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Properties

Label 24.0.93855357514...0001.1
Degree $24$
Signature $[0, 12]$
Discriminant $3^{12}\cdot 7^{20}\cdot 19^{12}$
Root discriminant $38.21$
Ramified primes $3, 7, 19$
Class number $84$ (GRH)
Class group $[84]$ (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![244140625, -48828125, 48828125, -27343750, 5468750, -5468750, 1109375, -1968750, 571250, -254875, 176750, -20230, 19671, -4046, 7070, -2039, 914, -630, 71, -70, 14, -14, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 5*x^22 - 14*x^21 + 14*x^20 - 70*x^19 + 71*x^18 - 630*x^17 + 914*x^16 - 2039*x^15 + 7070*x^14 - 4046*x^13 + 19671*x^12 - 20230*x^11 + 176750*x^10 - 254875*x^9 + 571250*x^8 - 1968750*x^7 + 1109375*x^6 - 5468750*x^5 + 5468750*x^4 - 27343750*x^3 + 48828125*x^2 - 48828125*x + 244140625)
 
gp: K = bnfinit(x^24 - x^23 + 5*x^22 - 14*x^21 + 14*x^20 - 70*x^19 + 71*x^18 - 630*x^17 + 914*x^16 - 2039*x^15 + 7070*x^14 - 4046*x^13 + 19671*x^12 - 20230*x^11 + 176750*x^10 - 254875*x^9 + 571250*x^8 - 1968750*x^7 + 1109375*x^6 - 5468750*x^5 + 5468750*x^4 - 27343750*x^3 + 48828125*x^2 - 48828125*x + 244140625, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} + 5 x^{22} - 14 x^{21} + 14 x^{20} - 70 x^{19} + 71 x^{18} - 630 x^{17} + 914 x^{16} - 2039 x^{15} + 7070 x^{14} - 4046 x^{13} + 19671 x^{12} - 20230 x^{11} + 176750 x^{10} - 254875 x^{9} + 571250 x^{8} - 1968750 x^{7} + 1109375 x^{6} - 5468750 x^{5} + 5468750 x^{4} - 27343750 x^{3} + 48828125 x^{2} - 48828125 x + 244140625 \)

