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Properties

Label 24.24.8035229565...0000.1
Degree $24$
Signature $[24, 0]$
Discriminant $2^{93}\cdot 5^{12}\cdot 7^{16}$
Root discriminant $120.05$
Ramified primes $2, 5, 7$
Class number $2$ (GRH)
Class group $[2]$ (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![854853761, 3801597744, 307199304, -14279404728, -4700840216, 21527186808, 4493317356, -15422126936, -1791337062, 6090345968, 317777564, -1448821552, -6961878, 217154504, -6900956, -20818864, 1258535, 1263752, -104880, -46592, 4690, 944, -108, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 8*x^23 - 108*x^22 + 944*x^21 + 4690*x^20 - 46592*x^19 - 104880*x^18 + 1263752*x^17 + 1258535*x^16 - 20818864*x^15 - 6900956*x^14 + 217154504*x^13 - 6961878*x^12 - 1448821552*x^11 + 317777564*x^10 + 6090345968*x^9 - 1791337062*x^8 - 15422126936*x^7 + 4493317356*x^6 + 21527186808*x^5 - 4700840216*x^4 - 14279404728*x^3 + 307199304*x^2 + 3801597744*x + 854853761)
 
gp: K = bnfinit(x^24 - 8*x^23 - 108*x^22 + 944*x^21 + 4690*x^20 - 46592*x^19 - 104880*x^18 + 1263752*x^17 + 1258535*x^16 - 20818864*x^15 - 6900956*x^14 + 217154504*x^13 - 6961878*x^12 - 1448821552*x^11 + 317777564*x^10 + 6090345968*x^9 - 1791337062*x^8 - 15422126936*x^7 + 4493317356*x^6 + 21527186808*x^5 - 4700840216*x^4 - 14279404728*x^3 + 307199304*x^2 + 3801597744*x + 854853761, 1)
 

