↑

Properties

Label 24.24.5605779889...0625.1
Degree $24$
Signature $[24, 0]$
Discriminant $5^{12}\cdot 7^{16}\cdot 17^{21}$
Root discriminant $97.62$
Ramified primes $5, 7, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_{24}$ (as 24T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![185809, 786453, -3123105, -10867585, 18427354, 48248857, -50769407, -86662186, 65224490, 77240969, -42438075, -37945411, 15636940, 10918179, -3523495, -1895700, 505390, 198718, -46313, -12139, 2601, 391, -80, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 5*x^23 - 80*x^22 + 391*x^21 + 2601*x^20 - 12139*x^19 - 46313*x^18 + 198718*x^17 + 505390*x^16 - 1895700*x^15 - 3523495*x^14 + 10918179*x^13 + 15636940*x^12 - 37945411*x^11 - 42438075*x^10 + 77240969*x^9 + 65224490*x^8 - 86662186*x^7 - 50769407*x^6 + 48248857*x^5 + 18427354*x^4 - 10867585*x^3 - 3123105*x^2 + 786453*x + 185809)
 
gp: K = bnfinit(x^24 - 5*x^23 - 80*x^22 + 391*x^21 + 2601*x^20 - 12139*x^19 - 46313*x^18 + 198718*x^17 + 505390*x^16 - 1895700*x^15 - 3523495*x^14 + 10918179*x^13 + 15636940*x^12 - 37945411*x^11 - 42438075*x^10 + 77240969*x^9 + 65224490*x^8 - 86662186*x^7 - 50769407*x^6 + 48248857*x^5 + 18427354*x^4 - 10867585*x^3 - 3123105*x^2 + 786453*x + 185809, 1)
 

Normalized defining polynomial

\( x^{24} - 5 x^{23} - 80 x^{22} + 391 x^{21} + 2601 x^{20} - 12139 x^{19} - 46313 x^{18} + 198718 x^{17} + 505390 x^{16} - 1895700 x^{15} - 3523495 x^{14} + 10918179 x^{13} + 15636940 x^{12} - 37945411 x^{11} - 42438075 x^{10} + 77240969 x^{9} + 65224490 x^{8} - 86662186 x^{7} - 50769407 x^{6} + 48248857 x^{5} + 18427354 x^{4} - 10867585 x^{3} - 3123105 x^{2} + 786453 x + 185809 \)

