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Properties

Label 24.0.673...625.1
Degree $24$
Signature $[0, 12]$
Discriminant $6.737\times 10^{31}$
Root discriminant \(21.19\)
Ramified primes $3,5,7$
Class number $1$ (GRH)
Class group trivial (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 - 2*x^22 + 5*x^21 - 4*x^20 + 8*x^19 + 15*x^18 - 59*x^17 + 26*x^16 + 114*x^15 + 34*x^14 - 119*x^13 - 10*x^12 - 196*x^11 - 198*x^10 + 289*x^9 + 559*x^8 - 307*x^7 + 22*x^6 + 46*x^5 - 22*x^4 + 12*x^3 - x^2 - 2*x + 1)
 
gp: K = bnfinit(y^24 - y^23 - 2*y^22 + 5*y^21 - 4*y^20 + 8*y^19 + 15*y^18 - 59*y^17 + 26*y^16 + 114*y^15 + 34*y^14 - 119*y^13 - 10*y^12 - 196*y^11 - 198*y^10 + 289*y^9 + 559*y^8 - 307*y^7 + 22*y^6 + 46*y^5 - 22*y^4 + 12*y^3 - y^2 - 2*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - x^23 - 2*x^22 + 5*x^21 - 4*x^20 + 8*x^19 + 15*x^18 - 59*x^17 + 26*x^16 + 114*x^15 + 34*x^14 - 119*x^13 - 10*x^12 - 196*x^11 - 198*x^10 + 289*x^9 + 559*x^8 - 307*x^7 + 22*x^6 + 46*x^5 - 22*x^4 + 12*x^3 - x^2 - 2*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - x^23 - 2*x^22 + 5*x^21 - 4*x^20 + 8*x^19 + 15*x^18 - 59*x^17 + 26*x^16 + 114*x^15 + 34*x^14 - 119*x^13 - 10*x^12 - 196*x^11 - 198*x^10 + 289*x^9 + 559*x^8 - 307*x^7 + 22*x^6 + 46*x^5 - 22*x^4 + 12*x^3 - x^2 - 2*x + 1)
 

\( x^{24} - x^{23} - 2 x^{22} + 5 x^{21} - 4 x^{20} + 8 x^{19} + 15 x^{18} - 59 x^{17} + 26 x^{16} + \cdots + 1 \) 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:   \(67372672480923938907623291015625\) \(\medspace = 3^{12}\cdot 5^{18}\cdot 7^{16}\) 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:  \(21.19\)
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:  $3^{1/2}5^{3/4}7^{2/3}\approx 21.192726037501743$
Ramified primes:   \(3\), \(5\), \(7\) 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:  \(105=3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{105}(64,·)$, $\chi_{105}(1,·)$, $\chi_{105}(2,·)$, $\chi_{105}(67,·)$, $\chi_{105}(4,·)$, $\chi_{105}(86,·)$, $\chi_{105}(71,·)$, $\chi_{105}(8,·)$, $\chi_{105}(74,·)$, $\chi_{105}(11,·)$, $\chi_{105}(79,·)$, $\chi_{105}(16,·)$, $\chi_{105}(22,·)$, $\chi_{105}(23,·)$, $\chi_{105}(88,·)$, $\chi_{105}(92,·)$, $\chi_{105}(29,·)$, $\chi_{105}(32,·)$, $\chi_{105}(37,·)$, $\chi_{105}(43,·)$, $\chi_{105}(44,·)$, $\chi_{105}(46,·)$, $\chi_{105}(53,·)$, $\chi_{105}(58,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{181}a^{18}+\frac{43}{181}a^{17}-\frac{48}{181}a^{15}-\frac{73}{181}a^{14}-\frac{23}{181}a^{13}+\frac{48}{181}a^{12}+\frac{65}{181}a^{11}+\frac{18}{181}a^{10}+\frac{49}{181}a^{9}+\frac{8}{181}a^{8}+\frac{62}{181}a^{7}+\frac{1}{181}a^{6}-\frac{22}{181}a^{5}-\frac{80}{181}a^{4}-\frac{1}{181}a^{3}+\frac{39}{181}a+\frac{48}{181}$, $\frac{1}{181}a^{19}-\frac{39}{181}a^{17}-\frac{48}{181}a^{16}+\frac{39}{181}a^{14}-\frac{49}{181}a^{13}-\frac{8}{181}a^{12}-\frac{62}{181}a^{11}-\frac{1}{181}a^{10}+\frac{73}{181}a^{9}+\frac{80}{181}a^{8}+\frac{50}{181}a^{7}-\frac{65}{181}a^{6}-\frac{39}{181}a^{5}+\frac{43}{181}a^{3}+\frac{39}{181}a^{2}-\frac{73}{181}$, $\frac{1}{2353}a^{20}-\frac{3}{2353}a^{19}+\frac{841}{2353}a^{17}+\frac{506}{2353}a^{16}-\frac{1109}{2353}a^{15}+\frac{969}{2353}a^{14}+\frac{509}{2353}a^{13}-\frac{881}{2353}a^{12}+\frac{186}{2353}a^{11}-\frac{127}{2353}a^{10}+\frac{505}{2353}a^{9}-\frac{59}{2353}a^{8}-\frac{1055}{2353}a^{7}+\frac{738}{2353}a^{6}-\frac{922}{2353}a^{5}+\frac{5}{13}a^{4}-\frac{310}{2353}a^{3}-\frac{9}{181}a^{2}+\frac{3}{13}a+\frac{100}{2353}$, $\frac{1}{184806973}a^{21}+\frac{26051}{184806973}a^{20}+\frac{32884}{184806973}a^{19}-\frac{111037}{184806973}a^{18}-\frac{28576544}{184806973}a^{17}+\frac{82830352}{184806973}a^{16}-\frac{79766532}{184806973}a^{15}-\frac{18675511}{184806973}a^{14}+\frac{31433071}{184806973}a^{13}-\frac{44362867}{184806973}a^{12}-\frac{45523675}{184806973}a^{11}+\frac{35106634}{184806973}a^{10}+\frac{28795990}{184806973}a^{9}+\frac{43507812}{184806973}a^{8}-\frac{9371057}{184806973}a^{7}+\frac{31125778}{184806973}a^{6}-\frac{54307542}{184806973}a^{5}+\frac{71515657}{184806973}a^{4}+\frac{22450640}{184806973}a^{3}+\frac{86998103}{184806973}a^{2}+\frac{86706103}{184806973}a+\frac{1004489}{184806973}$, $\frac{1}{184806973}a^{22}-\frac{27477}{184806973}a^{20}+\frac{31648}{184806973}a^{19}+\frac{3906}{14215921}a^{18}-\frac{7356750}{184806973}a^{17}+\frac{40972094}{184806973}a^{16}-\frac{41877687}{184806973}a^{15}-\frac{86032801}{184806973}a^{14}+\frac{63509435}{184806973}a^{13}+\frac{47239093}{184806973}a^{12}-\frac{75434023}{184806973}a^{11}+\frac{87678347}{184806973}a^{10}-\frac{74305633}{184806973}a^{9}+\frac{88509977}{184806973}a^{8}+\frac{8927168}{184806973}a^{7}+\frac{20183353}{184806973}a^{6}-\frac{42222595}{184806973}a^{5}-\frac{19460453}{184806973}a^{4}-\frac{1221923}{184806973}a^{3}-\frac{3667721}{14215921}a^{2}+\frac{15021030}{184806973}a+\frac{25166397}{184806973}$, $\frac{1}{184806973}a^{23}+\frac{12301}{184806973}a^{20}+\frac{225864}{184806973}a^{19}-\frac{326564}{184806973}a^{18}+\frac{60847156}{184806973}a^{17}+\frac{78740134}{184806973}a^{16}+\frac{34096128}{184806973}a^{15}-\frac{65004216}{184806973}a^{14}+\frac{91789795}{184806973}a^{13}+\frac{90467158}{184806973}a^{12}-\frac{62490772}{184806973}a^{11}-\frac{91927281}{184806973}a^{10}-\frac{91699176}{184806973}a^{9}+\frac{1223361}{184806973}a^{8}-\frac{53521058}{184806973}a^{7}-\frac{76466613}{184806973}a^{6}-\frac{92286394}{184806973}a^{5}-\frac{44212596}{184806973}a^{4}+\frac{5004964}{184806973}a^{3}+\frac{3851427}{14215921}a^{2}-\frac{64123171}{184806973}a+\frac{68503572}{184806973}$ 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

