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Properties

Label 24.0.14339436896...5625.1
Degree $24$
Signature $[0, 12]$
Discriminant $5^{18}\cdot 19^{20}$
Root discriminant $38.89$
Ramified primes $5, 19$
Class number $52$ (GRH)
Class group $[52]$ (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![2401, -1715, 1568, 1379, -3407, -3879, 3314, -7188, 1727, 2925, 2205, 2951, 1610, 1736, 1068, 483, 595, -283, 56, 61, -16, 11, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - x^22 + 11*x^21 - 16*x^20 + 61*x^19 + 56*x^18 - 283*x^17 + 595*x^16 + 483*x^15 + 1068*x^14 + 1736*x^13 + 1610*x^12 + 2951*x^11 + 2205*x^10 + 2925*x^9 + 1727*x^8 - 7188*x^7 + 3314*x^6 - 3879*x^5 - 3407*x^4 + 1379*x^3 + 1568*x^2 - 1715*x + 2401)
 
gp: K = bnfinit(x^24 - x^23 - x^22 + 11*x^21 - 16*x^20 + 61*x^19 + 56*x^18 - 283*x^17 + 595*x^16 + 483*x^15 + 1068*x^14 + 1736*x^13 + 1610*x^12 + 2951*x^11 + 2205*x^10 + 2925*x^9 + 1727*x^8 - 7188*x^7 + 3314*x^6 - 3879*x^5 - 3407*x^4 + 1379*x^3 + 1568*x^2 - 1715*x + 2401, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} - x^{22} + 11 x^{21} - 16 x^{20} + 61 x^{19} + 56 x^{18} - 283 x^{17} + 595 x^{16} + 483 x^{15} + 1068 x^{14} + 1736 x^{13} + 1610 x^{12} + 2951 x^{11} + 2205 x^{10} + 2925 x^{9} + 1727 x^{8} - 7188 x^{7} + 3314 x^{6} - 3879 x^{5} - 3407 x^{4} + 1379 x^{3} + 1568 x^{2} - 1715 x + 2401 \)

