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Properties

Label 24.0.724...625.1
Degree $24$
Signature $[0, 12]$
Discriminant $7.250\times 10^{34}$
Root discriminant \(28.35\)
Ramified primes $5,13$
Class number $16$ (GRH)
Class group [2, 2, 2, 2] (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 + 6*x^22 - 7*x^21 + 27*x^20 - 19*x^19 + 94*x^18 - 58*x^17 + 317*x^16 - 205*x^15 + 643*x^14 - 353*x^13 + 1114*x^12 - 86*x^11 + 1318*x^10 - 237*x^9 + 1501*x^8 - 593*x^7 + 614*x^6 - 193*x^5 + 250*x^4 + 59*x^3 + 15*x^2 + 3*x + 1)
 
gp: K = bnfinit(y^24 - y^23 + 6*y^22 - 7*y^21 + 27*y^20 - 19*y^19 + 94*y^18 - 58*y^17 + 317*y^16 - 205*y^15 + 643*y^14 - 353*y^13 + 1114*y^12 - 86*y^11 + 1318*y^10 - 237*y^9 + 1501*y^8 - 593*y^7 + 614*y^6 - 193*y^5 + 250*y^4 + 59*y^3 + 15*y^2 + 3*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - x^23 + 6*x^22 - 7*x^21 + 27*x^20 - 19*x^19 + 94*x^18 - 58*x^17 + 317*x^16 - 205*x^15 + 643*x^14 - 353*x^13 + 1114*x^12 - 86*x^11 + 1318*x^10 - 237*x^9 + 1501*x^8 - 593*x^7 + 614*x^6 - 193*x^5 + 250*x^4 + 59*x^3 + 15*x^2 + 3*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - x^23 + 6*x^22 - 7*x^21 + 27*x^20 - 19*x^19 + 94*x^18 - 58*x^17 + 317*x^16 - 205*x^15 + 643*x^14 - 353*x^13 + 1114*x^12 - 86*x^11 + 1318*x^10 - 237*x^9 + 1501*x^8 - 593*x^7 + 614*x^6 - 193*x^5 + 250*x^4 + 59*x^3 + 15*x^2 + 3*x + 1)
 

\( x^{24} - x^{23} + 6 x^{22} - 7 x^{21} + 27 x^{20} - 19 x^{19} + 94 x^{18} - 58 x^{17} + 317 x^{16} + \cdots + 1 \) 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:   \(72498183345339963679508209228515625\) \(\medspace = 5^{18}\cdot 13^{20}\) 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:  \(28.35\)
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:  $5^{3/4}13^{5/6}\approx 28.34742599459173$
Ramified primes:   \(5\), \(13\) 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:  \(65=5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{65}(64,·)$, $\chi_{65}(1,·)$, $\chi_{65}(3,·)$, $\chi_{65}(4,·)$, $\chi_{65}(9,·)$, $\chi_{65}(12,·)$, $\chi_{65}(14,·)$, $\chi_{65}(16,·)$, $\chi_{65}(17,·)$, $\chi_{65}(22,·)$, $\chi_{65}(23,·)$, $\chi_{65}(27,·)$, $\chi_{65}(29,·)$, $\chi_{65}(36,·)$, $\chi_{65}(38,·)$, $\chi_{65}(42,·)$, $\chi_{65}(43,·)$, $\chi_{65}(48,·)$, $\chi_{65}(49,·)$, $\chi_{65}(51,·)$, $\chi_{65}(53,·)$, $\chi_{65}(56,·)$, $\chi_{65}(61,·)$, $\chi_{65}(62,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{1445643759919}a^{21}-\frac{545276790517}{1445643759919}a^{20}+\frac{615546033350}{1445643759919}a^{19}-\frac{450642466093}{1445643759919}a^{18}-\frac{98378289150}{1445643759919}a^{17}+\frac{31198496828}{1445643759919}a^{16}+\frac{529838603505}{1445643759919}a^{15}+\frac{113352996699}{1445643759919}a^{14}-\frac{564197865009}{1445643759919}a^{13}+\frac{16331163075}{1445643759919}a^{12}-\frac{15567765193}{1445643759919}a^{11}-\frac{141867390675}{1445643759919}a^{10}-\frac{300826242972}{1445643759919}a^{9}-\frac{208465917818}{1445643759919}a^{8}+\frac{508367897020}{1445643759919}a^{7}-\frac{636551380589}{1445643759919}a^{6}-\frac{107303772139}{1445643759919}a^{5}-\frac{700355649864}{1445643759919}a^{4}+\frac{674570434851}{1445643759919}a^{3}+\frac{371462421767}{1445643759919}a^{2}-\frac{685643609770}{1445643759919}a-\frac{195689327083}{1445643759919}$, $\frac{1}{1445643759919}a^{22}-\frac