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Properties

Label 24.0.16176178662...5625.3
Degree $24$
Signature $[0, 12]$
Discriminant $3^{12}\cdot 5^{18}\cdot 7^{20}$
Root discriminant $29.31$
Ramified primes $3, 5, 7$
Class number $26$ (GRH)
Class group $[26]$ (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![1, 8, 104, -228, 1858, 151, 5722, -327, 10104, -456, 11097, -1296, 8400, -1119, 4524, -706, 1781, -254, 500, -67, 101, -10, 13, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 13*x^22 - 10*x^21 + 101*x^20 - 67*x^19 + 500*x^18 - 254*x^17 + 1781*x^16 - 706*x^15 + 4524*x^14 - 1119*x^13 + 8400*x^12 - 1296*x^11 + 11097*x^10 - 456*x^9 + 10104*x^8 - 327*x^7 + 5722*x^6 + 151*x^5 + 1858*x^4 - 228*x^3 + 104*x^2 + 8*x + 1)
 
gp: K = bnfinit(x^24 - x^23 + 13*x^22 - 10*x^21 + 101*x^20 - 67*x^19 + 500*x^18 - 254*x^17 + 1781*x^16 - 706*x^15 + 4524*x^14 - 1119*x^13 + 8400*x^12 - 1296*x^11 + 11097*x^10 - 456*x^9 + 10104*x^8 - 327*x^7 + 5722*x^6 + 151*x^5 + 1858*x^4 - 228*x^3 + 104*x^2 + 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} + 13 x^{22} - 10 x^{21} + 101 x^{20} - 67 x^{19} + 500 x^{18} - 254 x^{17} + 1781 x^{16} - 706 x^{15} + 4524 x^{14} - 1119 x^{13} + 8400 x^{12} - 1296 x^{11} + 11097 x^{10} - 456 x^{9} + 10104 x^{8} - 327 x^{7} + 5722 x^{6} + 151 x^{5} + 1858 x^{4} - 228 x^{3} + 104 x^{2} + 8 x + 1 \)

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:  \(161761786626698377317203521728515625=3^{12}\cdot 5^{18}\cdot 7^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.31$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(105=3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{105}(64,·)$, $\chi_{105}(1,·)$, $\chi_{105}(4,·)$, $\chi_{105}(71,·)$, $\chi_{105}(73,·)$, $\chi_{105}(74,·)$, $\chi_{105}(11,·)$, $\chi_{105}(13,·)$, $\chi_{105}(79,·)$, $\chi_{105}(16,·)$, $\chi_{105}(17,·)$, $\chi_{105}(82,·)$, $\chi_{105}(83,·)$, $\chi_{105}(86,·)$, $\chi_{105}(68,·)$, $\chi_{105}(29,·)$, $\chi_{105}(97,·)$, $\chi_{105}(38,·)$, $\chi_{105}(103,·)$, $\chi_{105}(44,·)$, $\chi_{105}(46,·)$, $\chi_{105}(47,·)$, $\chi_{105}(52,·)$, $\chi_{105}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{211} a^{22} - \frac{49}{211} a^{21} - \frac{59}{211} a^{20} + \frac{62}{211} a^{19} + \frac{37}{211} a^{18} + \frac{65}{211} a^{16} + \frac{2}{211} a^{15} + \frac{54}{211} a^{14} + \frac{83}{211} a^{13} + \frac{42}{211} a^{12} - \frac{79}{211} a^{11} + \frac{59}{211} a^{10} - \frac{44}{211} a^{8} - \frac{32}{211} a^{7} - \frac{75}{211} a^{6} + \frac{28}{211} a^{5} + \frac{76}{211} a^{4} - \frac{51}{211} a^{3} + \frac{65}{211} a^{2} + \frac{6}{211} a + \frac{84}{211}$, $\frac{1}{456832271377309710091322972489} a^{23} - \frac{39348608473360543506793371}{456832271377309710091322972489} a^{22} - \frac{121507200911501365025189784988}{456832271377309710091322972489} a^{21} + \frac{53648090487354196230041004832}{456832271377309710091322972489} a^{20} + \frac{221050939650122458216446500756}{456832271377309710091322972489} a^{19} - \frac{22773360831694992579304494116}{456832271377309710091322972489} a^{18} + \frac{169677021334490974149675163630}{456832271377309710091322972489} a^{17} - \frac{216470569985541892362251563773}{456832271377309710091322972489} a^{16} - \frac{86043745972463006213189558297}{456832271377309710091322972489} a^{15} + \frac{143408409339316275696327972457}{456832271377309710091322972489} a^{14} + \frac{69025913972221943531548501575}{456832271377309710091322972489} a^{13} - \frac{130577167417326380509643943115}{456832271377309710091322972489} a^{12} - \frac{85941747520005224858615939552}{456832271377309710091322972489} a^{11} - \frac{73838701041902782781964402638}{456832271377309710091322972489} a^{10} + \frac{114052605614368853270552455217}{456832271377309710091322972489} a^{9} - \frac{112567794797683929215338800937}{456832271377309710091322972489} a^{8} - \frac{128589491869690871719250799738}{456832271377309710091322972489} a^{7} + \frac{24271269959522118767884412436}{456832271377309710091322972489} a^{6} + \frac{227647043511136034184717946103}{456832271377309710091322972489} a^{5} - \frac{183989869913187551332864801037}{456832271377309710091322972489} a^{4} - \frac{221138664117409888520439011546}{456832271377309710091322972489} a^{3} + \frac{48813271409280084327921348215}{456832271377309710091322972489} a^{2} - \frac{160986104412054575659933938600}{456832271377309710091322972489} a - \frac{183643322256744948450947283762}{456832271377309710091322972489}$