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:  \(143394368963416893742964176177978515625=5^{18}\cdot 19^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.89$
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:  $5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(95=5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{95}(64,·)$, $\chi_{95}(1,·)$, $\chi_{95}(68,·)$, $\chi_{95}(69,·)$, $\chi_{95}(7,·)$, $\chi_{95}(8,·)$, $\chi_{95}(11,·)$, $\chi_{95}(12,·)$, $\chi_{95}(77,·)$, $\chi_{95}(18,·)$, $\chi_{95}(83,·)$, $\chi_{95}(84,·)$, $\chi_{95}(87,·)$, $\chi_{95}(88,·)$, $\chi_{95}(26,·)$, $\chi_{95}(27,·)$, $\chi_{95}(94,·)$, $\chi_{95}(31,·)$, $\chi_{95}(37,·)$, $\chi_{95}(39,·)$, $\chi_{95}(46,·)$, $\chi_{95}(49,·)$, $\chi_{95}(56,·)$, $\chi_{95}(58,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{11} a^{9} - \frac{2}{11} a^{8} + \frac{4}{11} a^{7} + \frac{3}{11} a^{6} + \frac{5}{11} a^{5} + \frac{1}{11} a^{4} - \frac{2}{11} a^{3} + \frac{4}{11} a^{2} + \frac{3}{11} a + \frac{5}{11}$, $\frac{1}{11} a^{10} - \frac{1}{11}$, $\frac{1}{11} a^{11} - \frac{1}{11} a$, $\frac{1}{11} a^{12} - \frac{1}{11} a^{2}$, $\frac{1}{11} a^{13} - \frac{1}{11} a^{3}$, $\frac{1}{121} a^{14} - \frac{2}{121} a^{13} + \frac{4}{121} a^{12} + \frac{3}{121} a^{11} + \frac{5}{121} a^{10} + \frac{2}{121} a^{9} - \frac{4}{121} a^{8} + \frac{8}{121} a^{7} - \frac{16}{121} a^{6} + \frac{32}{121} a^{5} + \frac{1}{121} a^{4} - \frac{2}{121} a^{3} + \frac{4}{121} a^{2} - \frac{19}{121} a + \frac{27}{121}$, $\frac{1}{121} a^{15} + \frac{1}{121} a^{10} - \frac{56}{121} a^{5} - \frac{56}{121}$, $\frac{1}{121} a^{16} + \frac{1}{121} a^{11} - \frac{56}{121} a^{6} - \frac{56}{121} a$, $\frac{1}{847} a^{17} + \frac{3}{847} a^{16} + \frac{2}{847} a^{15} + \frac{23}{847} a^{12} + \frac{2}{121} a^{11} + \frac{5}{121} a^{10} - \frac{3}{77} a^{9} + \frac{4}{11} a^{8} + \frac{25}{121} a^{7} + \frac{338}{847} a^{6} - \frac{5}{121} a^{5} - \frac{2}{11} a^{4} - \frac{38}{77} a^{3} + \frac{153}{847} a^{2} - \frac{157}{847} a - \frac{27}{121}$, $\frac{1}{161777} a^{18} + \frac{19}{161777} a^{17} + \frac{9}{14707} a^{16} - \frac{423}{161777} a^{15} - \frac{24}{23111} a^{14} + \frac{7135}{161777} a^{13} - \frac{6835}{161777} a^{12} - \frac{633}{23111} a^{11} - \frac{768}{161777} a^{10} + \frac{5296}{161777} a^{9} + \frac{4}{191} a^{8} - \frac{38323}{161777} a^{7} + \frac{66917}{161777} a^{6} - \frac{5392}{23111} a^{5} - \frac{7362}{161777} a^{4} - \frac{2657}{161777} a^{3} - \frac{1461}{161777} a^{2} - \frac{13495}{161777} a + \frac{11096}{23111}$, $\frac{1}{1779547} a^{19} - \frac{2}{1779547} a^{18} - \frac{491}{1779547} a^{17} + \frac{6284}{1779547} a^{16} + \frac{5659}{1779547} a^{15} - \frac{6718}{1779547} a^{14} + \frac{54576}{1779547} a^{13} - \frac{67176}{1779547} a^{12} - \frac{56124}{1779547} a^{11} + \frac{28109}{1779547} a^{10} - \frac{48045}{1779547} a^{9} + \frac{210072}{1779547} a^{8} + \frac{566864}{1779547} a^{7} - \frac{518179}{1779547} a^{6} - \frac{791061}{1779547} a^{5} + \frac{90443}{1779547} a^{4} + \frac{139522}{1779547} a^{3} + \frac{889101}{1779547} a^{2} - \frac{567575}{1779547} a + \frac{75258}{254221}$, $\frac{1}{1779547} a^{20} - \frac{3}{847} a^{16} + \frac{159}{254221} a^{15} + \frac{1}{847} a^{14} - \frac{5}{121} a^{13} - \frac{1}{121} a^{12} + \frac{2}{77} a^{11} + \frac{708}{23111} a^{10} + \frac{5}{121} a^{9} + \frac{414}{847} a^{8} + \frac{20}{121} a^{7} - \frac{16}{121} a^{6} + \frac{63465}{254221} a^{5} - \frac{47}{121} a^{4} - \frac{5}{121} a^{3} - \frac{216}{847} a^{2} - \frac{137}{847} a - \frac{86391}{254221}$, $\frac{1}{19575017} a^{21} + \frac{3}{19575017} a^{19} + \frac{60}{19575017} a^{18} + \frac{8185}{19575017} a^{17} + \frac{6420}{1779547} a^{16} - \frac{67668}{19575017} a^{15} + \frac{57000}{19575017} a^{14} + \frac{781708}{19575017} a^{13} + \frac{378953}{19575017} a^{12} - \frac{38627}{19575017} a^{11} + \frac{48346}{19575017} a^{10} - \frac{704332}{19575017} a^{9} - \frac{4190677}{19575017} a^{8} - \frac{5446724}{19575017} a^{7} - \frac{389090}{2796431} a^{6} - \frac{5820242}{19575017} a^{5} + \frac{6668313}{19575017} a^{4} + \frac{1407158}{19575017} a^{3} - \frac{8408949}{19575017} a^{2} + \frac{160818}{2796431} a + \frac{1174513}{2796431}$, $\frac{1}{137025119} a^{22} - \frac{1}{137025119} a^{21} - \frac{8}{137025119} a^{20} - \frac{31}{137025119} a^{19} + \frac{194}{137025119} a^{18} - \frac{25279}{137025119} a^{17} - \frac{8713}{19575017} a^{16} + \frac{177930}{137025119} a^{15} + \frac{78808}{19575017} a^{14} - \frac{409692}{19575017} a^{13} - \frac{5538872}{137025119} a^{12} - \frac{231780}{19575017} a^{11} - \frac{697105}{19575017} a^{10} + \frac{3636056}{137025119} a^{9} + \frac{1734307}{19575017} a^{8} + \frac{337251}{12456829} a^{7} + \frac{37476437}{137025119} a^{6} + \frac{11927511}{137025119} a^{5} + \frac{21711714}{137025119} a^{4} - \frac{6978602}{137025119} a^{3} + \frac{12656002}{137025119} a^{2} + \frac{4576122}{19575017} a + \frac{199493}{2796431}$, $\frac{1}{10550934163} a^{23} + \frac{6}{10550934163} a^{22} + \frac{41}{10550934163} a^{21} - \frac{241}{10550934163} a^{20} - \frac{1703}{10550934163} a^{19} + \frac{22097}{10550934163} a^{18} - \frac{495797}{1507276309} a^{17} + \frac{1043701}{959175833} a^{16} + \frac{4185585}{1507276309} a^{15} + \frac{257767}{1507276309} a^{14} - \frac{101399728}{10550934163} a^{13} - \frac{1241052}{30760741} a^{12} + \frac{62219961}{1507276309} a^{11} - \frac{412512565}{10550934163} a^{10} - \frac{24015449}{1507276309} a^{9} + \frac{3593856769}{10550934163} a^{8} + \frac{3599016587}{10550934163} a^{7} + \frac{3982767923}{10550934163} a^{6} + \frac{2032713931}{10550934163} a^{5} - \frac{5243289577}{10550934163} a^{4} - \frac{1037203711}{10550934163} a^{3} - \frac{110502046}{1507276309} a^{2} + \frac{28807937}{215325187} a - \frac{3668446}{30760741}$

