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Properties

Label 24.0.246...625.1
Degree $24$
Signature $[0, 12]$
Discriminant $2.466\times 10^{36}$
Root discriminant \(32.83\)
Ramified primes $3,5,13$
Class number $64$ (GRH)
Class group [4, 4, 4] (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 17*x^22 - 6*x^21 + 188*x^20 - 47*x^19 + 1066*x^18 + 111*x^17 + 4083*x^16 + 396*x^15 + 8622*x^14 + 1407*x^13 + 12820*x^12 + 2029*x^11 + 11865*x^10 + 2619*x^9 + 7679*x^8 + 1542*x^7 + 2678*x^6 + 721*x^5 + 616*x^4 + 85*x^3 + 36*x^2 - 3*x + 1)
 
gp: K = bnfinit(y^24 - y^23 + 17*y^22 - 6*y^21 + 188*y^20 - 47*y^19 + 1066*y^18 + 111*y^17 + 4083*y^16 + 396*y^15 + 8622*y^14 + 1407*y^13 + 12820*y^12 + 2029*y^11 + 11865*y^10 + 2619*y^9 + 7679*y^8 + 1542*y^7 + 2678*y^6 + 721*y^5 + 616*y^4 + 85*y^3 + 36*y^2 - 3*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - x^23 + 17*x^22 - 6*x^21 + 188*x^20 - 47*x^19 + 1066*x^18 + 111*x^17 + 4083*x^16 + 396*x^15 + 8622*x^14 + 1407*x^13 + 12820*x^12 + 2029*x^11 + 11865*x^10 + 2619*x^9 + 7679*x^8 + 1542*x^7 + 2678*x^6 + 721*x^5 + 616*x^4 + 85*x^3 + 36*x^2 - 3*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - x^23 + 17*x^22 - 6*x^21 + 188*x^20 - 47*x^19 + 1066*x^18 + 111*x^17 + 4083*x^16 + 396*x^15 + 8622*x^14 + 1407*x^13 + 12820*x^12 + 2029*x^11 + 11865*x^10 + 2619*x^9 + 7679*x^8 + 1542*x^7 + 2678*x^6 + 721*x^5 + 616*x^4 + 85*x^3 + 36*x^2 - 3*x + 1)
 

\( x^{24} - x^{23} + 17 x^{22} - 6 x^{21} + 188 x^{20} - 47 x^{19} + 1066 x^{18} + 111 x^{17} + 4083 x^{16} + \cdots + 1 \) 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:   \(2465824451534772200819297422119140625\) \(\medspace = 3^{12}\cdot 5^{12}\cdot 13^{20}\) 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:  \(32.83\)
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:  $3^{1/2}5^{1/2}13^{5/6}\approx 32.83460200257314$
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$$\chi_{195}(64,·)$, $\chi_{195}(1,·)$, $\chi_{195}(194,·)$, $\chi_{195}(131,·)$, $\chi_{195}(4,·)$, $\chi_{195}(134,·)$, $\chi_{195}(74,·)$, $\chi_{195}(139,·)$, $\chi_{195}(14,·)$, $\chi_{195}(79,·)$, $\chi_{195}(16,·)$, $\chi_{195}(146,·)$, $\chi_{195}(29,·)$, $\chi_{195}(94,·)$, $\chi_{195}(101,·)$, $\chi_{195}(166,·)$, $\chi_{195}(49,·)$, $\chi_{195}(179,·)$, $\chi_{195}(116,·)$, $\chi_{195}(181,·)$, $\chi_{195}(56,·)$, $\chi_{195}(121,·)$, $\chi_{195}(61,·)$, $\chi_{195}(191,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{15}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{6201996740150}a^{22}-\frac{1205449772479}{6201996740150}a^{21}-\frac{713667446537}{6201996740150}a^{20}-\frac{413023687803}{3100998370075}a^{19}+\frac{30674