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:  \(93855357514722449501524260642553280001=3^{12}\cdot 7^{20}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.21$
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:  $3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(399=3\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{399}(1,·)$, $\chi_{399}(134,·)$, $\chi_{399}(265,·)$, $\chi_{399}(398,·)$, $\chi_{399}(208,·)$, $\chi_{399}(20,·)$, $\chi_{399}(341,·)$, $\chi_{399}(151,·)$, $\chi_{399}(284,·)$, $\chi_{399}(286,·)$, $\chi_{399}(229,·)$, $\chi_{399}(227,·)$, $\chi_{399}(37,·)$, $\chi_{399}(113,·)$, $\chi_{399}(170,·)$, $\chi_{399}(172,·)$, $\chi_{399}(305,·)$, $\chi_{399}(115,·)$, $\chi_{399}(94,·)$, $\chi_{399}(248,·)$, $\chi_{399}(58,·)$, $\chi_{399}(379,·)$, $\chi_{399}(362,·)$, $\chi_{399}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{12} + \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{25} a^{14} - \frac{1}{25} a^{13} + \frac{1}{5} a^{12} + \frac{11}{25} a^{11} - \frac{11}{25} a^{10} + \frac{1}{5} a^{9} - \frac{4}{25} a^{8} - \frac{1}{5} a^{7} - \frac{11}{25} a^{6} + \frac{11}{25} a^{5} - \frac{1}{5} a^{4} + \frac{4}{25} a^{3} - \frac{4}{25} a^{2} - \frac{1}{5} a$, $\frac{1}{125} a^{15} - \frac{1}{125} a^{14} + \frac{1}{25} a^{13} - \frac{14}{125} a^{12} + \frac{14}{125} a^{11} + \frac{11}{25} a^{10} - \frac{54}{125} a^{9} - \frac{1}{25} a^{8} + \frac{39}{125} a^{7} - \frac{39}{125} a^{6} - \frac{11}{25} a^{5} - \frac{46}{125} a^{4} + \frac{46}{125} a^{3} + \frac{4}{25} a^{2}$, $\frac{1}{625} a^{16} - \frac{1}{625} a^{15} + \frac{1}{125} a^{14} - \frac{14}{625} a^{13} + \frac{14}{625} a^{12} - \frac{14}{125} a^{11} + \frac{71}{625} a^{10} - \frac{1}{125} a^{9} + \frac{289}{625} a^{8} - \frac{164}{625} a^{7} + \frac{39}{125} a^{6} - \frac{296}{625} a^{5} + \frac{296}{625} a^{4} - \frac{46}{125} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{3125} a^{17} - \frac{1}{3125} a^{16} + \frac{1}{625} a^{15} - \frac{14}{3125} a^{14} + \frac{14}{3125} a^{13} - \frac{14}{625} a^{12} + \frac{71}{3125} a^{11} - \frac{126}{625} a^{10} + \frac{914}{3125} a^{9} + \frac{1086}{3125} a^{8} + \frac{164}{625} a^{7} - \frac{921}{3125} a^{6} + \frac{921}{3125} a^{5} - \frac{296}{625} a^{4} - \frac{11}{25} a^{3} + \frac{11}{25} a^{2} - \frac{1}{5} a$, $\frac{1}{62500} a^{18} + \frac{1}{15625} a^{17} + \frac{11}{62500} a^{15} - \frac{14}{15625} a^{14} - \frac{279}{62500} a^{12} - \frac{159}{625} a^{11} - \frac{559}{15625} a^{10} + \frac{2531}{62500} a^{9} + \frac{1}{5} a^{8} - \frac{7799}{15625} a^{7} - \frac{559}{62500} a^{6} + \frac{131}{625} a^{4} + \frac{31}{500} a^{3} + \frac{1}{5} a + \frac{1}{4}$, $\frac{1}{18309062500} a^{19} + \frac{116339}{18309062500} a^{18} + \frac{82067}{915453125} a^{17} - \frac{1175989}{18309062500} a^{16} + \frac{7159629}{18309062500} a^{15} - \frac{1148938}{915453125} a^{14} + \frac{16463721}{18309062500} a^{13} + \frac{1712966327}{3661812500} a^{12} - \frac{1072105059}{4577265625} a^{11} - \frac{935377529}{18309062500} a^{10} - \frac{905373423}{3661812500} a^{9} - \frac{2200676924}{4577265625} a^{8} - \frac{7271742219}{18309062500} a^{7} - \frac{45875453}{3661812500} a^{6} + \frac{1644216}{183090625} a^{5} + \frac{63788099}{146472500} a^{4} + \frac{2650077}{29294500} a^{3} - \frac{117444}{1464725} a^{2} + \frac{46929}{1171780} a + \frac{70195}{234356}$, $\frac{1}{91545312500} a^{20} - \frac{1}{91545312500} a^{19} - \frac{128729}{18309062500} a^{18} + \frac{10315011}{91545312500} a^{17} - \frac{12889611}{91545312500} a^{16} + \frac{11473581}{18309062500} a^{15} - \frac{144410279}{91545312500} a^{14} + \frac{1772012699}{18309062500} a^{13} - \frac{9402303861}{91545312500} a^{12} - \frac{33388694289}{91545312500} a^{11} + \frac{8483264569}{18309062500} a^{10} + \frac{15720081929}{91545312500} a^{9} + \frac{6674402421}{91545312500} a^{8} + \frac{7604896159}{18309062500} a^{7} + \frac{857795299}{3661812500} a^{6} - \frac{168680657}{732362500} a^{5} - \frac{55647747}{146472500} a^{4} - \frac{9901191}{29294500} a^{3} + \frac{665413}{5858900} a^{2} + \frac{225183}{1171780} a + \frac{105471}{234356}$, $\frac{1}{457726562500} a^{21} - \frac{1}{457726562500} a^{20} + \frac{1}{91545312500} a^{19} + \frac{523111}{457726562500} a^{18} + \frac{46636889}{457726562500} a^{17} - \frac{8908889}{91545312500} a^{16} + \frac{228476321}{457726562500} a^{15} + \frac{1605338499}{91545312500} a^{14} - \frac{8055987211}{457726562500} a^{13} + \frac{40133986711}{457726562500} a^{12} + \frac{42477025289}{91545312500} a^{11} - \frac{78067745921}{457726562500} a^{10} + \frac{12884873421}{457726562500} a^{9} - \frac{20359668921}{91545312500} a^{8} + \frac{541129129}{3661812500} a^{7} + \frac{314040591}{3661812500} a^{6} - \frac{12655099}{146472500} a^{5} - \frac{70541281}{146472500} a^{4} + \frac{4649381}{29294500} a^{3} + \frac{237703}{1171780} a^{2} - \frac{31}{1171780} a + \frac{1044}{58589}$, $\frac{1}{2288632812500} a^{22} - \frac{1}{2288632812500} a^{21} + \frac{1}{457726562500} a^{20} - \frac{7}{1144316406250} a^{19} + \frac{9503139}{2288632812500} a^{18} - \frac{39041889}{457726562500} a^{17} + \frac{116610973}{1144316406250} a^{16} + \frac{1523606749}{457726562500} a^{15} - \frac{5946677211}{2288632812500} a^{14} + \frac{20066469293}{1144316406250} a^{13} - \frac{21568072961}{457726562500} a^{12} - \frac{1100011100921}{2288632812500} a^{11} - \frac{180892210477}{1144316406250} a^{10} + \frac{14976800329}{457726562500} a^{9} + \frac{7839397889}{18309062500} a^{8} - \frac{4493662657}{9154531250} a^{7} - \frac{29586801}{3661812500} a^{6} + \frac{1531859}{17862500} a^{5} - \frac{28919439}{73236250} a^{4} + \frac{986669}{29294500} a^{3} + \frac{937469}{5858900} a^{2} - \frac{472073}{1171780} a - \frac{29709}{117178}$, $\frac{1}{11443164062500} a^{23} - \frac{1}{11443164062500} a^{22} + \frac{1}{2288632812500} a^{21} - \frac{7}{5721582031250} a^{20} + \frac{7}{5721582031250} a^{19} + \frac{4405309}{572158203125} a^{18} + \frac{847642223}{5721582031250} a^{17} + \frac{733674937}{1144316406250} a^{16} - \frac{249035709}{2860791015625} a^{15} + \frac{9832031793}{5721582031250} a^{14} - \frac{10271449293}{1144316406250} a^{13} + \frac{733181801}{2860791015625} a^{12} + \frac{1764534252023}{5721582031250} a^{11} - \frac{420209653273}{1144316406250} a^{10} - \frac{1685957309}{22886328125} a^{9} + \frac{2581696043}{45772656250} a^{8} - \frac{2059667353}{9154531250} a^{7} - \frac{644304}{915453125} a^{6} - \frac{124355223}{366181250} a^{5} - \frac{10540119}{73236250} a^{4} - \frac{457004}{7323625} a^{3} - \frac{230161}{5858900} a^{2} - \frac{843}{1171780} a - \frac{57751}{234356}$

