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Properties

Label 24.0.80352295654...0000.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{93}\cdot 5^{12}\cdot 7^{16}$
Root discriminant $120.05$
Ramified primes $2, 5, 7$
Class number $25812562$ (GRH)
Class group $[25812562]$ (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![138265878241, -158776304176, 208299167304, -78964916728, 66295185944, -39972851752, 35133764236, -23276555096, 15378660938, -7693059152, 3963223884, -1555088112, 648555642, -203685496, 70097284, -17687984, 5077575, -1018488, 244320, -37632, 7490, -816, 132, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 8*x^23 + 132*x^22 - 816*x^21 + 7490*x^20 - 37632*x^19 + 244320*x^18 - 1018488*x^17 + 5077575*x^16 - 17687984*x^15 + 70097284*x^14 - 203685496*x^13 + 648555642*x^12 - 1555088112*x^11 + 3963223884*x^10 - 7693059152*x^9 + 15378660938*x^8 - 23276555096*x^7 + 35133764236*x^6 - 39972851752*x^5 + 66295185944*x^4 - 78964916728*x^3 + 208299167304*x^2 - 158776304176*x + 138265878241)
 
gp: K = bnfinit(x^24 - 8*x^23 + 132*x^22 - 816*x^21 + 7490*x^20 - 37632*x^19 + 244320*x^18 - 1018488*x^17 + 5077575*x^16 - 17687984*x^15 + 70097284*x^14 - 203685496*x^13 + 648555642*x^12 - 1555088112*x^11 + 3963223884*x^10 - 7693059152*x^9 + 15378660938*x^8 - 23276555096*x^7 + 35133764236*x^6 - 39972851752*x^5 + 66295185944*x^4 - 78964916728*x^3 + 208299167304*x^2 - 158776304176*x + 138265878241, 1)
 

Normalized defining polynomial

\( x^{24} - 8 x^{23} + 132 x^{22} - 816 x^{21} + 7490 x^{20} - 37632 x^{19} + 244320 x^{18} - 1018488 x^{17} + 5077575 x^{16} - 17687984 x^{15} + 70097284 x^{14} - 203685496 x^{13} + 648555642 x^{12} - 1555088112 x^{11} + 3963223884 x^{10} - 7693059152 x^{9} + 15378660938 x^{8} - 23276555096 x^{7} + 35133764236 x^{6} - 39972851752 x^{5} + 66295185944 x^{4} - 78964916728 x^{3} + 208299167304 x^{2} - 158776304176 x + 138265878241 \)

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:  \(80352295654101908234561020234985111552000000000000=2^{93}\cdot 5^{12}\cdot 7^{16}\)
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$$\chi_{1120}(1,·)$, $\chi_{1120}(961,·)$, $\chi_{1120}(459,·)$, $\chi_{1120}(641,·)$, $\chi_{1120}(841,·)$, $\chi_{1120}(779,·)$, $\chi_{1120}(499,·)$, $\chi_{1120}(401,·)$, $\chi_{1120}(659,·)$, $\chi_{1120}(921,·)$, $\chi_{1120}(281,·)$, $\chi_{1120}(219,·)$, $\chi_{1120}(361,·)$, $\chi_{1120}(1059,·)$, $\chi_{1120}(81,·)$, $\chi_{1120}(379,·)$, $\chi_{1120}(681,·)$, $\chi_{1120}(939,·)$, $\chi_{1120}(99,·)$, $\chi_{1120}(561,·)$, $\chi_{1120}(179,·)$, $\chi_{1120}(739,·)$, $\chi_{1120}(121,·)$, $\chi_{1120}(1019,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{7} - \frac{2}{5} a^{5} - \frac{1}{5} a^{3} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{12} + \frac{1}{25} a^{11} - \frac{2}{25} a^{10} - \frac{1}{25} a^{9} - \frac{1}{25} a^{8} - \frac{2}{25} a^{7} + \frac{2}{25} a^{6} - \frac{11}{25} a^{5} + \frac{2}{5} a^{4} + \frac{6}{25} a^{3} + \frac{1}{5} a^{2} - \frac{8}{25} a + \frac{6}{25}$, $\frac{1}{25} a^{13} + \frac{2}{25} a^{11} + \frac{1}{25} a^{10} - \frac{1}{25} a^{8} - \frac{1}{25} a^{7} + \frac{2}{25} a^{6} + \frac{6}{25} a^{5} + \frac{11}{25} a^{4} - \frac{1}{25} a^{3} + \frac{12}{25} a^{2} - \frac{11}{25} a - \frac{1}{25}$, $\frac{1}{25} a^{14} - \frac{1}{25} a^{11} - \frac{1}{25} a^{10} + \frac{1}{25} a^{9} + \frac{1}{25} a^{8} + \frac{1}{25} a^{7} + \frac{2}{25} a^{6} - \frac{7}{25} a^{5} - \frac{1}{25} a^{4} - \frac{1}{5} a^{3} - \frac{1}{25} a^{2} + \frac{1}{5} a + \frac{3}{25}$, $\frac{1}{25} a^{15} - \frac{1}{25} a^{10} + \frac{3}{25} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{11}{25}$, $\frac{1}{25} a^{16} - \frac{1}{25} a^{11} - \frac{2}{25} a^{6} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{6}{25} a - \frac{1}{5}$, $\frac{1}{25} a^{17} + \frac{1}{25} a^{11} - \frac{2}{25} a^{10} - \frac{1}{25} a^{9} - \frac{1}{25} a^{8} + \frac{1}{25} a^{7} + \frac{2}{25} a^{6} + \frac{4}{25} a^{5} - \frac{9}{25} a^{3} + \frac{11}{25} a^{2} + \frac{2}{25} a - \frac{9}{25}$, $\frac{1}{125} a^{18} - \frac{1}{125} a^{17} - \frac{2}{125} a^{16} + \frac{1}{125} a^{15} - \frac{1}{125} a^{13} + \frac{1}{125} a^{12} + \frac{2}{125} a^{11} + \frac{9}{125} a^{10} + \frac{2}{25} a^{9} + \frac{3}{125} a^{8} + \frac{7}{125} a^{7} - \frac{1}{125} a^{6} + \frac{48}{125} a^{5} + \frac{1}{25} a^{4} + \frac{26}{125} a^{3} - \frac{16}{125} a^{2} - \frac{27}{125} a + \frac{26}{125}$, $\frac{1}{125} a^{19} + \frac{2}{125} a^{17} - \frac{1}{125} a^{16} + \frac{1}{125} a^{15} - \frac{1}{125} a^{14} - \frac{2}{125} a^{12} + \frac{11}{125} a^{11} - \frac{6}{125} a^{10} - \frac{12}{125} a^{9} + \frac{2}{25} a^{8} - \frac{4}{125} a^{7} - \frac{3}{125} a^{6} + \frac{28}{125} a^{5} - \frac{19}{125} a^{4} - \frac{8}{25} a^{3} + \frac{62}{125} a^{2} + \frac{49}{125} a + \frac{1}{125}$, $\frac{1}{125} a^{20} + \frac{1}{125} a^{17} + \frac{2}{125} a^{15} - \frac{1}{125} a^{12} + \frac{2}{25} a^{11} + \frac{2}{25} a^{10} + \frac{3}{125} a^{7} - \frac{1}{25} a^{6} - \frac{8}{25} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{6}{125} a^{2} + \frac{11}{25} a - \frac{57}{125}$, $\frac{1}{125} a^{21} + \frac{1}{125} a^{17} - \frac{1}{125} a^{16} - \frac{1}{125} a^{15} - \frac{1}{125} a^{12} + \frac{3}{125} a^{11} + \frac{11}{125} a^{10} + \frac{2}{25} a^{8} + \frac{8}{125} a^{7} + \frac{1}{125} a^{6} + \frac{37}{125} a^{5} - \frac{6}{25} a^{4} + \frac{4}{25} a^{3} + \frac{46}{125} a^{2} - \frac{11}{25} a - \frac{11}{125}$, $\frac{1}{13885858676604917216385009975295692784940780125} a^{22} - \frac{7987001408115921457583165034260688822037897}{2777171735320983443277001995059138556988156025} a^{21} - \frac{13446291224706215310141536958114168412030733}{13885858676604917216385009975295692784940780125} a^{20} + \frac{51301707587099360591575907907143561515968109}{13885858676604917216385009975295692784940780125} a^{19} + \frac{6914201656413445224808990774198373487359572}{2777171735320983443277001995059138556988156025} a^{18} + \frac{752858668721357223820213492001412046784529}{2777171735320983443277001995059138556988156025} a^{17} - \frac{212365809036963713365088332230580461291544493}{13885858676604917216385009975295692784940780125} a^{16} + \frac{198882664333388945150816518218766086768065137}{13885858676604917216385009975295692784940780125} a^{15} + \frac{220124857431907557444368570776267942686620641}{13885858676604917216385009975295692784940780125} a^{14} - \frac{6319281011112741771546540833302416953872019}{555434347064196688655400399011827711397631205} a^{13} - \frac{5231885565445237366534801052441922130839283}{13885858676604917216385009975295692784940780125} a^{12} + \frac{767093377401386678578034580459468992909571148}{13885858676604917216385009975295692784940780125} a^{11} + \frac{1365280777225700727318816529872778251491478802}{13885858676604917216385009975295692784940780125} a^{10} - \frac{1287584068109091555960069668664393548557957143