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Properties

Label 24.0.22191215949...2073.2
Degree $24$
Signature $[0, 12]$
Discriminant $13^{22}\cdot 17^{21}$
Root discriminant $125.24$
Ramified primes $13, 17$
Class number $56464$ (GRH)
Class group $[2, 2, 14116]$ (GRH)
Galois group $C_{24}$ (as 24T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![484203487, -1194592532, 991948431, -898459740, 769210633, -20709065, 187309511, -54124340, 6068680, -36700642, 12300149, -1252303, 1952874, -520065, -97542, -66291, 29362, 5123, 1221, -1079, 81, -22, 18, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 18*x^22 - 22*x^21 + 81*x^20 - 1079*x^19 + 1221*x^18 + 5123*x^17 + 29362*x^16 - 66291*x^15 - 97542*x^14 - 520065*x^13 + 1952874*x^12 - 1252303*x^11 + 12300149*x^10 - 36700642*x^9 + 6068680*x^8 - 54124340*x^7 + 187309511*x^6 - 20709065*x^5 + 769210633*x^4 - 898459740*x^3 + 991948431*x^2 - 1194592532*x + 484203487)
 
gp: K = bnfinit(x^24 - x^23 + 18*x^22 - 22*x^21 + 81*x^20 - 1079*x^19 + 1221*x^18 + 5123*x^17 + 29362*x^16 - 66291*x^15 - 97542*x^14 - 520065*x^13 + 1952874*x^12 - 1252303*x^11 + 12300149*x^10 - 36700642*x^9 + 6068680*x^8 - 54124340*x^7 + 187309511*x^6 - 20709065*x^5 + 769210633*x^4 - 898459740*x^3 + 991948431*x^2 - 1194592532*x + 484203487, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} + 18 x^{22} - 22 x^{21} + 81 x^{20} - 1079 x^{19} + 1221 x^{18} + 5123 x^{17} + 29362 x^{16} - 66291 x^{15} - 97542 x^{14} - 520065 x^{13} + 1952874 x^{12} - 1252303 x^{11} + 12300149 x^{10} - 36700642 x^{9} + 6068680 x^{8} - 54124340 x^{7} + 187309511 x^{6} - 20709065 x^{5} + 769210633 x^{4} - 898459740 x^{3} + 991948431 x^{2} - 1194592532 x + 484203487 \)