675574}{3100998370075}a^{18}+\frac{2482592888}{620199674015}a^{17}-\frac{294193394217}{6201996740150}a^{16}-\frac{578797891043}{6201996740150}a^{15}-\frac{1906270011791}{6201996740150}a^{14}-\frac{867376474884}{3100998370075}a^{13}+\frac{200118271891}{3100998370075}a^{12}+\frac{599330056862}{3100998370075}a^{11}-\frac{585088476289}{6201996740150}a^{10}-\frac{784626013863}{6201996740150}a^{9}+\frac{929121692303}{6201996740150}a^{8}-\frac{922060619057}{6201996740150}a^{7}+\frac{1153340920727}{6201996740150}a^{6}+\frac{3026024584923}{6201996740150}a^{5}-\frac{2309913026223}{6201996740150}a^{4}-\frac{1685641053403}{6201996740150}a^{3}-\frac{2927334170207}{6201996740150}a^{2}-\frac{1448359772873}{3100998370075}a+\frac{336229445243}{3100998370075}$, $\frac{1}{17\!\cdots\!50}a^{23}+\frac{6755504845561}{35\!\cdots\!50}a^{22}+\frac{87\!\cdots\!77}{17\!\cdots\!50}a^{21}+\frac{57\!\cdots\!61}{17\!\cdots\!50}a^{20}+\frac{41\!\cdots\!19}{17\!\cdots\!50}a^{19}+\frac{20\!\cdots\!06}{88\!\cdots\!25}a^{18}+\frac{40\!\cdots\!03}{17\!\cdots\!50}a^{17}+\frac{77\!\cdots\!29}{17\!\cdots\!50}a^{16}-\frac{65\!\cdots\!53}{17\!\cdots\!50}a^{15}+\frac{17\!\cdots\!79}{37\!\cdots\!50}a^{14}+\frac{18\!\cdots\!69}{35\!\cdots\!50}a^{13}+\frac{40\!\cdots\!56}{88\!\cdots\!25}a^{12}-\frac{39\!\cdots\!73}{17\!\cdots\!50}a^{11}+\frac{42\!\cdots\!61}{17\!\cdots\!50}a^{10}+\frac{38\!\cdots\!11}{17\!\cdots\!50}a^{9}+\frac{31\!\cdots\!92}{17\!\cdots\!25}a^{8}-\frac{33\!\cdots\!43}{88\!\cdots\!25}a^{7}-\frac{87\!\cdots\!09}{17\!\cdots\!50}a^{6}-\frac{31\!\cdots\!33}{88\!\cdots\!25}a^{5}-\frac{10\!\cdots\!16}{17\!\cdots\!25}a^{4}+\frac{63\!\cdots\!41}{17\!\cdots\!50}a^{3}+\frac{73\!\cdots\!91}{17\!\cdots\!50}a^{2}+\frac{71\!\cdots\!97}{17\!\cdots\!50}a-\frac{62\!\cdots\!38}{88\!\cdots\!25}$ 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

$C_{4}\times C_{4}\times C_{4}$, which has order $64$ (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:   \( \frac{10316289652895263256781}{96906752646678726161750} a^{23} - \frac{702511008329429380021}{9690675264667872616175} a^{22} + \frac{171719279169693620878897}{96906752646678726161750} a^{21} - \frac{2907767028015160275802}{48453376323339363080875} a^{20} + \frac{1913697985924541771617479}{96906752646678726161750} a^{19} + \frac{132196697172520347171627}{96906752646678726161750} a^{18} + \frac{10774557893216937911354643}{96906752646678726161750} a^{17} + \frac{2314086029228867758909302}{48453376323339363080875} a^{16} + \frac{42102882135047228889049877}{96906752646678726161750} a^{15} + \frac{8624371902627437018808659}{48453376323339363080875} a^{14} + \frac{17740985810751019004126283}{19381350529335