{11518196924}{1445643759919}a^{20}-\frac{467276115929}{1445643759919}a^{19}+\frac{409685131303}{1445643759919}a^{18}-\frac{557375797327}{1445643759919}a^{17}+\frac{149281353153}{1445643759919}a^{16}+\frac{386299897432}{1445643759919}a^{15}+\frac{514143645886}{1445643759919}a^{14}-\frac{674526751747}{1445643759919}a^{13}+\frac{150765032714}{1445643759919}a^{12}-\frac{496141072407}{1445643759919}a^{11}+\frac{123635468964}{1445643759919}a^{10}-\frac{484124230646}{1445643759919}a^{9}-\frac{694772769238}{1445643759919}a^{8}+\frac{376286591203}{1445643759919}a^{7}+\frac{650871630636}{1445643759919}a^{6}+\frac{443994073545}{1445643759919}a^{5}-\frac{4893280191}{1445643759919}a^{4}-\frac{166106579447}{1445643759919}a^{3}-\frac{530482488819}{1445643759919}a^{2}+\frac{166490853936}{1445643759919}a-\frac{290199457394}{1445643759919}$, $\frac{1}{1445643759919}a^{23}-\frac{2921529482}{1445643759919}a^{20}+\frac{624776915251}{1445643759919}a^{19}+\frac{149734378737}{1445643759919}a^{18}-\frac{585348258765}{1445643759919}a^{17}-\frac{94937716659}{1445643759919}a^{16}-\frac{298676141967}{1445643759919}a^{15}-\frac{61454505577}{1445643759919}a^{14}+\frac{387113597936}{1445643759919}a^{13}+\frac{543085858416}{1445643759919}a^{12}-\frac{36957153463}{1445643759919}a^{11}+\frac{351554201768}{1445643759919}a^{10}+\frac{528222622044}{1445643759919}a^{9}+\frac{694909117606}{1445643759919}a^{8}+\frac{643944498019}{1445643759919}a^{7}+\frac{363349832697}{1445643759919}a^{6}-\frac{138138769758}{1445643759919}a^{5}-\frac{669101080326}{1445643759919}a^{4}-\frac{707601791392}{1445643759919}a^{3}-\frac{435431176893}{1445643759919}a^{2}-\frac{115414909970}{1445643759919}a+\frac{577837003617}{1445643759919}$ 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

$C_{2}\times C_{2}\times C_{2}\times C_{2}$, 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:   \( \frac{341728824381}{1445643759919} a^{23} - \frac{336966501786}{1445643759919} a^{22} + \frac{2049420481767}{1445643759919} a^{21} - \frac{2373052480287}{1445643759919} a^{20} + \frac{9219314197064}{1445643759919} a^{19} - \frac{6421412824314}{1445643759919} a^{18} + \frac{32156798214498}{1445643759919} a^{17} - \frac{149416667967}{11035448549} a^{16} + \frac{108434713354905}{1445643759919} a^{15} - \frac{69218274924038}{1445643759919} a^{14} + \frac{220007848787493}{1445643759919} a^{13} - \frac{119905449507534}{1445643759919} a^{12} + \frac{381665043885966}{1445643759919} a^{11} - \frac{28215242609358}{1445643759919} a^{10} + \frac{454112955994722}{1445643759919} a^{9} - \frac{79785816226281}{1445643759919} a^{8} + \frac{513799803179133}{1445643759919} a^{7} - \frac{202625191103034}{1445643759919} a^{6} + \frac{210222485732433}{1445643759919} a^{5} - \frac{73055463437345}{1445643759919} a^{4} + \frac{85726517631621}{1445643759919} a^{3} + \frac{20232483012885}{1445643759919} a^{2} + \frac{5143076727057}{1445643759919} a + \frac{1028996331219}{1445643759919} \)  (order $10$) 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:   $\frac{341728824381}{1445643759919}a^{23}-\frac{336966501786}{1445643759919}a^{22}+\frac{2049420481767}{1445643759919}a^{21}-\frac{2373052480287}{1445643759919}a^{20}+\frac{9219314197064}{1445643759919}a^{19}-\frac{6421412824314}{1445643759919}a^{18}+\frac{32156798214498}{1445643759919}a^{17}-\frac{149416667967}{11035448549}a^{16}+\frac{108434713354905}{1445643759919}a^{15}-\frac{69218274924038}{1445643759919}a^{14}+\frac{220007848787493}{1445643759919}a^{13}-\frac{119905449507534}{1445643759919}a^{12}+\frac{381665043885966}{1445643759919}a^{11}-\frac{28215242609358}{1445643759919}a^{10}+\frac{454112955994722}{1445643759919}a^{9}-\frac{79785816226281}{1445643759919}a^{8}+\frac{513799803179133}{1445643759919}a^{7}-\frac{202625191103034}{1445643759919}a^{6}+\frac{210222485732433}{1445643759919}a^{5}-\frac{73055463437345}{1445643759919}a^{4}+\frac{85726517631621}{1445643759919}a^{3}+\frac{20232483012885}{1445643759919}a^{2}+\frac{5143076727057}{1445643759919}a-\frac{416647428700}{1445643759919}$, $\frac{23800046124}{1445643759919}a^{23}-\frac{59500115310}{1445643759919}a^{22}+\frac{160598317872}{1445643759919}a^{21}-\frac{357000691860}{1445643759919}a^{20}+\frac{5950011531}{11035448549}a^{19}-\frac{1255452433041}{1445643759919}a^{18}+\frac{2391904635462}{1445643759919}a^{17}-\frac{4237084307444}{1445643759919}a^{16}+\frac{7824265163265}{1445643759919}a^{15}-\frac{14601328297074}{1445643759919}a^{14}+\frac{16618382206083}{1445643759919}a^{13}-\frac{25811150021478}{1445643759919}a^{12}+\frac{26448673765379}{1445643759919}a^{11}-\frac{31689761414106}{1445643759919}a^{10}+\frac{12453374134383}{1445643759919}a^{9}-\frac{44529886298004}{1445643759919}a^{8}+\frac{20146739043966}{1445643759919}a^{7}-\frac{55419840141852}{1445643759919}a^{6}+\frac{7580314690494}{1445643759919}a^{5}-\frac{6997213560456}{1445643759919}a^{4}-\frac{1648153194087}{1445643759919}a^{3}-\frac{422450818701}{1445643759919}a^{2}-\frac{8508246512854}{1445643759919}a-\frac{29750057655}{1445643759919}$, $\frac{366543209913}{1445643759919}a^{23}-\frac{487481277512}{1445643759919}a^{22}+\frac{2315536068182}{1445643759919}a^{21}-\frac{3290498623206}{1445643759919}a^{20}+\frac{10724588105264}{1445643759919}a^{19}-\frac{10221972158361}{1445643759919}a^{18}+\frac{36682966132778}{1445643759919}a^{17}-\frac{32661245593304}{1445643759919}a^{16}+\frac{122967152252402}{1445643759919}a^{15}-\frac{113583137660296}{1445643759919}a^{14}+\frac{259670999359786}{1445643759919}a^{13}-\frac{207423283581356}{1445643759919}a^{12}+\frac{450310830101710}{1445643759919}a^{11}-\frac{167206078186616}{1445643759919}a^{10}+\frac{492356090287120}{1445643759919}a^{9}-\frac{250067337946692}{1445643759919}a^{8}+\frac{577665313851720}{1445643759919}a^{7}-\frac{399734647559840}{1445643759919}a^{6}+\frac{296754227685430}{1445643759919}a^{5}-\frac{145391291017954}{1445643759919}a^{4}+\frac{119719212855547}{1445643759919}a^{3}-\frac{8896533314594}{1445643759919}a^{2}-\frac{1706184471292}{1445643759919}a-\frac{731221916268}{1445643759919}$, $\frac{61702345944}{1445643759919}a^{23}-\frac{154255864860}{1445643759919}a^{22}+\frac{416779709735}{1445643759919}a^{21}-\frac{925535189160}{1445643759919}a^{20}+\frac{15425586486}{11035448549}a^{19}-\frac{3254798748546}{1445643759919}a^{18}+\frac{6201085767372}{1445643759919}a^{17}-\frac{10978862175035}{1445643759919}a^{16}+\frac{20284646229090}{1445643759919}a^{15}-\frac{37854389236644}{1445643759919}a^{14}+\frac{43083663055398}{1445643759919}a^{13}-\frac{66916194176268}{1445643759919}a^{12}+\frac{68359141405126}{1445643759919}a^{11}-\frac{82156673624436}{1445643759919}a^{10}+