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

Class group and class number

$C_{84}$, which has order $84$ (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:  \( -\frac{155961}{91545312500} a^{22} + \frac{279}{390625} a^{15} - \frac{77836}{390625} a^{8} + \frac{4359375}{234356} a \) (order $42$)
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:  \( 445267011.8484134 \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{57}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-399}) \), \(\Q(\sqrt{133}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-3}, \sqrt{-19})\), \(\Q(\sqrt{-3}, \sqrt{133})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-7}, \sqrt{57})\), \(\Q(\sqrt{21}, \sqrt{57})\), \(\Q(\sqrt{-19}, \sqrt{21})\), \(\Q(\sqrt{-7}, \sqrt{-19})\), 6.0.64827.1, 6.6.444648393.1, 6.0.16468459.1, 6.0.3112538751.2, 6.6.115279213.1, \(\Q(\zeta_{7})\), \(\Q(\zeta_{21})^+\), 8.0.25344958401.1, 12.0.197712193397482449.1, 12.0.9687897476476640001.2, \(\Q(\zeta_{21})\), 12.0.9687897476476640001.3, 12.12.9687897476476640001.1, 12.0.9687897476476640001.1, 12.0.13289296949899369.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$19$19.12.6.1$x^{12} + 41154 x^{6} - 2476099 x^{2} + 423412929$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
19.12.6.1$x^{12} + 41154 x^{6} - 2476099 x^{2} + 423412929$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$