745232350} a^{13} + \frac{41869685808686134819061117}{96906752646678726161750} a^{12} + \frac{133574998648687952010455427}{96906752646678726161750} a^{11} + \frac{30447145078871001743141263}{48453376323339363080875} a^{10} + \frac{123966758091005610355196461}{96906752646678726161750} a^{9} + \frac{2516337352019889058505081}{3876270105867149046470} a^{8} + \frac{82974011953599154204950889}{96906752646678726161750} a^{7} + \frac{19150428489822785453530108}{48453376323339363080875} a^{6} + \frac{29322866671213926872900259}{96906752646678726161750} a^{5} + \frac{2916669400240715495399667}{19381350529335745232350} a^{4} + \frac{3757269742102012624289333}{48453376323339363080875} a^{3} + \frac{1257654603331363798135838}{48453376323339363080875} a^{2} + \frac{337333081682921137866397}{96906752646678726161750} a + \frac{78137336706218675139329}{96906752646678726161750} \)  (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:   $\frac{39\!\cdots\!47}{88\!\cdots\!25}a^{23}-\frac{76\!\cdots\!07}{17\!\cdots\!25}a^{22}+\frac{65\!\cdots\!74}{88\!\cdots\!25}a^{21}-\frac{20\!\cdots\!68}{88\!\cdots\!25}a^{20}+\frac{72\!\cdots\!63}{88\!\cdots\!25}a^{19}-\frac{15\!\cdots\!71}{88\!\cdots\!25}a^{18}+\frac{40\!\cdots\!91}{88\!\cdots\!25}a^{17}+\frac{64\!\cdots\!78}{88\!\cdots\!25}a^{16}+\frac{15\!\cdots\!44}{88\!\cdots\!25}a^{15}+\frac{22\!\cdots\!06}{88\!\cdots\!25}a^{14}+\frac{24\!\cdots\!14}{70\!\cdots\!01}a^{13}+\frac{72\!\cdots\!99}{88\!\cdots\!25}a^{12}+\frac{43\!\cdots\!39}{88\!\cdots\!25}a^{11}+\frac{10\!\cdots\!72}{88\!\cdots\!25}a^{10}+\frac{37\!\cdots\!52}{88\!\cdots\!25}a^{9}+\frac{26\!\cdots\!01}{17\!\cdots\!25}a^{8}+\frac{22\!\cdots\!98}{88\!\cdots\!25}a^{7}+\frac{75\!\cdots\!37}{88\!\cdots\!25}a^{6}+\frac{12\!\cdots\!79}{18\!\cdots\!75}a^{5}+\frac{71\!\cdots\!23}{17\!\cdots\!25}a^{4}+\frac{99\!\cdots\!87}{88\!\cdots\!25}a^{3}+\frac{28\!\cdots\!42}{88\!\cdots\!25}a^{2}-\frac{64\!\cdots\!96}{88\!\cdots\!25}a+\frac{90\!\cdots\!83}{88\!\cdots\!25}$, $\frac{35\!\cdots\!21}{17\!\cdots\!50}a^{23}-\frac{34\!\cdots\!47}{35\!\cdots\!50}a^{22}+\frac{58\!\cdots\!77}{17\!\cdots\!50}a^{21}+\frac{51\!\cdots\!93}{88\!\cdots\!25}a^{20}+\frac{32\!\cdots\!57}{88\!\cdots\!25}a^{19}+\frac{17\!\cdots\!57}{17\!\cdots\!50}a^{18}+\frac{36\!\cdots\!63}{17\!\cdots\!50}a^{17}+\frac{23\!\cdots\!89}{17\!\cdots\!50}a^{16}+\frac{14\!\cdots\!07}{17\!\cdots\!50}a^{15}+\frac{43\!\cdots\!44}{88\!\cdots\!25}a^{14}+\frac{30\!\cdots\!64}{17\!\cdots\!25}a^{13}+\frac{20\!\cdots\!47}{17\!\cdots\!50}a^{12}+\frac{45\!\cdots\!07}{17\!\cdots\!50}a^{11}+\frac{29\!\cdots\!91}{17\!\cdots\!50}a^{10}+\frac{41\!\cdots\!01}{17\!\cdots\!50}a^{9}+\frac{11\!\cdots\!31}{70\!