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Properties

Label 24.0.39786976753...5625.1
Degree $24$
Signature $[0, 12]$
Discriminant $5^{18}\cdot 7^{16}\cdot 11^{12}$
Root discriminant $40.58$
Ramified primes $5, 7, 11$
Class number $18$ (GRH)
Class group $[3, 6]$ (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25411681, -1431644, -10298763, 91235, 2830245, 356623, -937497, -52357, 355120, -20248, 33907, 2486, -19199, -4490, 1363, 640, 166, 160, 153, -73, -18, 4, -3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 3*x^22 + 4*x^21 - 18*x^20 - 73*x^19 + 153*x^18 + 160*x^17 + 166*x^16 + 640*x^15 + 1363*x^14 - 4490*x^13 - 19199*x^12 + 2486*x^11 + 33907*x^10 - 20248*x^9 + 355120*x^8 - 52357*x^7 - 937497*x^6 + 356623*x^5 + 2830245*x^4 + 91235*x^3 - 10298763*x^2 - 1431644*x + 25411681)
 
gp: K = bnfinit(x^24 - x^23 - 3*x^22 + 4*x^21 - 18*x^20 - 73*x^19 + 153*x^18 + 160*x^17 + 166*x^16 + 640*x^15 + 1363*x^14 - 4490*x^13 - 19199*x^12 + 2486*x^11 + 33907*x^10 - 20248*x^9 + 355120*x^8 - 52357*x^7 - 937497*x^6 + 356623*x^5 + 2830245*x^4 + 91235*x^3 - 10298763*x^2 - 1431644*x + 25411681, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} - 3 x^{22} + 4 x^{21} - 18 x^{20} - 73 x^{19} + 153 x^{18} + 160 x^{17} + 166 x^{16} + 640 x^{15} + 1363 x^{14} - 4490 x^{13} - 19199 x^{12} + 2486 x^{11} + 33907 x^{10} - 20248 x^{9} + 355120 x^{8} - 52357 x^{7} - 937497 x^{6} + 356623 x^{5} + 2830245 x^{4} + 91235 x^{3} - 10298763 x^{2} - 1431644 x + 25411681 \)