\frac{32285752515198}{1445643759919}a^{9}-\frac{115445089261224}{1445643759919}a^{8}+\frac{52231035841596}{1445643759919}a^{7}-\frac{144875792187714}{1445643759919}a^{6}+\frac{19652197183164}{1445643759919}a^{5}-\frac{18140489707536}{1445643759919}a^{4}-\frac{4272887456622}{1445643759919}a^{3}-\frac{1095216640506}{1445643759919}a^{2}-\frac{16752187654821}{1445643759919}a-\frac{77127932430}{1445643759919}$, $\frac{107121345224}{1445643759919}a^{23}-\frac{140813873260}{1445643759919}a^{22}+\frac{668865897985}{1445643759919}a^{21}-\frac{950493644505}{1445643759919}a^{20}+\frac{3097905211720}{1445643759919}a^{19}-\frac{2932515970529}{1445643759919}a^{18}+\frac{10596243962815}{1445643759919}a^{17}-\frac{9434529508420}{1445643759919}a^{16}+\frac{35520299529835}{1445643759919}a^{15}-\frac{32809632469580}{1445643759919}a^{14}+\frac{74474774202475}{1445643759919}a^{13}-\frac{59916303072130}{1445643759919}a^{12}+\frac{130076815423925}{1445643759919}a^{11}-\frac{48299158528180}{1445643759919}a^{10}+\frac{142222011992600}{1445643759919}a^{9}-\frac{76003908413335}{1445643759919}a^{8}+\frac{166864439813100}{1445643759919}a^{7}-\frac{115467376073200}{1445643759919}a^{6}+\frac{85720445347025}{1445643759919}a^{5}-\frac{41997737699795}{1445643759919}a^{4}+\frac{41250025664750}{1445643759919}a^{3}-\frac{2569853186995}{1445643759919}a^{2}-\frac{492848556410}{1445643759919}a-\frac{211220809890}{1445643759919}$, $\frac{162724182402}{1445643759919}a^{23}-\frac{82165998893}{1445643759919}a^{22}+\frac{867862306144}{1445643759919}a^{21}-\frac{623776032541}{1445643759919}a^{20}+\frac{3661294104045}{1445643759919}a^{19}-\frac{705138123742}{1445643759919}a^{18}+\frac{13005369817665}{1445643759919}a^{17}-\frac{1274672762149}{1445643759919}a^{16}+\frac{44315219007478}{1445643759919}a^{15}-\frac{5993674051807}{1445643759919}a^{14}+\frac{79355159618042}{1445643759919}a^{13}+\frac{785431184874}{1445643759919}a^{12}+\frac{135115312787794}{1445643759919}a^{11}+\frac{86216695975993}{1445643759919}a^{10}+\frac{177260876029912}{1445643759919}a^{9}+\frac{71001984921406}{1445643759919}a^{8}+\frac{191408561809412}{1445643759919}a^{7}+\frac{32056663933194}{1445643759919}a^{6}+\frac{10956761615068}{1445643759919}a^{5}+\frac{34633130154559}{1445643759919}a^{4}+\frac{8326053999569}{1445643759919}a^{3}+\frac{30126493681165}{1445643759919}a^{2}+\frac{461051850139}{1445643759919}a+\frac{108482788268}{1445643759919}$, $\frac{449330934273}{1445643759919}a^{23}-\frac{390503474820}{1445643759919}a^{22}+\frac{2623298401191}{1445643759919}a^{21}-\frac{2785527234873}{1445643759919}a^{20}+\frac{11640361669634}{1445643759919}a^{19}-\frac{6887688633846}{1445643759919}a^{18}+\frac{40769036854133}{1445643759919}a^{17}-\frac{20416466697831}{1445643759919}a^{16}+\frac{137738354615493}{1445643759919}a^{15}-\frac{73181619305060}{1445643759919}a^{14}+\frac{272481811044825}{1445643759919}a^{13}-\frac{333527879938}{4026862841}a^{12}+\frac{471010662466290}{1445643759919}a^{11}+\frac{28795941948420}{1445643759919}a^{10}+\frac{571327521037074}{1445643759919}a^{9}-\frac{32835428943405}{1445643759919}a^{8}+\frac{638211076963946}{1445643759919}a^{7}-\frac{181427575454310}{1445643759919}a^{6}+\frac{217467694465161}{1445643759919}a^{5}-\frac{50154147715331}{1445643759919}a^{4}+\frac{91232158921095}{1445643759919}a^{3}+\frac