Normalized defining polynomial

\( x^{24} - 8 x^{23} - 108 x^{22} + 944 x^{21} + 4690 x^{20} - 46592 x^{19} - 104880 x^{18} + 1263752 x^{17} + 1258535 x^{16} - 20818864 x^{15} - 6900956 x^{14} + 217154504 x^{13} - 6961878 x^{12} - 1448821552 x^{11} + 317777564 x^{10} + 6090345968 x^{9} - 1791337062 x^{8} - 15422126936 x^{7} + 4493317356 x^{6} + 21527186808 x^{5} - 4700840216 x^{4} - 14279404728 x^{3} + 307199304 x^{2} + 3801597744 x + 854853761 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $24$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[24, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(80352295654101908234561020234985111552000000000000=2^{93}\cdot 5^{12}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $120.05$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1120=2^{5}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{1120}(1,·)$, $\chi_{1120}(961,·)$, $\chi_{1120}(389,·)$, $\chi_{1120}(641,·)$, $\chi_{1120}(841,·)$, $\chi_{1120}(989,·)$, $\chi_{1120}(589,·)$, $\chi_{1120}(109,·)$, $\chi_{1120}(401,·)$, $\chi_{1120}(149,·)$, $\chi_{1120}(921,·)$, $\chi_{1120}(281,·)$, $\chi_{1120}(669,·)$, $\chi_{1120}(709,·)$, $\chi_{1120}(869,·)$, $\chi_{1120}(81,·)$, $\chi_{1120}(681,·)$, $\chi_{1120}(429,·)$, $\chi_{1120}(29,·)$, $\chi_{1120}(561,·)$, $\chi_{1120}(949,·)$, $\chi_{1120}(361,·)$, $\chi_{1120}(121,·)$, $\chi_{1120}(309,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{7} - \frac{2}{5} a^{5} - \frac{1}{5} a^{3} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{12} + \frac{1}{25} a^{11} - \frac{2}{25} a^{10} - \frac{1}{25} a^{9} - \frac{1}{25} a^{8} - \frac{2}{25} a^{7} + \frac{2}{25} a^{6} - \frac{11}{25} a^{5} + \frac{2}{5} a^{4} + \frac{6}{25} a^{3} + \frac{1}{5} a^{2} - \frac{8}{25} a + \frac{6}{25}$, $\frac{1}{25} a^{13} + \frac{2}{25} a^{11} + \frac{1}{25} a^{10} - \frac{1}{25} a^{8} - \frac{1}{25} a^{7} + \frac{2}{25} a^{6} + \frac{6}{25} a^{5} + \frac{11}{25} a^{4} - \frac{1}{25} a^{3} + \frac{12}{25} a^{2} - \frac{11}{25} a - \frac{1}{25}$, $\frac{1}{25} a^{14} - \frac{1}{25} a^{11} - \frac{1}{25} a^{10} + \frac{1}{25} a^{9} + \frac{1}{25} a^{8} + \frac{1}{25} a^{7} + \frac{2}{25} a^{6} - \frac{7}{25} a^{5} - \frac{1}{25} a^{4} - \frac{1}{5} a^{3} - \frac{1}{25} a^{2} + \frac{1}{5} a + \frac{3}{25}$, $\frac{1}{25} a^{15} - \frac{1}{25} a^{10} + \frac{3}{25} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{11}{25}$, $\frac{1}{25} a^{16} - \frac{1}{25} a^{11} - \frac{2}{25} a^{6} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{6}{25} a - \frac{1}{5}$, $\frac{1}{25} a^{17} + \frac{1}{25} a^{11} - \frac{2}{25} a^{10} - \frac{1}{25} a^{9} - \frac{1}{25} a^{8} + \frac{1}{25} a^{7} + \frac{2}{25} a^{6} + \frac{4}{25} a^{5} - \frac{9}{25} a^{3} + \frac{11}{25} a^{2} + \frac{2}{25} a - \frac{9}{25}$, $\frac{1}{125} a^{18} - \frac{1}{125} a^{17} - \frac{2}{125} a^{16} + \frac{1}{125} a^{15} - \frac{1}{125} a^{13} + \frac{1}{125} a^{12} + \frac{2}{125} a^{11} + \frac{9}{125} a^{10} + \frac{2}{25} a^{9} + \frac{3}{125} a^{8} + \frac{7}{125} a^{7} - \frac{1}{125} a^{6} + \frac{48}{125} a^{5} + \frac{1}{25} a^{4} + \frac{26}{125} a^{3} - \frac{16}{125} a^{2} - \frac{27}{125} a + \frac{26}{125}$, $\frac{1}{125} a^{19} + \frac{2}{125} a^{17} - \frac{1}{125} a^{16} + \frac{1}{125} a^{15} - \frac{1}{125} a^{14} - \frac{2}{125} a^{12} + \frac{11}{125} a^{11} - \frac{6}{125} a^{10} - \frac{12}{125} a^{9} + \frac{2}{25} a^{8} - \frac{4}{125} a^{7} - \frac{3}{125} a^{6} + \frac{28}{125} a^{5} - \frac{19}{125} a^{4} - \frac{8}{25} a^{3} + \frac{62}{125} a^{2} + \frac{49}{125} a + \frac{1}{125}$, $\frac{1}{14125} a^{20} - \frac{16}{14125} a^{19} + \frac{31}{14125} a^{18} - \frac{72}{14125} a^{17} + \frac{179}{14125} a^{16} + \frac{37}{14125} a^{15} - \frac{84}{14125} a^{14} + \frac{139}{14125} a^{13} + \frac{222}{14125} a^{12} + \frac{1061}{14125} a^{11} - \frac{23}{2825} a^{10} + \frac{377}{14125} a^{9} - \frac{1187}{14125} a^{8} - \frac{816}{14125} a^{7} - \frac{148}{14125} a^{6} + \frac{701}{2825} a^{5} - \frac{1471}{14125} a^{4} - \frac{1274}{14125} a^{3} + \frac{3048}{14125} a^{2} - \frac{11}{14125} a + \frac{3358}{14125}$, $\frac{1}{14125} a^{21} + \frac{1}{14125} a^{19} - \frac{28}{14125} a^{18} - \frac{69}{14125} a^{17} + \frac{189}{14125} a^{16} + \frac{282}{14125} a^{15} + \frac{264}{14125} a^{14} + \frac{73}{14125} a^{13} - \frac{246}{14125} a^{12} - \frac{767}{14125} a^{11} - \frac{672}{14125} a^{10} + \frac{438}{14125} a^{9} + \frac{871}{14125} a^{8} + \frac{808}{14125} a^{7} + \frac{346}{14125} a^{6} - \frac{5394}{14125} a^{5} + \frac{841}{14125} a^{4} + \frac{5942}{14125} a^{3} + \frac{506}{14125} a^{2} - \frac{923}{2825} a - \frac{1868}{14125}$, $\frac{1}{66238025277002533390533688962125} a^{22} - \frac{1162106659335267330148827579}{66238025277002533390533688962125} a^{21} + \frac{787205577583924514773035881}{66238025277002533390533688962125} a^{20} + \frac{198524358016770487433580798013}{66238025277002533390533688962125} a^{19} - \frac{15329177150006195705880661452}{13247605055400506678106737792425} a^{18} - \frac{595652658016920973513849216682}{66238025277002533390533688962125} a^{17} - \frac{213221869090260024511724092148}{66238025277002533390533688962125} a^{16} + \frac{1118338720937022107847353200023}{66238025277002533390533688962125} a^{15} - \frac{1105432371732691468056156625148}{66238025277002533390533688962125} a^{14} + \frac{237393687816168124490406155903}{13247605055400506678106737792425} a^{13} - \frac{799620177341466202270552359671}{66238025277002533390533688962125} a^{12} + \frac{464467257971984102969681132991}{13247605055400506678106737792425} a^{11} + \frac{5554308061962785793707925006884}{66238025277002533390533688962125} a^{10} - \frac{2922032983153578498217573080436}{66238025277002533390533688962125} a^{9} + \frac{204281305765758463120514352961}{13247605055400506678106737792425} a^{8} - \frac{1289736415866343773306204393482}{66238025277002533390533688962125} a^{7} + \frac{256075394399422592453053952541}{2649521011080101335621347558485} a^{6} + \frac{14817790362905135687678213128138}{66238025277002533390533688962125} a^{5} + \frac{12298830127007530156411743096553}{66238025277002533390533688962125} a^{4} - \frac{1141950194094995847411573683279}{2649521011080101335621347558485} a^{3} + \frac{11060955669983068048565093089179}{66238025277002533390533688962125} a^{2} + \frac{26372924227742006318276245444223}{66238025277002533390533688962125} a + \frac{28629329942854156781376891428619}{66238025277002533390533688962125}$, $\frac{1}{33822894691229105979768231758358611736867377875} a^{23} + \frac{3241103565069}{1352915787649164239190729270334344469474695115} a^{22} + \frac{127662745676317981690391948190344479675793}{33822894691229105979768231758358611736867377875} a^{21} + \frac{26892626348066856908308720962972349668824}{33822894691229105979768231758358611736867377875} a^{20} - \frac{9670813102679158151110624716822614591120749}{33822894691229105979768231758358611736867377875} a^{19} + \frac{3066191604091467605141023112641363053359019}{33822894691229105979768231758358611736867377875} a^{18} + \frac{606897199285351088073921694087796458060582431}{33822894691229105979768231758358611736867377875} a^{17} - \frac{529249960091751380183731136763806909799228993}{33822894691229105979768231758358611736867377875} a^{16} + \frac{102261277069280974137738384401485567222866009}{6764578938245821195953646351671722347373475575} a^{15} + \frac{619889521280197269954873070731754285631229759}{33822894691229105979768231758358611736867377875} a^{14} - \frac{115290620092599796967686727783003456924953367}{33822894691229105979768231758358611736867377875} a^{13} + \frac{6661694418313227389742569743064920407008949}{33822894691229105979768231758358611736867377875} a^{12} + \frac{39768284574039224200297714226983282975015529}{33822894691229105979768231758358611736867377875} a^{11} + \frac{2571932459057794287276374567133741702649921058}{33822894691229105979768231758358611736867377875} a^{10} + \frac{1444583282270027069602340308253747195372036073}{33822894691229105979768231758358611736867377875} a^{9} - \frac{2253642928705710576710445983770557256445696994}{33822894691229105979768231758358611736867377875} a^{8} - \frac{2377438122714285348632132561902950881148517507}{33822894691229105979768231758358611736867377875} a^{7} + \frac{839467248265318635428947891249581449794838453}{33822894691229105979768231758358611736867377875} a^{6} - \frac{3111748835038253597151679232984092130020837419}{33822894691229105979768231758358611736867377875} a^{5} - \frac{10634216607329242812157830975161436790852268104}{33822894691229105979768231758358611736867377875} a^{4} + \frac{653628938977853943395761078785291234152285839}{6764578938245821195953646351671722347373475575} a^{3} + \frac{8388500803023261232473456662291970316364499641}{33822894691229105979768231758358611736867377875} a^{2} + \frac{807662229013138421695436419775724956090220444}{6764578938245821195953646351671722347373475575} a + \frac{4787681853216985950853604694701925482927665874}{33822894691229105979768231758358611736867377875}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $23$
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:  \( 78355829164159620 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{16})^+\), 6.6.1229312.1, 8.8.1342177280000.1, 12.12.49519263525896192.1

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 R $24$ R R $24$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{8}$ $24$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{8}$ $24$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{8}$ $24$ $24$

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
2Data not computed
5Data not computed
$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}$