\cdots\!10}a^{8}+\frac{28\!\cdots\!99}{17\!\cdots\!50}a^{7}+\frac{11\!\cdots\!07}{11\!\cdots\!75}a^{6}+\frac{98\!\cdots\!69}{17\!\cdots\!50}a^{5}+\frac{12\!\cdots\!97}{35\!\cdots\!50}a^{4}+\frac{13\!\cdots\!78}{88\!\cdots\!25}a^{3}+\frac{51\!\cdots\!33}{88\!\cdots\!25}a^{2}+\frac{51\!\cdots\!51}{88\!\cdots\!25}a-\frac{33\!\cdots\!11}{17\!\cdots\!50}$, $\frac{40\!\cdots\!09}{88\!\cdots\!25}a^{23}-\frac{11\!\cdots\!88}{17\!\cdots\!25}a^{22}+\frac{70\!\cdots\!58}{88\!\cdots\!25}a^{21}-\frac{52\!\cdots\!56}{88\!\cdots\!25}a^{20}+\frac{77\!\cdots\!06}{88\!\cdots\!25}a^{19}-\frac{50\!\cdots\!22}{88\!\cdots\!25}a^{18}+\frac{43\!\cdots\!27}{88\!\cdots\!25}a^{17}-\frac{12\!\cdots\!44}{88\!\cdots\!25}a^{16}+\frac{16\!\cdots\!03}{88\!\cdots\!25}a^{15}-\frac{49\!\cdots\!48}{88\!\cdots\!25}a^{14}+\frac{68\!\cdots\!62}{17\!\cdots\!25}a^{13}-\frac{75\!\cdots\!87}{88\!\cdots\!25}a^{12}+\frac{49\!\cdots\!03}{88\!\cdots\!25}a^{11}-\frac{10\!\cdots\!86}{88\!\cdots\!25}a^{10}+\frac{43\!\cdots\!29}{88\!\cdots\!25}a^{9}-\frac{21\!\cdots\!01}{35\!\cdots\!05}a^{8}+\frac{25\!\cdots\!96}{88\!\cdots\!25}a^{7}-\frac{32\!\cdots\!51}{88\!\cdots\!25}a^{6}+\frac{76\!\cdots\!51}{88\!\cdots\!25}a^{5}+\frac{87\!\cdots\!38}{17\!\cdots\!25}a^{4}+\frac{91\!\cdots\!99}{88\!\cdots\!25}a^{3}-\frac{99\!\cdots\!36}{88\!\cdots\!25}a^{2}+\frac{19\!\cdots\!08}{88\!\cdots\!25}a+\frac{65\!\cdots\!81}{88\!\cdots\!25}$, $\frac{48\!\cdots\!31}{17\!\cdots\!50}a^{23}-\frac{97\!\cdots\!71}{17\!\cdots\!25}a^{22}+\frac{78\!\cdots\!97}{17\!\cdots\!50}a^{21}+\frac{18\!\cdots\!98}{88\!\cdots\!25}a^{20}+\frac{43\!\cdots\!02}{88\!\cdots\!25}a^{19}+\frac{50\!\cdots\!77}{17\!\cdots\!50}a^{18}+\frac{48\!\cdots\!93}{17\!\cdots\!50}a^{17}+\frac{23\!\cdots\!02}{88\!\cdots\!25}a^{16}+\frac{19\!\cdots\!27}{17\!\cdots\!50}a^{15}+\frac{88\!\cdots\!59}{88\!\cdots\!25}a^{14}+\frac{39\!\cdots\!79}{17\!\cdots\!25}a^{13}+\frac{39\!\cdots\!67}{17\!\cdots\!50}a^{12}+\frac{60\!\cdots\!77}{17\!\cdots\!50}a^{11}+\frac{28\!\cdots\!13}{88\!\cdots\!25}a^{10}+\frac{54\!\cdots\!11}{17\!\cdots\!50}a^{9}+\frac{22\!\cdots\!91}{70\!\cdots\!10}a^{8}+\frac{18\!\cdots\!32}{88\!\cdots\!25}a^{7}+\frac{16\!\cdots\!58}{88\!\cdots\!25}a^{6}+\frac{12\!\cdots\!09}{17\!\cdots\!50}a^{5}+\frac{11\!\cdots\!21}{17\!\cdots\!25}a^{4}+\frac{19\!\cdots\!58}{88\!\cdots\!25}a^{3}+\frac{88\!\cdots\!38}{88\!\cdots\!25}a^{2}+\frac{55\!\cdots\!36}{88\!\cdots\!25}a+\frac{18\!\cdots\!79}{17\!\cdots\!50}$, $\frac{74\!\cdots\!43}{70\!\cdots\!10}a^{23}+\frac{12\!\cdots\!42}{70\!\cdots\!01}a^{22}+\frac{49\!\cdots\!83}{35\!\cdots\!05}a^{21}+\frac{15\!\cdots\!94}{35\!\cdots\!05}a^{20}+\frac{57\!\cdots\!81}{35\!\cdots\!05}a^{19}+\frac{17\!