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:  \(397869767537058122019418033599853515625=5^{18}\cdot 7^{16}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.58$
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:  $5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(385=5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{385}(1,·)$, $\chi_{385}(67,·)$, $\chi_{385}(197,·)$, $\chi_{385}(263,·)$, $\chi_{385}(331,·)$, $\chi_{385}(142,·)$, $\chi_{385}(144,·)$, $\chi_{385}(274,·)$, $\chi_{385}(78,·)$, $\chi_{385}(23,·)$, $\chi_{385}(219,·)$, $\chi_{385}(221,·)$, $\chi_{385}(351,·)$, $\chi_{385}(32,·)$, $\chi_{385}(296,·)$, $\chi_{385}(298,·)$, $\chi_{385}(43,·)$, $\chi_{385}(109,·)$, $\chi_{385}(177,·)$, $\chi_{385}(309,·)$, $\chi_{385}(186,·)$, $\chi_{385}(232,·)$, $\chi_{385}(254,·)$, $\chi_{385}(373,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{3613} a^{20} - \frac{317}{3613} a^{19} + \frac{313}{3613} a^{18} + \frac{952}{3613} a^{17} - \frac{1282}{3613} a^{16} + \frac{158}{3613} a^{15} + \frac{785}{3613} a^{14} - \frac{311}{3613} a^{13} + \frac{618}{3613} a^{12} - \frac{128}{3613} a^{11} - \frac{861}{3613} a^{10} - \frac{365}{3613} a^{9} + \frac{427}{3613} a^{8} + \frac{1646}{3613} a^{7} + \frac{1022}{3613} a^{6} + \frac{940}{3613} a^{5} - \frac{1780}{3613} a^{4} + \frac{331}{3613} a^{3} - \frac{1149}{3613} a^{2} + \frac{1471}{3613} a - \frac{1252}{3613}$, $\frac{1}{64642281770425802423948323} a^{21} + \frac{6659172969473172480059}{64642281770425802423948323} a^{20} - \frac{23989222710322829149828182}{64642281770425802423948323} a^{19} + \frac{23962586018444936459907943}{64642281770425802423948323} a^{18} + \frac{7332045533512158198016256}{64642281770425802423948323} a^{17} + \frac{22764866624337526933006448}{64642281770425802423948323} a^{16} + \frac{1835574915765425612290725}{64642281770425802423948323} a^{15} - \frac{28387657844918645174561490}{64642281770425802423948323} a^{14} - \frac{9982010982808952920837811}{64642281770425802423948323} a^{13} - \frac{10347141969337060882313068}{64642281770425802423948323} a^{12} + \frac{30786173780575110636770292}{64642281770425802423948323} a^{11} - \frac{4747446125137935364949509}{64642281770425802423948323} a^{10} + \frac{27577522539924421005720998}{64642281770425802423948323} a^{9} - \frac{7907253314766711630129137}{64642281770425802423948323} a^{8} + \frac{27720936632291735840285349}{64642281770425802423948323} a^{7} + \frac{21083520355399372241574807}{64642281770425802423948323} a^{6} - \frac{6253817891155890089266363}{64642281770425802423948323} a^{5} + \frac{9413141449171270334441011}{64642281770425802423948323} a^{4} + \frac{1687435765536293273728478}{64642281770425802423948323} a^{3} + \frac{12457445935211490300509464}{64642281770425802423948323} a^{2} + \frac{6403305178001195856741650}{64642281770425802423948323} a + \frac{152676607354917141133438}{910454672822898625689413}$, $\frac{1}{4589602005700231972100330933} a^{22} - \frac{1}{4589602005700231972100330933} a^{21} + \frac{413831657546074651623273}{4589602005700231972100330933} a^{20} - \frac{1488340520020369805870217048}{4589602005700231972100330933} a^{19} + \frac{1486685193390185507263723930}{4589602005700231972100330933} a^{18} - \frac{1750693235895871677337621218}{4589602005700231972100330933} a^{17} + \frac{256972904327402900996301658}{4589602005700231972100330933} a^{16} + \frac{2232186152636938845984781013}{4589602005700231972100330933} a^{15} - \frac{1074484856656681053191208057}{4589602005700231972100330933} a^{14} - \frac{407300654714681338325549882}{4589602005700231972100330933} a^{13} - \frac{1163282772324603252064547654}{4589602005700231972100330933} a^{12} + \frac{299441586176175108962988760}{4589602005700231972100330933} a^{11} + \frac{1415269072099929893287220407}{4589602005700231972100330933} a^{10} + \frac{2085014301701458941404158794}{4589602005700231972100330933} a^{9} + \frac{2037168364861767212100566313}{4589602005700231972100330933} a^{8} - \frac{985965734366747866763005303}{4589602005700231972100330933} a^{7} - \frac{602855264941926083052185338}{4589602005700231972100330933} a^{6} - \frac{1974322245923442424426551058}{4589602005700231972100330933} a^{5} + \frac{2208552239459340472527382136}{4589602005700231972100330933} a^{4} - \frac{764727896348726803026403471}{4589602005700231972100330933} a^{3} - \frac{484172898777394195735251223}{4589602005700231972100330933} a^{2} - \frac{27057676932754676729555618}{64642281770425802423948323} a + \frac{413294188241921375594593}{910454672822898625689413}$, $\frac{1}{325861742404716470019123496243} a^{23} - \frac{1}{325861742404716470019123496243} a^{22} - \frac{3}{325861742404716470019123496243} a^{21} + \frac{31607685723950243156130861}{325861742404716470019123496243} a^{20} + \frac{72966145604710280302071348294}{325861742404716470019123496243} a^{19} + \frac{131812987286990062272319455528}{325861742404716470019123496243} a^{18} - \frac{97910650358286564190949745024}{325861742404716470019123496243} a^{17} + \frac{2567126319633857182177075630}{325861742404716470019123496243} a^{16} + \frac{133700783788606810711729129799}{325861742404716470019123496243} a^{15} + \frac{111997074500123485280486941399}{325861742404716470019123496243} a^{14} + \frac{145422266734707097847187246748}{325861742404716470019123496243} a^{13} + \frac{65674939232659076634488426130}{325861742404716470019123496243} a^{12} + \frac{125539256190056374209139615236}{325861742404716470019123496243} a^{11} - \frac{4687718802992311392765402174}{325861742404716470019123496243} a^{10} - \frac{56917512240132346996891651519}{325861742404716470019123496243} a^{9} + \frac{81206153618439790315199531505}{325861742404716470019123496243} a^{8} - \frac{55737149452600363080323445930}{325861742404716470019123496243} a^{7} + \frac{86319085182458855621136161227}{325861742404716470019123496243} a^{6} - \frac{10536895808838485136014967714}{325861742404716470019123496243} a^{5} + \frac{87951694099341683234724662130}{325861742404716470019123496243} a^{4} + \frac{17044707620467656315188507702}{325861742404716470019123496243} a^{3} + \frac{1256658120298293149018972120}{4589602005700231972100330933} a^{2} - \frac{17477595102529108032687843}{64642281770425802423948323} a - \frac{293816029118185767398099}{910454672822898625689413}$