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{1800399}{184806973} a^{23} - \frac{2416974}{184806973} a^{22} + \frac{7818171}{184806973} a^{21} - \frac{7596204}{184806973} a^{20} - \frac{9634676}{184806973} a^{19} + \frac{17930001}{184806973} a^{18} - \frac{88860789}{184806973} a^{17} + \frac{59832438}{184806973} a^{16} + \frac{150000366}{184806973} a^{15} - \frac{436806393}{184806973} a^{14} - \frac{180705801}{184806973} a^{13} + \frac{32900442}{184806973} a^{12} - \frac{271588956}{184806973} a^{11} - \frac{201003450}{184806973} a^{10} + \frac{1771395312}{184806973} a^{9} + \frac{671795457}{184806973} a^{8} - \frac{652163709}{184806973} a^{7} + \frac{152072058}{184806973} a^{6} + \frac{55985010}{184806973} a^{5} - \frac{4737629686}{184806973} a^{4} + \frac{25550868}{184806973} a^{3} - \frac{6141087}{184806973} a^{2} - \frac{2416974}{184806973} a + \frac{2195007}{184806973} \)  (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:   $\frac{6266110}{14215921}a^{23}-\frac{3762082}{14215921}a^{22}-\frac{13785442}{14215921}a^{21}+\frac{25064440}{14215921}a^{20}-\frac{15038664}{14215921}a^{19}+\frac{46369214}{14215921}a^{18}+\frac{108983051}{14215921}a^{17}-\frac{322078054}{14215921}a^{16}+\frac{33836994}{14215921}a^{15}+\frac{705563986}{14215921}a^{14}+\frac{531366128}{14215921}a^{13}-\frac{518490542}{14215921}a^{12}-\frac{318318388}{14215921}a^{11}-\frac{1402355418}{14215921}a^{10}-\frac{1744485024}{14215921}a^{9}+\frac{1068998366}{14215921}a^{8}+\frac{3978486835}{14215921}a^{7}-\frac{160412416}{14215921}a^{6}+\frac{68927210}{14215921}a^{5}-\frac{41356326}{14215921}a^{4}+\frac{5012888}{14215921}a^{3}+\frac{77691157}{14215921}a^{2}-\frac{3759666}{14215921}a-1$, $\frac{72534337}{184806973}a^{23}-\frac{72374193}{184806973}a^{22}-\frac{161108546}{184806973}a^{21}+\frac{371905348}{184806973}a^{20}-\frac{254069505}{184806973}a^{19}+\frac{515723556}{184806973}a^{18}+\frac{1127374555}{184806973}a^{17}-\frac{4392950009}{184806973}a^{16}+\frac{8824587}{1021033}a^{15}+\frac{9094169086}{184806973}a^{14}+\frac{2403692715}{184806973}a^{13}-\frac{10432742742}{184806973}a^{12}-\frac{2066405585}{184806973}a^{11}-\frac{12887907150}{184806973}a^{10}-\frac{13598151732}{184806973}a^{9}+\frac{24531675932}{184806973}a^{8}+\frac{45089279343}{184806973}a^{7}-\frac{25034239411}{184806973}a^{6}-\frac{8651898109}{184806973}a^{5}+\frac{3753093834}{184806973}a^{4}-\frac{1775207193}{184806973}a^{3}+\frac{977458641}{184806973}a^{2}+\frac{329969306}{184806973}a-\frac{160478427}{184806973}$, $\frac{45222210}{184806973}a^{23}-\frac{27202670}{184806973}a^{22}-\frac{99488862}{184806973}a^{21}+\frac{180888840}{184806973}a^{20}-\frac{108533304}{184806973}a^{19}+\frac{334644354}{184806973}a^{18}+\frac{785509912}{184806973}a^{17}-\frac{2324421594}{184806973}a^{16}+\frac{244199934}{184806973}a^{15}+\frac{5092020846}{184806973}a^{14}+\frac{3834843408}{184806973}a^{13}-\frac{3760998495}{184806973}a^{12}-\frac{2297288268}{184806973}a^{11}-\frac{10120730598}{184806973}a^{10}-\frac{12589863264}{184806973}a^{9}+\frac{7714909026}{184806973}a^{8}+\frac{28701952840}{184806973}a^{7}-\frac{1157688576}{184806973}a^{6}+\frac{497444310}{184806973}a^{5}-\frac{298466586}{184806973}a^{4}+\frac{36177768}{184806973}a^{3}+\frac{560508366}{184806973}a^{2}-