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

Class group and class number

$C_{26}$, which has order $26$ (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:  \( -\frac{40275169973073951974765831376}{456832271377309710091322972489} a^{23} + \frac{44176154617248139694770867886}{456832271377309710091322972489} a^{22} - \frac{526254136804407595493161055898}{456832271377309710091322972489} a^{21} + \frac{452133593815321818010938992798}{456832271377309710091322972489} a^{20} - \frac{4091053168577426392206461453599}{456832271377309710091322972489} a^{19} + \frac{3079183561562760803343853354932}{456832271377309710091322972489} a^{18} - \frac{20277701330267994781750663802336}{456832271377309710091322972489} a^{17} + \frac{12091439390537731797470379537475}{456832271377309710091322972489} a^{16} - \frac{72127968105567091836197407765794}{456832271377309710091322972489} a^{15} + \frac{35044226390447899636051562128720}{456832271377309710091322972489} a^{14} - \frac{182877981497119048911104545357783}{456832271377309710091322972489} a^{13} + \frac{61768276953422934216913385727128}{456832271377309710091322972489} a^{12} - \frac{337496456365936640068246521325888}{456832271377309710091322972489} a^{11} + \frac{83435537821838624839465578082000}{456832271377309710091322972489} a^{10} - \frac{442631703717544316978098014698024}{456832271377309710091322972489} a^{9} + \frac{59790576973265951487110728395544}{456832271377309710091322972489} a^{8} - \frac{396794199050430128674137048944060}{456832271377309710091322972489} a^{7} + \frac{51817097237324061116845908716956}{456832271377309710091322972489} a^{6} - \frac{221463053870749105378871015549672}{456832271377309710091322972489} a^{5} + \frac{74299008220157707971271501025}{2165081854868766398537075699} a^{4} - \frac{68856893084681350999602063672938}{456832271377309710091322972489} a^{3} + \frac{17068157777565733184268471093594}{456832271377309710091322972489} a^{2} - \frac{3585159392982813046942108051302}{456832271377309710091322972489} a + \frac{185853642695523601601844065182}{456832271377309710091322972489} \) (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:  \( 3391665.6012423597 \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-3}, \sqrt{5})\), 4.0.55125.1, 4.4.6125.1, 6.0.64827.1, 6.6.300125.1, 6.0.8103375.1, 8.0.3038765625.1, 12.0.65664686390625.1, 12.0.402196204142578125.1, \(\Q(\zeta_{35})^+\)

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
3Data not computed
5Data not computed
$7$7.12.10.5$x^{12} + 56 x^{6} + 1323$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
7.12.10.5$x^{12} + 56 x^{6} + 1323$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$