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

Class group and class number

$C_{52}$, which has order $52$ (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:  \( -\frac{2879221}{10550934163} a^{23} - \frac{1034794}{10550934163} a^{22} + \frac{6208764}{10550934163} a^{21} - \frac{24835056}{10550934163} a^{20} + \frac{1552191}{10550934163} a^{19} - \frac{123440243}{10550934163} a^{18} - \frac{52257097}{1507276309} a^{17} + \frac{48117921}{959175833} a^{16} - \frac{73470374}{1507276309} a^{15} - \frac{486353180}{1507276309} a^{14} - \frac{6007987014}{10550934163} a^{13} - \frac{1194152276}{1507276309} a^{12} - \frac{1493207742}{1507276309} a^{11} - \frac{13465256925}{10550934163} a^{10} - \frac{2341738822}{1507276309} a^{9} - \frac{16622448743}{10550934163} a^{8} - \frac{15061426670}{10550934163} a^{7} + \frac{12340435847}{10550934163} a^{6} + \frac{17460079162}{10550934163} a^{5} - \frac{1967660791}{10550934163} a^{4} - \frac{3929642378}{10550934163} a^{3} + \frac{179019362}{215325187} a^{2} - \frac{26387247}{30760741} a - \frac{3621779}{30760741} \) (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:  \( 694660924.0250753 \) (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

\(\Q(\sqrt{-19}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-95}) \), 3.3.361.1, \(\Q(\sqrt{5}, \sqrt{-19})\), \(\Q(\zeta_{5})\), 4.4.45125.1, 6.0.2476099.1, 6.6.16290125.1, 6.0.309512375.1, 8.0.2036265625.1, 12.0.95797910278140625.1, 12.0.33171021564453125.1, 12.12.11974738784767578125.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/13.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ R ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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}$
$19$19.12.10.1$x^{12} - 171 x^{6} + 23104$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
19.12.10.1$x^{12} - 171 x^{6} + 23104$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$