\cdots\!53}{35\!\cdots\!05}a^{18}+\frac{55\!\cdots\!29}{70\!\cdots\!10}a^{17}+\frac{11\!\cdots\!56}{35\!\cdots\!05}a^{16}+\frac{12\!\cdots\!03}{35\!\cdots\!05}a^{15}+\frac{42\!\cdots\!52}{35\!\cdots\!05}a^{14}+\frac{42\!\cdots\!77}{70\!\cdots\!01}a^{13}+\frac{91\!\cdots\!63}{35\!\cdots\!05}a^{12}+\frac{63\!\cdots\!81}{70\!\cdots\!10}a^{11}+\frac{13\!\cdots\!14}{35\!\cdots\!05}a^{10}+\frac{21\!\cdots\!79}{35\!\cdots\!05}a^{9}+\frac{48\!\cdots\!75}{14\!\cdots\!02}a^{8}+\frac{17\!\cdots\!71}{35\!\cdots\!05}a^{7}+\frac{70\!\cdots\!24}{35\!\cdots\!05}a^{6}+\frac{71\!\cdots\!97}{70\!\cdots\!10}a^{5}+\frac{44\!\cdots\!33}{70\!\cdots\!01}a^{4}+\frac{36\!\cdots\!74}{35\!\cdots\!05}a^{3}+\frac{29\!\cdots\!64}{35\!\cdots\!05}a^{2}-\frac{99\!\cdots\!17}{35\!\cdots\!05}a+\frac{37\!\cdots\!81}{35\!\cdots\!05}$, $\frac{42\!\cdots\!08}{88\!\cdots\!25}a^{23}-\frac{12\!\cdots\!81}{17\!\cdots\!25}a^{22}+\frac{74\!\cdots\!96}{88\!\cdots\!25}a^{21}-\frac{57\!\cdots\!72}{88\!\cdots\!25}a^{20}+\frac{81\!\cdots\!97}{88\!\cdots\!25}a^{19}-\frac{55\!\cdots\!14}{88\!\cdots\!25}a^{18}+\frac{46\!\cdots\!74}{88\!\cdots\!25}a^{17}-\frac{14\!\cdots\!03}{88\!\cdots\!25}a^{16}+\frac{17\!\cdots\!61}{88\!\cdots\!25}a^{15}-\frac{57\!\cdots\!26}{88\!\cdots\!25}a^{14}+\frac{71\!\cdots\!19}{17\!\cdots\!25}a^{13}-\frac{90\!\cdots\!44}{88\!\cdots\!25}a^{12}+\frac{51\!\cdots\!36}{88\!\cdots\!25}a^{11}-\frac{12\!\cdots\!07}{88\!\cdots\!25}a^{10}+\frac{45\!\cdots\!48}{88\!\cdots\!25}a^{9}-\frac{28\!\cdots\!02}{35\!\cdots\!05}a^{8}+\frac{26\!\cdots\!02}{88\!\cdots\!25}a^{7}-\frac{41\!\cdots\!37}{88\!\cdots\!25}a^{6}+\frac{77\!\cdots\!12}{88\!\cdots\!25}a^{5}+\frac{61\!\cdots\!06}{17\!\cdots\!25}a^{4}+\frac{94\!\cdots\!88}{88\!\cdots\!25}a^{3}-\frac{11\!\cdots\!32}{88\!\cdots\!25}a^{2}+\frac{21\!\cdots\!71}{88\!\cdots\!25}a+\frac{13\!\cdots\!97}{88\!\cdots\!25}$, $\frac{23\!\cdots\!57}{17\!\cdots\!50}a^{23}-\frac{24\!\cdots\!37}{17\!\cdots\!25}a^{22}+\frac{40\!\cdots\!09}{17\!\cdots\!50}a^{21}-\frac{81\!\cdots\!44}{88\!\cdots\!25}a^{20}+\frac{44\!\cdots\!13}{17\!\cdots\!50}a^{19}-\frac{13\!\cdots\!31}{17\!\cdots\!50}a^{18}+\frac{24\!\cdots\!21}{17\!\cdots\!50}a^{17}+\frac{58\!\cdots\!69}{88\!\cdots\!25}a^{16}+\frac{11\!\cdots\!61}{22\!\cdots\!50}a^{15}+\frac{15\!\cdots\!48}{88\!\cdots\!25}a^{14}+\frac{39\!\cdots\!51}{35\!\cdots\!50}a^{13}+\frac{19\!\cdots\!49}{17\!\cdots\!50}a^{12}+\frac{28\!\cdots\!19}{17\!\cdots\!50}a^{11}+\frac{12\!\cdots\!86}{88\!\cdots\!25}a^{10}+\frac{26\!\cdots\!67}{17\!\cdots\!50}a^{9}+\frac{15\!\cdots\!67}{70\!\cdots\!10}a^{8}+\frac{16\!\cdots\!83}{17\!\cdots\!50}a^{7}+\frac{89\!\cdots\!76}{88\!\cdots\!25}a^{6}+\frac{53\!