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

Invariants

Degree:  $24$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[24, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(560577988928339833496317505446258210551025390625=5^{12}\cdot 7^{16}\cdot 17^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.62$
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:  $5, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(595=5\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{595}(256,·)$, $\chi_{595}(1,·)$, $\chi_{595}(389,·)$, $\chi_{595}(134,·)$, $\chi_{595}(519,·)$, $\chi_{595}(9,·)$, $\chi_{595}(526,·)$, $\chi_{595}(144,·)$, $\chi_{595}(529,·)$, $\chi_{595}(274,·)$, $\chi_{595}(86,·)$, $\chi_{595}(219,·)$, $\chi_{595}(506,·)$, $\chi_{595}(16,·)$, $\chi_{595}(484,·)$, $\chi_{595}(421,·)$, $\chi_{595}(81,·)$, $\chi_{595}(361,·)$, $\chi_{595}(106,·)$, $\chi_{595}(359,·)$, $\chi_{595}(179,·)$, $\chi_{595}(569,·)$, $\chi_{595}(186,·)$, $\chi_{595}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13} a^{10} + \frac{5}{13} a^{9} - \frac{3}{13} a^{8} + \frac{6}{13} a^{7} - \frac{4}{13} a^{6} + \frac{5}{13} a^{5} - \frac{2}{13} a^{4} - \frac{1}{13} a^{3} - \frac{2}{13} a^{2} - \frac{5}{13} a$, $\frac{1}{13} a^{11} - \frac{2}{13} a^{9} - \frac{5}{13} a^{8} + \frac{5}{13} a^{7} - \frac{1}{13} a^{6} - \frac{1}{13} a^{5} - \frac{4}{13} a^{4} + \frac{3}{13} a^{3} + \frac{5}{13} a^{2} - \frac{1}{13} a$, $\frac{1}{13} a^{12} + \frac{5}{13} a^{9} - \frac{1}{13} a^{8} - \frac{2}{13} a^{7} + \frac{4}{13} a^{6} + \frac{6}{13} a^{5} - \frac{1}{13} a^{4} + \frac{3}{13} a^{3} - \frac{5}{13} a^{2} + \frac{3}{13} a$, $\frac{1}{13} a^{13} - \frac{1}{13} a$, $\frac{1}{13} a^{14} - \frac{1}{13} a^{2}$, $\frac{1}{13} a^{15} - \frac{1}{13} a^{3}$, $\frac{1}{13} a^{16} - \frac{1}{13} a^{4}$, $\frac{1}{13} a^{17} - \frac{1}{13} a^{5}$, $\frac{1}{169} a^{18} - \frac{2}{169} a^{17} + \frac{4}{169} a^{16} - \frac{1}{169} a^{15} - \frac{5}{169} a^{14} - \frac{6}{169} a^{13} - \frac{4}{169} a^{12} - \frac{4}{169} a^{10} + \frac{64}{169} a^{9} + \frac{81}{169} a^{8} + \frac{10}{169} a^{7} + \frac{25}{169} a^{6} - \frac{29}{169} a^{5} + \frac{47}{169} a^{4} + \frac{71}{169} a^{3} + \frac{59}{169} a^{2} + \frac{53}{169} a - \frac{4}{13}$, $\frac{1}{169} a^{19} - \frac{6}{169} a^{16} + \frac{6}{169} a^{15} - \frac{3}{169} a^{14} - \frac{3}{169} a^{13} + \frac{5}{169} a^{12} - \frac{4}{169} a^{11} + \frac{4}{169} a^{10} + \frac{14}{169} a^{9} - \frac{23}{169} a^{8} + \frac{45}{169} a^{7} - \frac{57}{169} a^{6} - \frac{24}{169} a^{5} - \frac{69}{169} a^{4} - \frac{59}{169} a^{3} + \frac{28}{169} a^{2} + \frac{2}{169} a + \frac{5}{13}$, $\frac{1}{169} a^{20} - \frac{6}{169} a^{17} + \frac{6}{169} a^{16} - \frac{3}{169} a^{15} - \frac{3}{169} a^{14} + \frac{5}{169} a^{13} - \frac{4}{169} a^{12} + \frac{4}{169} a^{11} + \frac{1}{169} a^{10} + \frac{81}{169} a^{9} + \frac{84}{169} a^{8} + \frac{34}{169} a^{7} + \frac{28}{169} a^{6} + \frac{35}{169} a^{5} - \frac{33}{169} a^{4} + \frac{41}{169} a^{3} + \frac{28}{169} a^{2} - \frac{3}{13} a$, $\frac{1}{7943} a^{21} + \frac{3}{7943} a^{20} + \frac{19}{7943} a^{19} - \frac{22}{7943} a^{18} - \frac{71}{7943} a^{17} + \frac{149}{7943} a^{16} - \frac{25}{7943} a^{15} + \frac{45}{7943} a^{14} + \frac{154}{7943} a^{13} - \frac{226}{7943} a^{12} + \frac{184}{7943} a^{11} - \frac{101}{7943} a^{10} - \frac{62}{169} a^{9} - \frac{992}{7943} a^{8} - \frac{1502}{7943} a^{7} + \frac{716}{7943} a^{6} - \frac{1259}{7943} a^{5} + \frac{2507}{7943} a^{4} - \frac{9}{47} a^{3} - \frac{1693}{7943} a^{2} - \frac{1122}{7943} a + \frac{224}{611}$, $\frac{1}{103259} a^{22} - \frac{4}{103259} a^{21} - \frac{2}{103259} a^{20} - \frac{108}{103259} a^{19} - \frac{11}{103259} a^{18} - \frac{2832}{103259} a^{17} - \frac{2337}{103259} a^{16} + \frac{2429}{103259} a^{15} + \frac{2001}{103259} a^{14} - \frac{3936}{103259} a^{13} - \frac{3733}{103259} a^{12} - \frac{2188}{103259} a^{11} - \frac{1032}{103259} a^{10} + \frac{3661}{103259} a^{9} - \frac{27693}{103259} a^{8} + \frac{25236}{103259} a^{7} + \frac{2142}{103259} a^{6} + \frac{39191}{103259} a^{5} + \frac{28870}{103259} a^{4} + \frac{8672}{103259} a^{3} - \frac{24662}{103259} a^{2} + \frac{19931}{103259} a + \frac{2098}{7943}$, $\frac{1}{42452831516767208244233762645001469965867290689846556771} a^{23} - \frac{100866017329493877317027302569376968956858698378375}{42452831516767208244233762645001469965867290689846556771} a^{22} - \frac{39269375235818931499279226821912115411675457091879}{42452831516767208244233762645001469965867290689846556771} a^{21} + \frac{77435539038837208380946882299577700869856706421444623}{42452831516767208244233762645001469965867290689846556771} a^{20} - \frac{7934046598178192088888113738105753461895438687507195}{3265602424366708326479520203461651535835945437680504367} a^{19} - \frac{384087665456526326005287081219634714244146840352744}{177626910111996687214367207719671422451327576108144589} a^{18} + \frac{813609469433984263239069858748123942100642230901796724}{42452831516767208244233762645001469965867290689846556771} a^{17} - \frac{1491652928249930605978889516295300151201220517885537743}{42452831516767208244233762645001469965867290689846556771} a^{16} - \frac{1543204801814983753276966297313566265250153689769105105}{42452831516767208244233762645001469965867290689846556771} a^{15} + \frac{1410314027832414952614238318853929140501965529776722221}{42452831516767208244233762645001469965867290689846556771} a^{14} - \frac{16763406446773914275945787865839440394472953056216363}{42452831516767208244233762645001469965867290689846556771} a^{13} - \frac{1278362417820047378442097887341518057108255873660759973}{42452831516767208244233762645001469965867290689846556771} a^{12} - \frac{787192847926371740334499889919487473383081259596224721}{42452831516767208244233762645001469965867290689846556771} a^{11} + \frac{226398032451561910640508728581209800745035813061252838}{42452831516767208244233762645001469965867290689846556771} a^{10} + \frac{15910228086880988441014634696475244751676207573327431895}{42452831516767208244233762645001469965867290689846556771} a^{9} + \frac{9295143430598985523800259574301517060731655576799598256}{42452831516767208244233762645001469965867290689846556771} a^{8} + \frac{13862771340781025524292086579527190222358121487448010695}{42452831516767208244233762645001469965867290689846556771} a^{7} + \frac{19803982033746223283998544674213365113901058621083815144}{42452831516767208244233762645001469965867290689846556771} a^{6} - \frac{15683050841998733392645318820211919662070863211088836983}{42452831516767208244233762645001469965867290689846556771} a^{5} + \frac{3705327876653672353265638002596480957528285876136523218}{42452831516767208244233762645001469965867290689846556771} a^{4} + \frac{5961849099661521580119224024104239340150898016448375156}{42452831516767208244233762645001469965867290689846556771} a^{3} - \frac{2026680761410021316865818198324694442853072243120132588}{42452831516767208244233762645001469965867290689846556771} a^{2} + \frac{10953119617549966635704600362284988261374497714341800518}{42452831516767208244233762645001469965867290689846556771} a - \frac{435179897702275547052768655880173995684400981941518848}{3265602424366708326479520203461651535835945437680504367}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $23$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 32766500146226612 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{24}$ (as 24T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 24
The 24 conjugacy class representatives for $C_{24}$
Character table for $C_{24}$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\zeta_{7})^+\), 4.4.4913.1, 6.6.11796113.1, 8.8.256461670625.1, 12.12.683635509017782097.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ $24$ R R $24$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{24}$ R ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ $24$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ $24$ $24$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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
5Data not computed
7Data not computed
17Data not computed