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:  \(221912159494888970129181006529110805060115683312073=13^{22}\cdot 17^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $125.24$
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:  $13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(221=13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{221}(64,·)$, $\chi_{221}(1,·)$, $\chi_{221}(4,·)$, $\chi_{221}(138,·)$, $\chi_{221}(140,·)$, $\chi_{221}(206,·)$, $\chi_{221}(16,·)$, $\chi_{221}(145,·)$, $\chi_{221}(213,·)$, $\chi_{221}(151,·)$, $\chi_{221}(152,·)$, $\chi_{221}(120,·)$, $\chi_{221}(219,·)$, $\chi_{221}(93,·)$, $\chi_{221}(30,·)$, $\chi_{221}(161,·)$, $\chi_{221}(162,·)$, $\chi_{221}(35,·)$, $\chi_{221}(38,·)$, $\chi_{221}(166,·)$, $\chi_{221}(110,·)$, $\chi_{221}(118,·)$, $\chi_{221}(202,·)$, $\chi_{221}(189,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{359} a^{21} - \frac{164}{359} a^{20} + \frac{114}{359} a^{19} + \frac{15}{359} a^{18} - \frac{30}{359} a^{17} - \frac{118}{359} a^{16} + \frac{41}{359} a^{15} + \frac{32}{359} a^{14} - \frac{93}{359} a^{13} - \frac{126}{359} a^{12} - \frac{6}{359} a^{11} + \frac{95}{359} a^{10} + \frac{67}{359} a^{9} + \frac{155}{359} a^{8} - \frac{13}{359} a^{7} - \frac{62}{359} a^{6} + \frac{138}{359} a^{5} - \frac{94}{359} a^{4} + \frac{131}{359} a^{3} - \frac{118}{359} a^{2} - \frac{82}{359} a - \frac{33}{359}$, $\frac{1}{359} a^{22} + \frac{143}{359} a^{20} + \frac{43}{359} a^{19} - \frac{83}{359} a^{18} - \frac{12}{359} a^{17} + \frac{75}{359} a^{16} - \frac{65}{359} a^{15} + \frac{129}{359} a^{14} + \frac{59}{359} a^{13} + \frac{152}{359} a^{12} - \frac{171}{359} a^{11} - \frac{149}{359} a^{10} + \frac{14}{359} a^{9} - \frac{82}{359} a^{8} - \frac{40}{359} a^{7} + \frac{22}{359} a^{6} - \frac{79}{359} a^{5} + \frac{152}{359} a^{4} - \frac{174}{359} a^{3} - \frac{48}{359} a^{2} + \frac{161}{359} a - \frac{27}{359}$, $\frac{1}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{23} - \frac{985895262983025033078946378512488217607358629067679769515394565183359245302532619954678264748237310807}{2435379710649741684351901919155276124548623816301983901734464695999159881394356074310120002536741724592901} a^{22} - \frac{1010825583074025010454702011427204766591532759509505468630527032911135341850252209101159194202253839665934}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{21} + \frac{709268611252182690481566293831806569048702355665853710645654266526729615047509063956116150891907926747820160}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{20} - \frac{421507585885586184395070278870249205696102103328482962080303300077912484894165088914356558134705573252988879}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{19} + \frac{140351530133385866624081172519638073917527833470392593949683981762984672151113459920881407707883496331892098}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{18} + \frac{305252929738485789158953899279469181788312428495964109061171214011448106333441575431813800952764524262114490}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{17} - \frac{582788526839719290475538585403224695396905206557623813777862633032159188414715992217738629129593218827001240}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{16} + \frac{1990385549417602463735782387457823368793175754352405633347727187625301126361650955734549898390882152854819}{4022785984443723729305509298214704016315693379016926054954143634338445152275356969542900171321136052043427} a^{15} - \frac{236789919530656517481661413385994219526186783316181954830115766343951892142952966893488686603995534754429885}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{14} + \frac{307710094361192053092081389883563658313709590144507131758158875223252253028685503248786079782847498539030467}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{13} - \frac{29251974578490880848140256914289108572818246460927112808488067354380212438178778241393864006776164701801361}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{12} - \frac{41879095594569431676887420836240615601345010453495737077801699303762463536977775455771210323407269812882209}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{11} - \frac{391055467464330803662221351625575310942445039600507605173987300708542080873612452120126868285002529365545433}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{10} + \frac{132195046060993475382973633441629635072432698168325508138661395643285249842428479222811522343565316114381843}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{9} + \frac{185350067734586748606661877183862752371268846777333136625627086257728717675042202249743444157815581523256733}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{8} - \frac{431750246253642513322421309658193236798787133627709200097669178855199337693251323672358931981833385185493875}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{7} + \frac{376289143429959786038449342116298114645670897525093702518412233930203234507893217446824343586543379469691055}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{6} - \frac{2236121678751629022418350667396819792822720327365026185951580337682602122121874087747122101893474442611235}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{5} + \frac{323555463328507723230774073371464341701678448331125306433283280354126092562854479675579325162798030522541983}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{4} + \frac{258250586861253265507823832863527151153351017926704390398697564875860124257874113191598824357055858159181299}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{3} + \frac{447527846763606731410894670234235768066490551688808123804785545345112580935300020118707932243700113671338127}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a^{2} + \frac{487858604560003471377906887024294329453411454813856759512178505505807828997295713164578372864180413457587072}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293} a + \frac{141744963632140803030456052252542875198961829212485578547193556620021584473933324913061967784727927865755863}{1444180168415296818820677838059078741857333923067076453728537564727501809666853152065901161504287842683590293}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{14116}$, which has order $56464$ (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:  \( -1 \) (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:  \( 20400053551.79016 \) (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

\(\Q(\sqrt{17}) \), 3.3.169.1, 4.4.830297.1, 6.6.140320193.1, 8.0.1980626399884457.2, 12.12.16348345805451893172953.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{4}$ $24$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{3}$ $24$ $24$ R R ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ $24$ $24$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{3}$ $24$ $24$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
$13$13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
17Data not computed