\cdots\!23}{17\!\cdots\!50}a^{5}+\frac{18\!\cdots\!99}{35\!\cdots\!50}a^{4}+\frac{61\!\cdots\!51}{88\!\cdots\!25}a^{3}-\frac{91\!\cdots\!39}{88\!\cdots\!25}a^{2}+\frac{82\!\cdots\!09}{17\!\cdots\!50}a-\frac{20\!\cdots\!37}{17\!\cdots\!50}$, $\frac{19\!\cdots\!59}{17\!\cdots\!50}a^{23}-\frac{15\!\cdots\!44}{17\!\cdots\!25}a^{22}+\frac{32\!\cdots\!33}{17\!\cdots\!50}a^{21}-\frac{30\!\cdots\!28}{88\!\cdots\!25}a^{20}+\frac{35\!\cdots\!31}{17\!\cdots\!50}a^{19}-\frac{30\!\cdots\!47}{17\!\cdots\!50}a^{18}+\frac{20\!\cdots\!27}{17\!\cdots\!50}a^{17}+\frac{27\!\cdots\!28}{88\!\cdots\!25}a^{16}+\frac{78\!\cdots\!03}{17\!\cdots\!50}a^{15}+\frac{99\!\cdots\!26}{88\!\cdots\!25}a^{14}+\frac{33\!\cdots\!37}{35\!\cdots\!50}a^{13}+\frac{50\!\cdots\!63}{17\!\cdots\!50}a^{12}+\frac{24\!\cdots\!53}{17\!\cdots\!50}a^{11}+\frac{35\!\cdots\!07}{88\!\cdots\!25}a^{10}+\frac{22\!\cdots\!29}{17\!\cdots\!50}a^{9}+\frac{28\!\cdots\!69}{70\!\cdots\!10}a^{8}+\frac{14\!\cdots\!71}{17\!\cdots\!50}a^{7}+\frac{18\!\cdots\!62}{88\!\cdots\!25}a^{6}+\frac{50\!\cdots\!01}{17\!\cdots\!50}a^{5}+\frac{24\!\cdots\!63}{35\!\cdots\!50}a^{4}+\frac{58\!\cdots\!62}{88\!\cdots\!25}a^{3}+\frac{35\!\cdots\!57}{88\!\cdots\!25}a^{2}+\frac{52\!\cdots\!83}{17\!\cdots\!50}a-\frac{33\!\cdots\!69}{17\!\cdots\!50}$, $\frac{37\!\cdots\!53}{88\!\cdots\!25}a^{23}-\frac{95\!\cdots\!68}{17\!\cdots\!25}a^{22}+\frac{24\!\cdots\!27}{17\!\cdots\!50}a^{21}-\frac{78\!\cdots\!82}{88\!\cdots\!25}a^{20}+\frac{24\!\cdots\!49}{17\!\cdots\!50}a^{19}-\frac{17\!\cdots\!83}{17\!\cdots\!50}a^{18}+\frac{91\!\cdots\!84}{88\!\cdots\!25}a^{17}-\frac{47\!\cdots\!53}{88\!\cdots\!25}a^{16}+\frac{54\!\cdots\!87}{17\!\cdots\!50}a^{15}-\frac{17\!\cdots\!06}{88\!\cdots\!25}a^{14}+\frac{12\!\cdots\!77}{14\!\cdots\!02}a^{13}-\frac{73\!\cdots\!73}{17\!\cdots\!50}a^{12}+\frac{11\!\cdots\!61}{88\!\cdots\!25}a^{11}-\frac{52\!\cdots\!22}{88\!\cdots\!25}a^{10}+\frac{28\!\cdots\!71}{17\!\cdots\!50}a^{9}-\frac{93\!\cdots\!76}{17\!\cdots\!25}a^{8}+\frac{14\!\cdots\!29}{17\!\cdots\!50}a^{7}-\frac{27\!\cdots\!12}{88\!\cdots\!25}a^{6}+\frac{36\!\cdots\!37}{88\!\cdots\!25}a^{5}-\frac{32\!\cdots\!21}{35\!\cdots\!50}a^{4}-\frac{39\!\cdots\!87}{88\!\cdots\!25}a^{3}-\frac{11\!\cdots\!17}{88\!\cdots\!25}a^{2}+\frac{22\!\cdots\!67}{17\!\cdots\!50}a-\frac{43\!\cdots\!91}{17\!\cdots\!50}$, $\frac{41\!\cdots\!73}{17\!\cdots\!50}a^{23}-\frac{23\!\cdots\!63}{70\!\cdots\!10}a^{22}+\frac{72\!\cdots\!81}{17\!\cdots\!50}a^{21}-\frac{53\!\cdots\!17}{17\!\cdots\!50}a^{20}+\frac{39\!\cdots\!01}{88\!\cdots\!25}a^{19}-\frac{32\!\cdots\!43}{11\!\cdots\!75}a^{18}+\frac{45\!\cdots\!19}{17\!\cdots\!50}a^{17}-\frac{13\!\cdots\!03}{17\!