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

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (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:  \( \frac{18832211760643333}{1270302243481935226155641} a^{23} + \frac{8434654599732333}{1270302243481935226155641} a^{22} - \frac{12987732248719540}{1270302243481935226155641} a^{21} + \frac{21429758210387241}{1270302243481935226155641} a^{20} - \frac{466908974341467463}{1270302243481935226155641} a^{19} - \frac{1237081496690536185}{1270302243481935226155641} a^{18} - \frac{1165945463726580736}{1270302243481935226155641} a^{17} + \frac{44807676258082413}{1270302243481935226155641} a^{16} + \frac{10596041355117836709}{1270302243481935226155641} a^{15} + \frac{19860840154741920568}{1270302243481935226155641} a^{14} + \frac{34984404971763388921}{1270302243481935226155641} a^{13} + \frac{204605395172600545975}{1270302243481935226155641} a^{12} - \frac{310379526506674694966}{1270302243481935226155641} a^{11} - \frac{210071374030139071684}{1270302243481935226155641} a^{10} - \frac{240196418981044044714}{1270302243481935226155641} a^{9} - \frac{178618982843414961666}{1270302243481935226155641} a^{8} - \frac{1068665174153916664171}{1270302243481935226155641} a^{7} - \frac{997120805049806399937}{1270302243481935226155641} a^{6} - \frac{19969287719018728727}{17891580894111763748671} a^{5} + \frac{1866812475141301033164}{1270302243481935226155641} a^{4} + \frac{154950139593348471970}{17891580894111763748671} a^{3} + \frac{492027212572704163428682}{1270302243481935226155641} a^{2} - \frac{12540954259363587824}{251994097100165686601} a - \frac{9820673739869280171}{251994097100165686601} \) (order $10$)
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:  \( 486434529.8152184 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{12}$ (as 24T2):

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 $C_2\times C_{12}$
Character table for $C_2\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-11}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{5}, \sqrt{-11})\), 4.4.15125.1, \(\Q(\zeta_{5})\), 6.0.399466375.2, 6.6.300125.1, 6.0.3195731.1, 8.0.228765625.1, 12.0.159573384755640625.1, 12.12.19946673094455078125.1, 12.0.11259376953125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }^{2}$ R R R ${\href{/LocalNumberField/13.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
$5$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$