\frac{27133326}{184806973}a$, $\frac{7693632}{184806973}a^{23}-\frac{7693632}{184806973}a^{22}-\frac{2564544}{184806973}a^{21}+\frac{30721419}{184806973}a^{20}-\frac{58984512}{184806973}a^{19}+\frac{112839936}{184806973}a^{18}+\frac{84629952}{184806973}a^{17}-\frac{359036160}{184806973}a^{16}+\frac{422429888}{184806973}a^{15}+\frac{217986240}{184806973}a^{14}+\frac{330826176}{184806973}a^{13}+\frac{528296064}{184806973}a^{12}+\frac{1010430336}{184806973}a^{11}-\frac{2581191040}{184806973}a^{10}-\frac{2174733312}{184806973}a^{9}-\frac{646265088}{184806973}a^{8}+\frac{730895040}{184806973}a^{7}-\frac{174388992}{184806973}a^{6}+\frac{8386424877}{184806973}a^{5}+\frac{25645440}{184806973}a^{4}-\frac{28209984}{184806973}a^{3}+\frac{7693632}{184806973}a^{2}+\frac{2564544}{184806973}a-\frac{3284416}{184806973}$, $\frac{1800399}{184806973}a^{23}+\frac{2416974}{184806973}a^{22}-\frac{7818171}{184806973}a^{21}+\frac{7596204}{184806973}a^{20}+\frac{9634676}{184806973}a^{19}-\frac{17930001}{184806973}a^{18}+\frac{88860789}{184806973}a^{17}-\frac{59832438}{184806973}a^{16}-\frac{150000366}{184806973}a^{15}+\frac{436806393}{184806973}a^{14}+\frac{180705801}{184806973}a^{13}-\frac{32900442}{184806973}a^{12}+\frac{271588956}{184806973}a^{11}+\frac{201003450}{184806973}a^{10}-\frac{1771395312}{184806973}a^{9}-\frac{671795457}{184806973}a^{8}+\frac{652163709}{184806973}a^{7}-\frac{152072058}{184806973}a^{6}-\frac{55985010}{184806973}a^{5}+\frac{4737629686}{184806973}a^{4}-\frac{25550868}{184806973}a^{3}+\frac{6141087}{184806973}a^{2}+\frac{2416974}{184806973}a-\frac{187001980}{184806973}$, $\frac{174158095}{184806973}a^{23}-\frac{92175535}{184806973}a^{22}-\frac{31884486}{14215921}a^{21}+\frac{697218540}{184806973}a^{20}-\frac{321975139}{184806973}a^{19}+\frac{1129124488}{184806973}a^{18}+\frac{3230232853}{184806973}a^{17}-\frac{8930608743}{184806973}a^{16}-\frac{31516791}{184806973}a^{15}+\frac{21164292537}{184806973}a^{14}+\frac{15348963148}{184806973}a^{13}-\frac{16133765114}{184806973}a^{12}-\frac{10193791455}{184806973}a^{11}-\frac{36301888722}{184806973}a^{10}-\frac{51375864480}{184806973}a^{9}+\frac{30501906784}{184806973}a^{8}+\frac{116603023345}{184806973}a^{7}-\frac{4739111469}{184806973}a^{6}-\frac{11168758131}{184806973}a^{5}+\frac{9405979618}{184806973}a^{4}+\frac{130461573}{184806973}a^{3}+\frac{13384075}{14215921}a^{2}+\frac{410213718}{184806973}a-\frac{230427192}{184806973}$, $\frac{81963963}{184806973}a^{23}-\frac{3658758}{14215921}a^{22}-\frac{182117562}{184806973}a^{21}+\frac{327515660}{184806973}a^{20}-\frac{189134859}{184806973}a^{19}+\frac{45475527}{14215921}a^{18}+\frac{1450593897}{184806973}a^{17}-\frac{4190553573}{184806973}a^{16}+\frac{354779216}{184806973}a^{15}+\frac{9341112093}{184806973}a^{14}+\frac{6990758899}{184806973}a^{13}-\frac{6784242786}{184806973}a^{12}-\frac{3958542255}{184806973}a^{11}-\frac{18070150900}{184806973}a^{10}-\frac{23472904800}{184806973}a^{9}+\frac{13453534251}{184806