{15\!\cdots\!29}{47184545236369}a^{2}+\frac{1416909885283}{1522082104399}a-\frac{46242328411883}{47184545236369}$, $\frac{14706561848633}{47184545236369}a^{23}-\frac{20217400345340}{47184545236369}a^{22}-\frac{54183199520459}{47184545236369}a^{21}+\frac{243862804835442}{47184545236369}a^{20}-\frac{315065376974897}{47184545236369}a^{19}+\frac{985826814536763}{47184545236369}a^{18}+\frac{66950630605674}{1522082104399}a^{17}-\frac{12\!\cdots\!09}{47184545236369}a^{16}+\frac{14\!\cdots\!44}{47184545236369}a^{15}+\frac{46\!\cdots\!66}{47184545236369}a^{14}+\frac{21\!\cdots\!93}{47184545236369}a^{13}-\frac{58\!\cdots\!98}{47184545236369}a^{12}-\frac{53\!\cdots\!36}{47184545236369}a^{11}-\frac{14\!\cdots\!59}{47184545236369}a^{10}-\frac{10\!\cdots\!46}{47184545236369}a^{9}+\frac{20\!\cdots\!37}{47184545236369}a^{8}+\frac{47\!\cdots\!83}{47184545236369}a^{7}-\frac{22\!\cdots\!13}{1522082104399}a^{6}-\frac{76\!\cdots\!58}{47184545236369}a^{5}-\frac{72\!\cdots\!52}{47184545236369}a^{4}-\frac{26\!\cdots\!46}{47184545236369}a^{3}+\frac{854253280128427}{47184545236369}a^{2}+\frac{260330213700834}{47184545236369}a+\frac{166262375129177}{47184545236369}$, $\frac{29081816889515}{47184545236369}a^{23}-\frac{37240803331479}{47184545236369}a^{22}-\frac{107522864877936}{47184545236369}a^{21}+\frac{468950472160405}{47184545236369}a^{20}-\frac{19073504891331}{1522082104399}a^{19}+\frac{19\!\cdots\!07}{47184545236369}a^{18}+\frac{42\!\cdots\!97}{47184545236369}a^{17}-\frac{24\!\cdots\!10}{47184545236369}a^{16}+\frac{26\!\cdots\!72}{47184545236369}a^{15}+\frac{91\!\cdots\!13}{47184545236369}a^{14}+\frac{53\!\cdots\!98}{47184545236369}a^{13}-\frac{10\!\cdots\!74}{47184545236369}a^{12}-\frac{10\!\cdots\!65}{47184545236369}a^{11}-\frac{29\!\cdots\!30}{47184545236369}a^{10}-\frac{24\!\cdots\!25}{47184545236369}a^{9}+\frac{33\!\cdots\!81}{47184545236369}a^{8}+\frac{92\!\cdots\!67}{47184545236369}a^{7}-\frac{25\!\cdots\!08}{47184545236369}a^{6}-\frac{45\!\cdots\!37}{47184545236369}a^{5}+\frac{96\!\cdots\!58}{47184545236369}a^{4}-\frac{29\!\cdots\!55}{47184545236369}a^{3}-\frac{69445137277373}{1522082104399}a^{2}+\frac{258246325034767}{47184545236369}a+\frac{119406998516247}{47184545236369}$, $\frac{17681638838230}{47184545236369}a^{23}-\frac{21317880662785}{47184545236369}a^{22}-\frac{66962128930794}{47184545236369}a^{21}+\frac{279347824064074}{47184545236369}a^{20}-\frac{337584163432212}{47184545236369}a^{19}+\frac{11\!\cdots\!27}{47184545236369}a^{18}+\frac{26\!\cdots\!03}{47184545236369}a^{17}-\frac{14\!\cdots\!74}{47184545236369}a^{16}+\frac{14\!\cdots\!66}{47184545236369}a^{15}+\frac{56\!\cdots\!53}{47184545236369}a^{14}+\frac{12\!\cdots\!33}{1522082104399}a^{13}-\frac{58\!\cdots\!18}{47184545236369}a^{12}-\frac{71\!\cdots\!15}{47184545236369}a^{11}-\frac{18\!\cdots\!95}{47184545236369}a^{10}-\frac{16\!\cdots\!95}{47184545236369}a^{9}+\frac{19\!\cdots\!05}{47184545236369}a^{8}+\frac{58\!\cdots\!48}{47184545236369}a^{7}+\frac{34\!\cdots\!52}{47184545236369}a^{6}-\frac{34\!\cdots\!98}{47184545236369}a^{5}-\frac{20\!\cdots\!61}{47184545236369}a^{4}-\frac{369360458103010}{47184545236369}a^{3}+\frac{847594582461239}{47184545236369}a^{2}+\frac{194748050310103}{47184545236369}a-\frac{10650494274975}{47184545236369}$, $\frac{5820406395990}{47184545236369}a^{23}-\frac{7856837276265}{47184545236369}a^{22}-\frac{19613971007635}{47184545236369}a^{21}+\frac{93112556140467}{47184545236369}a^{20}-\frac{129291920866840}{47184545236369}a^{19}+\frac{420535637601068}{47184545236369}a^{18}+\frac{784959033902761}{47184545236369}a^{17}-\frac{49\!\cdots\!32}{47184545236369}a^{16}+\frac{57\!\cdots\!20}{47184545236369}a^{15}+\frac{16\!\cdots\!66}{47184545236369}a^{14}+\frac{11\!\cdots\!63}{47184545236369}a^{13}-\frac{17\!\cdots\!35}{47184545236369}a^{12}-\frac{18\!\cdots\!05}{47184545236369}a^{11}-\frac{63\!\cdots\!90}{47184545236369}a^{10}-\frac{48\!\cdots\!50}{47184545236369}a^{9}+\frac{58\!\cdots\!01}{47184545236369}a^{8}+\frac{17\!\cdots\!80}{47184545236369}a^{7}+\frac{10\!\cdots\!69}{47184545236369}a^{6}+\frac{29\!\cdots\!87}{47184545236369}a^{5}-\frac{11\!\cdots\!64}{47184545236369}a^{4}+\frac{518991786349466}{47184545236369}a^{3}+\frac{166648034145941}{47184545236369}a^{2}-\frac{165596094864507}{47184545236369}a+\frac{67093450869534}{47184545236369}$ 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:  \( 183169998.253836 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 183169998.253836 \cdot 16}{30\cdot\sqrt{1348990128329918967746280670166015625}}\cr\approx \mathstrut & 0.318424960576314 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 4*x^22 + 15*x^21 - 16*x^20 + 62*x^19 + 163*x^18 - 809*x^17 + 670*x^16 + 3344*x^15 + 2796*x^14 - 2771*x^13 - 4594*x^12 - 11446*x^11 - 11444*x^10 + 8739*x^9 + 34835*x^8 + 8891*x^7 - 162*x^6 - 698*x^5 - 196*x^4 - 50*x^3 + x^2 + 4*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^24 - x^23 - 4*x^22 + 15*x^21 - 16*x^20 + 62*x^19 + 163*x^18 - 809*x^17 + 670*x^16 + 3344*x^15 + 2796*x^14 - 2771*x^13 - 4594*x^12 - 11446*x^11 - 11444*x^10 + 8739*x^9 + 34835*x^8 + 8891*x^7 - 162*x^6 - 698*x^5 - 196*x^4 - 50*x^3 + x^2 + 4*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^24 - x^23 - 4*x^22 + 15*x^21 - 16*x^20 + 62*x^19 + 163*x^18 - 809*x^17 + 670*x^16 + 3344*x^15 + 2796*x^14 - 2771*x^13 - 4594*x^12 - 11446*x^11 - 11444*x^10 + 8739*x^9 + 34835*x^8 + 8891*x^7 - 162*x^6 - 698*x^5 - 196*x^4 - 50*x^3 + x^2 + 4*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^24 - x^23 - 4*x^22 + 15*x^21 - 16*x^20 + 62*x^19 + 163*x^18 - 809*x^17 + 670*x^16 + 3344*x^15 + 2796*x^14 - 2771*x^13 - 4594*x^12 - 11446*x^11 - 11444*x^10 + 8739*x^9 + 34835*x^8 + 8891*x^7 - 162*x^6 - 698*x^5 - 196*x^4 - 50*x^3 + x^2 + 4*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