{44897013477400}{1445643759919}a^{2}+\frac{5447949371751}{1445643759919}a-\frac{344912688772}{1445643759919}$, $\frac{357376924601}{1445643759919}a^{23}-\frac{398274967290}{1445643759919}a^{22}+\frac{2181959889701}{1445643759919}a^{21}-\frac{2736673179334}{1445643759919}a^{20}+\frac{9903024616414}{1445643759919}a^{19}-\frac{7825190757220}{1445643759919}a^{18}+\frac{34199243620634}{1445643759919}a^{17}-\frac{24285693198590}{1445643759919}a^{16}+\frac{115071779799288}{1445643759919}a^{15}-\frac{85296378470578}{1445643759919}a^{14}+\frac{236201906407687}{1445643759919}a^{13}-\frac{149321473375322}{1445643759919}a^{12}+\frac{407202600143037}{1445643759919}a^{11}-\frac{69792497470282}{1445643759919}a^{10}+\frac{3548930298488}{11035448549}a^{9}-\frac{128864946526849}{1445643759919}a^{8}+\frac{535171630213517}{1445643759919}a^{7}-\frac{263881937446232}{1445643759919}a^{6}+\frac{229428642945704}{1445643759919}a^{5}-\frac{81088425284909}{1445643759919}a^{4}+\frac{83832052140567}{1445643759919}a^{3}+\frac{19748764716907}{1445643759919}a^{2}-\frac{916180910172}{1445643759919}a+\frac{3546391886609}{1445643759919}$, $\frac{346667404252}{1445643759919}a^{23}-\frac{436806386315}{1445643759919}a^{22}+\frac{2170722096075}{1445643759919}a^{21}-\frac{2962297525642}{1445643759919}a^{20}+\frac{9987123652285}{1445643759919}a^{19}-\frac{8990523077029}{1445643759919}a^{18}+\frac{34266970102885}{1445643759919}a^{17}-\frac{28436837701038}{1445643759919}a^{16}+\frac{115038875652038}{1445643759919}a^{15}-\frac{99161471353000}{1445643759919}a^{14}+\frac{241127287431320}{1445643759919}a^{13}-\frac{178745037923465}{1445643759919}a^{12}+\frac{416830653972967}{1445643759919}a^{11}-\frac{127122980292854}{1445643759919}a^{10}+\frac{462714525896584}{1445643759919}a^{9}-\frac{195472084673788}{1445643759919}a^{8}+\frac{541230979764683}{1445643759919}a^{7}-\frac{336016028905999}{1445643759919}a^{6}+\frac{264358915771587}{1445643759919}a^{5}-\frac{117007274336706}{1445643759919}a^{4}+\frac{100101757335047}{1445643759919}a^{3}-\frac{1222016242382}{1445643759919}a^{2}+\frac{84930785563}{1445643759919}a+\frac{1185064518230}{1445643759919}$, $\frac{560164821745}{1445643759919}a^{23}-\frac{494850650501}{1445643759919}a^{22}+\frac{3264316792216}{1445643759919}a^{21}-\frac{3502946761759}{1445643759919}a^{20}+\frac{14487320350432}{1445643759919}a^{19}-\frac{8696790476311}{1445643759919}a^{18}+\frac{50627261278294}{1445643759919}a^{17}-\frac{25932820501777}{1445643759919}a^{16}+\frac{171048174413815}{1445643759919}a^{15}-\frac{92936601688847}{1445643759919}a^{14}+\frac{337539574511999}{1445643759919}a^{13}-\frac{151393194908788}{1445643759919}a^{12}+\frac{583126498928860}{1445643759919}a^{11}+\frac{30771370451185}{1445643759919}a^{10}+\frac{702533302841311}{1445643759919}a^{9}-\frac{52656886496247}{1445643759919}a^{8}+\frac{789315108970479}{1445643759919}a^{7}-\frac{236572211551579}{1445643759919}a^{6}+\frac{264076980464182}{1445643759919}a^{5}-\frac{59805933138279}{1445643759919}a^{4}+\frac{115357046798361}{1445643759919}a^{3}+\frac{49051145493662}{1445643759919}a^{2}+\frac{6893890708866}{1445643759919}a-\frac{415102848675}{1445643759919}$, $\frac{652843465257}{1445643759919}a^{23}-\frac{874500626972}{1445643759919}a^{22}+\frac{4194935107816}{1445643759919}a^{21}-\frac{5943054192800}{1445643759919