973}a^{8}+\frac{52075523064}{184806973}a^{7}-\frac{2152526421}{184806973}a^{6}+\frac{859639676}{184806973}a^{5}+\frac{1576472536}{184806973}a^{4}+\frac{51229068}{184806973}a^{3}+\frac{79654155}{184806973}a^{2}+\frac{367974975}{184806973}a-\frac{1181493}{184806973}$, $\frac{95446170}{184806973}a^{23}-\frac{85343162}{184806973}a^{22}-\frac{202147068}{184806973}a^{21}+\frac{448343170}{184806973}a^{20}-\frac{324131111}{184806973}a^{19}+\frac{738385538}{184806973}a^{18}+\frac{1480866195}{184806973}a^{17}-\frac{5467058268}{184806973}a^{16}+\frac{1793436030}{184806973}a^{15}+\frac{11035755253}{184806973}a^{14}+\frac{4842743088}{184806973}a^{13}-\frac{11164117600}{184806973}a^{12}-\frac{3248184006}{184806973}a^{11}-\frac{19631082216}{184806973}a^{10}-\frac{20269387013}{184806973}a^{9}+\frac{26255203275}{184806973}a^{8}+\frac{58856435792}{184806973}a^{7}-\frac{20934882532}{184806973}a^{6}-\frac{2746910050}{184806973}a^{5}-\frac{1023274454}{184806973}a^{4}-\frac{2315032966}{184806973}a^{3}+\frac{67344048}{184806973}a^{2}-\frac{91455135}{184806973}a-\frac{210183668}{184806973}$, $\frac{163290223}{184806973}a^{23}-\frac{163240218}{184806973}a^{22}-\frac{362212886}{184806973}a^{21}+\frac{837035941}{184806973}a^{20}-\frac{572777992}{184806973}a^{19}+\frac{1163730805}{184806973}a^{18}+\frac{194578228}{14215921}a^{17}-\frac{9887504943}{184806973}a^{16}+\frac{3612580636}{184806973}a^{15}+\frac{20444861334}{184806973}a^{14}+\frac{5421496180}{184806973}a^{13}-\frac{1809790532}{14215921}a^{12}-\frac{4690623722}{184806973}a^{11}-\frac{28952592233}{184806973}a^{10}-\frac{2341750310}{14215921}a^{9}+\frac{54988251355}{184806973}a^{8}+\frac{101350815814}{184806973}a^{7}-\frac{56242396151}{184806973}a^{6}-\frac{19452365429}{184806973}a^{5}+\frac{8438396158}{184806973}a^{4}-\frac{3057248439}{184806973}a^{3}+\frac{1780659505}{184806973}a^{2}-\frac{191190881}{184806973}a-\frac{693730677}{184806973}$, $\frac{162435569}{184806973}a^{23}-\frac{98240416}{184806973}a^{22}-\frac{400074463}{184806973}a^{21}+\frac{691661219}{184806973}a^{20}-\frac{311499492}{184806973}a^{19}+\frac{5521884}{1021033}a^{18}+\frac{2994455458}{184806973}a^{17}-\frac{8729558592}{184806973}a^{16}+\frac{270099308}{184806973}a^{15}+\frac{20726524265}{184806973}a^{14}+\frac{12607720641}{184806973}a^{13}-\frac{18068644167}{184806973}a^{12}-\frac{10097473112}{184806973}a^{11}-\frac{32123976878}{184806973}a^{10}-\frac{44810615333}{184806973}a^{9}+\frac{37025160477}{184806973}a^{8}+\frac{112342981984}{184806973}a^{7}-\frac{14743619921}{184806973}a^{6}-\frac{21102165348}{184806973}a^{5}+\frac{9148070237}{184806973}a^{4}-\frac{4415601883}{184806973}a^{3}+\frac{565933915}{184806973}a^{2}-\frac{130297540}{184806973}a-\frac{481595425}{184806973}$, $\frac{19988895}{14215921}a^{23}-\frac{161399061}{184806973}a^{22}-\frac{602523590}{184806973}a^{21}+\frac{1091054836}{184806973}a^{20}-\frac{588224756}{184806973}a^{19}+\frac{1757459702}{184806973}a^{18}+\frac{4655681943}{184806