}{13885858676604917216385009975295692784940780125} a^{9} - \frac{10436632913915376735526396308602790324045744}{2777171735320983443277001995059138556988156025} a^{8} + \frac{657502418426947509984107688093549085847418484}{13885858676604917216385009975295692784940780125} a^{7} + \frac{366696161039752522817992308110293625453079971}{13885858676604917216385009975295692784940780125} a^{6} - \frac{6836839098282172162057649733184156104645738146}{13885858676604917216385009975295692784940780125} a^{5} + \frac{5364659310291942282515091058512761015639388704}{13885858676604917216385009975295692784940780125} a^{4} - \frac{220478372755964802893035853526742121674089681}{2777171735320983443277001995059138556988156025} a^{3} - \frac{3672242348012347902638067496444681948497175599}{13885858676604917216385009975295692784940780125} a^{2} - \frac{1494493912696753878254798016921630536517985368}{13885858676604917216385009975295692784940780125} a - \frac{6802703091103435022633705868913177334757970191}{13885858676604917216385009975295692784940780125}$, $\frac{1}{14320646185603515073149911099972371322886344924882924499866400125} a^{23} + \frac{397495542267170923}{14320646185603515073149911099972371322886344924882924499866400125} a^{22} + \frac{44165490586060263410990716295507663752577492072506126059949326}{14320646185603515073149911099972371322886344924882924499866400125} a^{21} + \frac{53698166099739247445977287308728389124618246227737363559951678}{14320646185603515073149911099972371322886344924882924499866400125} a^{20} - \frac{35329840489786402983104005137767926484563112461283586343192371}{14320646185603515073149911099972371322886344924882924499866400125} a^{19} - \frac{1956790389656330112184960739487043893738240412112298473857581}{2864129237120703014629982219994474264577268984976584899973280025} a^{18} + \frac{24959177010139034503423229017463240863564975246557340558708463}{14320646185603515073149911099972371322886344924882924499866400125} a^{17} - \frac{251249025105322996160781806958787112987681829117394169823429123}{14320646185603515073149911099972371322886344924882924499866400125} a^{16} + \frac{274948990972673024615787412095139691789493658252784626067864131}{14320646185603515073149911099972371322886344924882924499866400125} a^{15} + \frac{91249857346440575564701953089911367980616964726093853934217826}{14320646185603515073149911099972371322886344924882924499866400125} a^{14} + \frac{120300928595290597929030887392944106940719548229806742495057392}{14320646185603515073149911099972371322886344924882924499866400125} a^{13} + \frac{49388409858318816153885571427423537376307178448963669746723268}{14320646185603515073149911099972371322886344924882924499866400125} a^{12} + \frac{238223849922182900697968017612102746848652188759085575617136591}{2864129237120703014629982219994474264577268984976584899973280025} a^{11} + \frac{95582120809972443784685275803013086745356599882756694304126389}{2864129237120703014629982219994474264577268984976584899973280025} a^{10} - \frac{715038288296651140829346637667316156538039287702218834279679738}{14320646185603515073149911099972371322886344924882924499866400125} a^{9} + \frac{192093442723011653699416548775228392237735419742389999322719739}{14320646185603515073149911099972371322886344924882924499866400125} a^{8} - \frac{1215047414223871719452165665888420224688728086018349038766076354}{14320646185603515073149911099972371322886344924882924499866400125} a^{7} - \frac{255090538719802631255635501253609899395017073118880851048902334}{2864129237120703014629982219994474264577268984976584899973280025} a^{6} + \frac{652937070794737281922205592715814495978539126035831252340024572}{2864129237120703014629982219994474264577268984976584899973280025} a^{5} - \frac{3934040065611495743048335318496204387560573897391592124966940786}{14320646185603515073149911099972371322886344924882924499866400125} a^{4} - \frac{7060469055047201153047399031837043114338387230329759721989568234}{14320646185603515073149911099972371322886344924882924499866400125} a^{3} + \frac{4306825899557539747041563658439339076634729195529209855071221246}{14320646185603515073149911099972371322886344924882924499866400125} a^{2} + \frac{26326530077529203963155889147955434791488501769393012259513496}{126731382173482434275662930088251073653861459512238269910322125} a + \frac{5787170198637107573129316163489419164873818618503074434817617574}{14320646185603515073149911099972371322886344924882924499866400125}$

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

Class group and class number

$C_{25812562}$, which has order $25812562$ (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:  \( 39019312.21180779 \) (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, 8.0.1342177280000.1, 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 $24$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{8}$ $24$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ $24$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ $24$ $24$

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}$