}a^{20}+\frac{19489756665423}{1445643759919}a^{19}-\frac{18692616041505}{1445643759919}a^{18}+\frac{66937130556756}{1445643759919}a^{17}-\frac{59345415392459}{1445643759919}a^{16}+\frac{224550506038737}{1445643759919}a^{15}-\frac{205989749661902}{1445643759919}a^{14}+\frac{481297933410834}{1445643759919}a^{13}-\frac{379932716570478}{1445643759919}a^{12}+\frac{835639552283012}{1445643759919}a^{11}-\frac{312882134144923}{1445643759919}a^{10}+\frac{931738643311827}{1445643759919}a^{9}-\frac{435498262576355}{1445643759919}a^{8}+\frac{10\!\cdots\!84}{1445643759919}a^{7}-\frac{719289580628795}{1445643759919}a^{6}+\frac{603684732434376}{1445643759919}a^{5}-\frac{275578508047918}{1445643759919}a^{4}+\frac{220017114745168}{1445643759919}a^{3}-\frac{16863497328513}{1445643759919}a^{2}+\frac{9964898381878}{1445643759919}a+\frac{1769886870677}{1445643759919}$ 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:  \( 7346081.887826216 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 7346081.887826216 \cdot 16}{10\cdot\sqrt{72498183345339963679508209228515625}}\cr\approx \mathstrut & 0.165261087399787 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 6*x^22 - 7*x^21 + 27*x^20 - 19*x^19 + 94*x^18 - 58*x^17 + 317*x^16 - 205*x^15 + 643*x^14 - 353*x^13 + 1114*x^12 - 86*x^11 + 1318*x^10 - 237*x^9 + 1501*x^8 - 593*x^7 + 614*x^6 - 193*x^5 + 250*x^4 + 59*x^3 + 15*x^2 + 3*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 + 6*x^22 - 7*x^21 + 27*x^20 - 19*x^19 + 94*x^18 - 58*x^17 + 317*x^16 - 205*x^15 + 643*x^14 - 353*x^13 + 1114*x^12 - 86*x^11 + 1318*x^10 - 237*x^9 + 1501*x^8 - 593*x^7 + 614*x^6 - 193*x^5 + 250*x^4 + 59*x^3 + 15*x^2 + 3*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 + 6*x^22 - 7*x^21 + 27*x^20 - 19*x^19 + 94*x^18 - 58*x^17 + 317*x^16 - 205*x^15 + 643*x^14 - 353*x^13 + 1114*x^12 - 86*x^11 + 1318*x^10 - 237*x^9 + 1501*x^8 - 593*x^7 + 614*x^6 - 193*x^5 + 250*x^4 + 59*x^3 + 15*x^2 + 3*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 + 6*x^22 - 7*x^21 + 27*x^20 - 19*x^19 + 94*x^18 - 58*x^17 + 317*x^16 - 205*x^15 + 643*x^14 - 353*x^13 + 1114*x^12 - 86*x^11 + 1318*x^10 - 237*x^9 + 1501*x^8 - 593*x^7 + 614*x^6 - 193*x^5 + 250*x^4 + 59*x^3 + 15*x^2 + 3*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

\(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\zeta_{5})\), 4.0.21125.1, 6.6.46411625.1, 6.6.3570125.1, \(\Q(\zeta_{13})^+\), 8.0.446265625.1, 12.12.2154038935140625.1, 12.0.1593224064453125.1, 12.0.269254866892578125.1

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 ${\href{/padicField/2.12.0.1}{12} }^{2}$ ${\href{/padicField/3.12.0.1}{12} }^{2}$ R ${\href{/padicField/7.12.0.1}{12} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{4}$ R ${\href{/padicField/17.12.0.1}{12} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{4}$ ${\href{/padicField/23.12.0.1}{12} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{12}$ ${\href{/padicField/37.12.0.1}{12} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }^{4}$ ${\href{/padicField/43.12.0.1}{12} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{6}$ ${\href{/padicField/53.4.0.1}{4} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

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
\(5\) 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 Deg $24$$6$$4$$20$