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Properties

Label 24.0.24658244515...0625.1
Degree $24$
Signature $[0, 12]$
Discriminant $3^{12}\cdot 5^{12}\cdot 13^{20}$
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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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 36, 85, 616, 721, 2678, 1542, 7679, 2619, 11865, 2029, 12820, 1407, 8622, 396, 4083, 111, 1066, -47, 188, -6, 17, -1, 1]);
 
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(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)
 

Normalized defining polynomial

\( x^{24} - x^{23} + 17 x^{22} - 6 x^{21} + 188 x^{20} - 47 x^{19} + 1066 x^{18} + 111 x^{17} + 4083 x^{16} + 396 x^{15} + 8622 x^{14} + 1407 x^{13} + 12820 x^{12} + 2029 x^{11} + 11865 x^{10} + 2619 x^{9} + 7679 x^{8} + 1542 x^{7} + 2678 x^{6} + 721 x^{5} + 616 x^{4} + 85 x^{3} + 36 x^{2} - 3 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $24$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 12]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2465824451534772200819297422119140625=3^{12}\cdot 5^{12}\cdot 13^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.83$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
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.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{6201996740150} a^{22} - \frac{1205449772479}{6201996740150} a^{21} - \frac{713667446537}{6201996740150} a^{20} - \frac{413023687803}{3100998370075} a^{19} + \frac{30674675574}{3100998370075} a^{18} + \frac{2482592888}{620199674015} a^{17} - \frac{294193394217}{6201996740150} a^{16} - \frac{578797891043}{6201996740150} a^{15} - \frac{1906270011791}{6201996740150} a^{14} - \frac{867376474884}{3100998370075} a^{13} + \frac{200118271891}{3100998370075} a^{12} + \frac{599330056862}{3100998370075} a^{11} - \frac{585088476289}{6201996740150} a^{10} - \frac{784626013863}{6201996740150} a^{9} + \frac{929121692303}{6201996740150} a^{8} - \frac{922060619057}{6201996740150} a^{7} + \frac{1153340920727}{6201996740150} a^{6} + \frac{3026024584923}{6201996740150} a^{5} - \frac{2309913026223}{6201996740150} a^{4} - \frac{1685641053403}{6201996740150} a^{3} - \frac{2927334170207}{6201996740150} a^{2} - \frac{1448359772873}{3100998370075} a + \frac{336229445243}{3100998370075}$, $\frac{1}{1771746158639227150415275250} a^{23} + \frac{6755504845561}{354349231727845430083055050} a^{22} + \frac{87363032942531573816672677}{1771746158639227150415275250} a^{21} + \frac{57439294557037664934514961}{1771746158639227150415275250} a^{20} + \frac{413865594187380152658402419}{1771746158639227150415275250} a^{19} + \frac{20404588822057511982486906}{885873079319613575207637625} a^{18} + \frac{404780486088759760929920503}{1771746158639227150415275250} a^{17} + \frac{772497736677368882349464429}{1771746158639227150415275250} a^{16} - \frac{659382022313595848292126253}{1771746158639227150415275250} a^{15} + \frac{17063560475306527543253179}{37696726779558024476920750} a^{14} + \frac{1848234356027138805620769}{354349231727845430083055050} a^{13} + \frac{40351031137440514631847456}{885873079319613575207637625} a^{12} - \frac{392541234078879380205978173}{1771746158639227150415275250} a^{11} + \frac{427571037911399588458410761}{1771746158639227150415275250} a^{10} + \frac{382575924370175196110462611}{1771746158639227150415275250} a^{9} + \frac{31831047547460091035742692}{177174615863922715041527525} a^{8} - \frac{333741879711842422256795343}{885873079319613575207637625} a^{7} - \frac{87546556790375026545378809}{1771746158639227150415275250} a^{6} - \frac{310305408429968199516571233}{885873079319613575207637625} a^{5} - \frac{10425558319581447115508116}{177174615863922715041527525} a^{4} + \frac{634114638122560934622654941}{1771746158639227150415275250} a^{3} + \frac{738331837091770043148504291}{1771746158639227150415275250} a^{2} + \frac{717143969330646941681421397}{1771746158639227150415275250} a - \frac{62233219262214666672158038}{885873079319613575207637625}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{4}\times C_{4}\times C_{4}$, which has order $64$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
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 $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 7346081.887826216 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{4}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$