\cdots\!50}a^{16}+\frac{16\!\cdots\!01}{17\!\cdots\!50}a^{15}-\frac{50\!\cdots\!61}{17\!\cdots\!50}a^{14}+\frac{34\!\cdots\!43}{17\!\cdots\!25}a^{13}-\frac{38\!\cdots\!27}{88\!\cdots\!25}a^{12}+\frac{50\!\cdots\!11}{17\!\cdots\!50}a^{11}-\frac{10\!\cdots\!37}{17\!\cdots\!50}a^{10}+\frac{44\!\cdots\!23}{17\!\cdots\!50}a^{9}-\frac{55\!\cdots\!61}{17\!\cdots\!25}a^{8}+\frac{26\!\cdots\!27}{17\!\cdots\!50}a^{7}-\frac{32\!\cdots\!37}{17\!\cdots\!50}a^{6}+\frac{38\!\cdots\!56}{88\!\cdots\!25}a^{5}+\frac{88\!\cdots\!53}{35\!\cdots\!50}a^{4}+\frac{96\!\cdots\!13}{17\!\cdots\!50}a^{3}-\frac{10\!\cdots\!77}{17\!\cdots\!50}a^{2}+\frac{10\!\cdots\!23}{88\!\cdots\!25}a+\frac{13\!\cdots\!56}{88\!\cdots\!25}$, $\frac{72\!\cdots\!07}{88\!\cdots\!25}a^{23}-\frac{38\!\cdots\!43}{35\!\cdots\!50}a^{22}+\frac{12\!\cdots\!84}{88\!\cdots\!25}a^{21}-\frac{16\!\cdots\!51}{17\!\cdots\!50}a^{20}+\frac{13\!\cdots\!88}{88\!\cdots\!25}a^{19}-\frac{76\!\cdots\!81}{88\!\cdots\!25}a^{18}+\frac{77\!\cdots\!71}{88\!\cdots\!25}a^{17}-\frac{31\!\cdots\!99}{17\!\cdots\!50}a^{16}+\frac{29\!\cdots\!19}{88\!\cdots\!25}a^{15}-\frac{12\!\cdots\!83}{17\!\cdots\!50}a^{14}+\frac{12\!\cdots\!56}{17\!\cdots\!25}a^{13}-\frac{80\!\cdots\!26}{88\!\cdots\!25}a^{12}+\frac{88\!\cdots\!69}{88\!\cdots\!25}a^{11}-\frac{23\!\cdots\!31}{17\!\cdots\!50}a^{10}+\frac{79\!\cdots\!67}{88\!\cdots\!25}a^{9}-\frac{29\!\cdots\!93}{70\!\cdots\!10}a^{8}+\frac{94\!\cdots\!91}{17\!\cdots\!50}a^{7}-\frac{24\!\cdots\!98}{88\!\cdots\!25}a^{6}+\frac{28\!\cdots\!21}{17\!\cdots\!50}a^{5}+\frac{55\!\cdots\!83}{35\!\cdots\!50}a^{4}+\frac{20\!\cdots\!52}{88\!\cdots\!25}a^{3}-\frac{28\!\cdots\!31}{17\!\cdots\!50}a^{2}+\frac{32\!\cdots\!09}{88\!\cdots\!25}a+\frac{35\!\cdots\!13}{88\!\cdots\!25}$ 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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 7346081.887826216 \cdot 64}{6\cdot\sqrt{2465824451534772200819297422119140625}}\cr\approx \mathstrut & 0.188912994284165 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 17*x^22 - 6*x^21 + 188*x^20 - 47*x^19 + 1066*x^18 + 111*x^17 + 4083*x^16 + 396*x^15 + 8622*x^14 + 1407*x^13 + 12820*x^12 + 2029*x^11 + 11865*x^10 + 2619*x^9 + 7679*x^8 + 1542*x^7 + 2678*x^6 + 721*x^5 + 616*x^4 + 85*x^3 + 36*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 + 17*x^22 - 6*x^21 + 188*x^20 - 47*x^19 + 1066*x^18 + 111*x^17 + 4083*x^16 + 396*x^15 + 8622*x^14 + 1407*x^13 + 12820*x^12 + 2029*x^11 + 11865*x^10 + 2619*x^9 + 7679*x^8 + 1542*x^7 + 2678*x^6 + 721*x^5 + 616*x^4 + 85*x^3 + 36*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 + 17*x^22 - 6*x^21 + 188*x^20 - 47*x^19 + 1066*x^18 + 111*x^17 + 4083*x^16 + 396*x^15 + 8622*x^14 + 1407*x^13 + 12820*x^12 + 2029*x^11 + 11865*x^10 + 2619*x^9 + 7679*x^8 + 1542*x^7 + 2678*x^6 + 721*x^5 + 616*x^4 + 85*x^3 + 36*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 + 17*x^22 - 6*x^21 + 188*x^20 - 47*x^19 + 1066*x^18 + 111*x^17 + 4083*x^16 + 396*x^15 + 8622*x^14 + 1407*x^13 + 12820*x^12 + 2029*x^11 + 11865*x^10 + 2619*x^9 + 7679*x^8 + 1542*x^7 + 2678*x^6 + 721*x^5 + 616*x^4 + 85*x^3 + 36*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

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
An abelian group of order 24
The 24 conjugacy class representatives for $C_2^2\times C_6$
Character table for $C_2^2\times C_6$ is not computed

Intermediate fields

\(\Q(\sqrt{65}) \), \(\Q(\sqrt{-195}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{-15}) \), 3.3.169.1, \(\Q(\sqrt{-3}, \sqrt{65})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-15}, \sqrt{-39})\), \(\Q(\sqrt{5}, \sqrt{-39})\), \(\Q(\sqrt{13}, \sqrt{-15})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{13})\), 6.6.46411625.1, 6.0.1253113875.1, 6.0.771147.1, 6.6.3570125.1, \(\Q(\zeta_{13})^+\), 6.0.10024911.1, 6.0.96393375.1, 8.0.1445900625.1, 12.0.1570294383717515625.1, 12.12.2154038935140625.1, 12.0.1570294383717515625.3, 12.0.1570294383717515625.2, 12.0.1570294383717515625.4, 12.0.9291682743890625.1, 12.0.100498840557921.1

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 ${\href{/padicField/2.6.0.1}{6} }^{4}$ R R ${\href{/padicField/7.6.0.1}{6} }^{4}$ ${\href{/padicField/11.6.0.1}{6} }^{4}$ R ${\href{/padicField/17.6.0.1}{6} }^{4}$ ${\href{/padicField/19.6.0.1}{6} }^{4}$ ${\href{/padicField/23.6.0.1}{6} }^{4}$ ${\href{/padicField/29.6.0.1}{6} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{12}$ ${\href{/padicField/37.6.0.1}{6} }^{4}$ ${\href{/padicField/41.6.0.1}{6} }^{4}$ ${\href{/padicField/43.6.0.1}{6} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{12}$ ${\href{/padicField/53.2.0.1}{2} }^{12}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

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 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}$
\(13\) Copy content Toggle raw display 13.12.10.1$x^{12} + 72 x^{11} + 2172 x^{10} + 35280 x^{9} + 328380 x^{8} + 1703232 x^{7} + 4282170 x^{6} + 3407400 x^{5} + 1340820 x^{4} + 712800 x^{3} + 3855192 x^{2} + 18082080 x + 35650393$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
13.12.10.1$x^{12} + 72 x^{11} + 2172 x^{10} + 35280 x^{9} + 328380 x^{8} + 1703232 x^{7} + 4282170 x^{6} + 3407400 x^{5} + 1340820 x^{4} + 712800 x^{3} + 3855192 x^{2} + 18082080 x + 35650393$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$