$C_2\times C_{12}$ (as 24T2):

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 $C_2\times C_{12}$
Character table for $C_2\times C_{12}$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), 3.3.169.1, \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), 6.0.771147.1, 6.6.3570125.1, 6.0.96393375.1, \(\Q(\zeta_{15})\), 12.0.9291682743890625.1, 12.12.1161460342986328125.1, 12.0.1593224064453125.1

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 ${\href{/padicField/2.12.0.1}{12} }^{2}$ R R ${\href{/padicField/7.12.0.1}{12} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{4}$ R ${\href{/padicField/17.12.0.1}{12} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{4}$ ${\href{/padicField/23.12.0.1}{12} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{4}$ ${\href{/padicField/31.1.0.1}{1} }^{24}$ ${\href{/padicField/37.12.0.1}{12} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }^{4}$ ${\href{/padicField/43.12.0.1}{12} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{6}$ ${\href{/padicField/53.4.0.1}{4} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display Deg $24$$2$$12$$12$
\(5\) Copy content Toggle raw display 5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
\(13\) Copy content Toggle raw display 13.12.8.1$x^{12} + 9 x^{10} + 88 x^{9} + 33 x^{8} + 216 x^{7} - 1299 x^{6} - 78 x^{5} - 1797 x^{4} - 15494 x^{3} + 21687 x^{2} - 41586 x + 201846$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
13.12.8.1$x^{12} + 9 x^{10} + 88 x^{9} + 33 x^{8} + 216 x^{7} - 1299 x^{6} - 78 x^{5} - 1797 x^{4} - 15494 x^{3} + 21687 x^{2} - 41586 x + 201846$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$