973}a^{17}-\frac{13766877455}{184806973}a^{16}+\frac{94606286}{14215921}a^{15}+\frac{31281577434}{184806973}a^{14}+\frac{20117626803}{184806973}a^{13}-\frac{25419400915}{184806973}a^{12}-\frac{13311730993}{184806973}a^{11}-\frac{54188140752}{184806973}a^{10}-\frac{71966894719}{184806973}a^{9}+\frac{52842581672}{184806973}a^{8}+\frac{13074530741}{14215921}a^{7}-\frac{19824113496}{184806973}a^{6}-\frac{12779249216}{184806973}a^{5}+\frac{11150354301}{184806973}a^{4}-\frac{5515413129}{184806973}a^{3}+\frac{1693859380}{184806973}a^{2}+\frac{44959197}{184806973}a-\frac{605784256}{184806973}$ 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:  \( 8057321.833968681 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 8057321.833968681 \cdot 1}{30\cdot\sqrt{67372672480923938907623291015625}}\cr\approx \mathstrut & 0.123875662215965 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 2*x^22 + 5*x^21 - 4*x^20 + 8*x^19 + 15*x^18 - 59*x^17 + 26*x^16 + 114*x^15 + 34*x^14 - 119*x^13 - 10*x^12 - 196*x^11 - 198*x^10 + 289*x^9 + 559*x^8 - 307*x^7 + 22*x^6 + 46*x^5 - 22*x^4 + 12*x^3 - x^2 - 2*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 - 2*x^22 + 5*x^21 - 4*x^20 + 8*x^19 + 15*x^18 - 59*x^17 + 26*x^16 + 114*x^15 + 34*x^14 - 119*x^13 - 10*x^12 - 196*x^11 - 198*x^10 + 289*x^9 + 559*x^8 - 307*x^7 + 22*x^6 + 46*x^5 - 22*x^4 + 12*x^3 - x^2 - 2*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 - 2*x^22 + 5*x^21 - 4*x^20 + 8*x^19 + 15*x^18 - 59*x^17 + 26*x^16 + 114*x^15 + 34*x^14 - 119*x^13 - 10*x^12 - 196*x^11 - 198*x^10 + 289*x^9 + 559*x^8 - 307*x^7 + 22*x^6 + 46*x^5 - 22*x^4 + 12*x^3 - x^2 - 2*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 - 2*x^22 + 5*x^21 - 4*x^20 + 8*x^19 + 15*x^18 - 59*x^17 + 26*x^16 + 114*x^15 + 34*x^14 - 119*x^13 - 10*x^12 - 196*x^11 - 198*x^10 + 289*x^9 + 559*x^8 - 307*x^7 + 22*x^6 + 46*x^5 - 22*x^4 + 12*x^3 - x^2 - 2*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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), 6.0.64827.1, 6.6.300125.1, 6.0.8103375.1, \(\Q(\zeta_{15})\), 12.0.65664686390625.1, 12.12.8208085798828125.1, 12.0.11259376953125.1

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 R ${\href{/padicField/11.6.0.1}{6} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{6}$ ${\href{/padicField/17.12.0.1}{12} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{4}$ ${\href{/padicField/23.12.0.1}{12} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{12}$ ${\href{/padicField/31.3.0.1}{3} }^{8}$ ${\href{/padicField/37.12.0.1}{12} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{12}$ ${\href{/padicField/43.4.0.1}{4} }^{6}$ ${\href{/padicField/47.12.0.1}{12} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

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 Deg $24$$4$$6$$18$
\(7\) Copy content Toggle raw display 7.12.8.1$x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
7.12.8.1$x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$