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Properties

Label 24.0.24548764029...0441.1
Degree $24$
Signature $[0, 12]$
Discriminant $7^{16}\cdot 41^{21}$
Root discriminant $94.32$
Ramified primes $7, 41$
Class number $6436$ (GRH)
Class group $[2, 3218]$ (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![40608077, 5264033, -63782464, -483717, 44954508, -43925395, 48792467, -34272758, 25647354, -14846103, 7176468, -3573246, 1615995, -616456, 207606, -72096, 22384, -7779, 3090, -700, -39, 61, 1, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 5*x^23 + x^22 + 61*x^21 - 39*x^20 - 700*x^19 + 3090*x^18 - 7779*x^17 + 22384*x^16 - 72096*x^15 + 207606*x^14 - 616456*x^13 + 1615995*x^12 - 3573246*x^11 + 7176468*x^10 - 14846103*x^9 + 25647354*x^8 - 34272758*x^7 + 48792467*x^6 - 43925395*x^5 + 44954508*x^4 - 483717*x^3 - 63782464*x^2 + 5264033*x + 40608077)
 
gp: K = bnfinit(x^24 - 5*x^23 + x^22 + 61*x^21 - 39*x^20 - 700*x^19 + 3090*x^18 - 7779*x^17 + 22384*x^16 - 72096*x^15 + 207606*x^14 - 616456*x^13 + 1615995*x^12 - 3573246*x^11 + 7176468*x^10 - 14846103*x^9 + 25647354*x^8 - 34272758*x^7 + 48792467*x^6 - 43925395*x^5 + 44954508*x^4 - 483717*x^3 - 63782464*x^2 + 5264033*x + 40608077, 1)
 

Normalized defining polynomial

\( x^{24} - 5 x^{23} + x^{22} + 61 x^{21} - 39 x^{20} - 700 x^{19} + 3090 x^{18} - 7779 x^{17} + 22384 x^{16} - 72096 x^{15} + 207606 x^{14} - 616456 x^{13} + 1615995 x^{12} - 3573246 x^{11} + 7176468 x^{10} - 14846103 x^{9} + 25647354 x^{8} - 34272758 x^{7} + 48792467 x^{6} - 43925395 x^{5} + 44954508 x^{4} - 483717 x^{3} - 63782464 x^{2} + 5264033 x + 40608077 \)

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:  \(245487640296656844759247184649542028256755290441=7^{16}\cdot 41^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $94.32$
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:  $7, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(287=7\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{287}(1,·)$, $\chi_{287}(260,·)$, $\chi_{287}(9,·)$, $\chi_{287}(247,·)$, $\chi_{287}(204,·)$, $\chi_{287}(79,·)$, $\chi_{287}(81,·)$, $\chi_{287}(85,·)$, $\chi_{287}(214,·)$, $\chi_{287}(155,·)$, $\chi_{287}(284,·)$, $\chi_{287}(32,·)$, $\chi_{287}(163,·)$, $\chi_{287}(165,·)$, $\chi_{287}(232,·)$, $\chi_{287}(44,·)$, $\chi_{287}(109,·)$, $\chi_{287}(50,·)$, $\chi_{287}(219,·)$, $\chi_{287}(137,·)$, $\chi_{287}(120,·)$, $\chi_{287}(249,·)$, $\chi_{287}(114,·)$, $\chi_{287}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{23074} a^{19} - \frac{423}{11537} a^{18} + \frac{6945}{23074} a^{17} - \frac{7211}{23074} a^{16} - \frac{3643}{23074} a^{15} + \frac{5239}{11537} a^{14} + \frac{6669}{23074} a^{13} + \frac{2925}{11537} a^{12} + \frac{2286}{11537} a^{11} - \frac{2875}{11537} a^{10} - \frac{1666}{11537} a^{9} - \frac{1116}{11537} a^{8} - \frac{3484}{11537} a^{7} + \frac{4280}{11537} a^{6} - \frac{5443}{11537} a^{5} + \frac{8309}{23074} a^{4} - \frac{8807}{23074} a^{3} + \frac{483}{23074} a^{2} + \frac{9609}{23074} a - \frac{3315}{11537}$, $\frac{1}{23074} a^{20} - \frac{2507}{11537} a^{18} + \frac{7463}{23074} a^{17} - \frac{538}{11537} a^{16} + \frac{8879}{23074} a^{15} - \frac{448}{11537} a^{14} - \frac{2653}{11537} a^{13} + \frac{4299}{23074} a^{12} + \frac{4402}{11537} a^{11} + \frac{391}{11537} a^{10} - \frac{3038}{11537} a^{9} - \frac{1586}{11537} a^{8} - \frac{1249}{11537} a^{7} + \frac{4356}{11537} a^{6} + \frac{5279}{23074} a^{5} + \frac{6111}{23074} a^{4} - \frac{4437}{11537} a^{3} - \frac{4321}{11537} a^{2} - \frac{11001}{23074} a + \frac{9539}{23074}$, $\frac{1}{853738} a^{21} - \frac{3}{426869} a^{20} + \frac{9}{853738} a^{19} + \frac{33705}{853738} a^{18} + \frac{389399}{853738} a^{17} - \frac{81965}{426869} a^{16} - \frac{170647}{853738} a^{15} - \frac{365}{426869} a^{14} - \frac{88296}{426869} a^{13} - \frac{83580}{426869} a^{12} + \frac{34927}{426869} a^{11} - \frac{163703}{426869} a^{10} + \frac{1253}{3071} a^{9} + \frac{1818}{11537} a^{8} - \frac{171245}{426869} a^{7} - \frac{13901}{853738} a^{6} - \frac{274375}{853738} a^{5} - \frac{4151}{853738} a^{4} + \frac{247565}{853738} a^{3} + \frac{137465}{426869} a^{2} + \frac{956}{11537} a + \frac{118010}{426869}$, $\frac{1}{853738} a^{22} + \frac{5}{426869} a^{20} + \frac{15}{853738} a^{19} - \frac{73557}{853738} a^{18} + \frac{313251}{853738} a^{17} - \frac{327647}{853738} a^{16} + \frac{148769}{853738} a^{15} - \frac{168112}{426869} a^{14} - \frac{110600}{426869} a^{13} - \frac{109227}{853738} a^{12} - \frac{93631}{426869} a^{11} + \frac{174891}{426869} a^{10} + \frac{16920}{426869} a^{9} - \frac{159368}{426869} a^{8} - \frac{103549}{853738} a^{7} - \frac{320633}{853738} a^{6} - \frac{185748}{426869} a^{5} + \frac{47967}{426869} a^{4} - \frac{96455}{426869} a^{3} - \frac{192118}{426869} a^{2} - \frac{425947}{853738} a + \frac{104951}{853738}$, $\frac{1}{766210979607372118004502324099345009224024590117487258827936246728514801809662134} a^{23} - \frac{117814807488682976325964665236446080073323137245782501767977873433088781412}{383105489803686059002251162049672504612012295058743629413968123364257400904831067} a^{22} - \frac{30948891074172577879517357149455327036806396739326121358565118969965891114}{383105489803686059002251162049672504612012295058743629413968123364257400904831067} a^{21} - \frac{7174173686775414682246981664784276157330837806907020202814265644560614951097}{766210979607372118004502324099345009224024590117487258827936246728514801809662134} a^{20} + \frac{4692080451471203823145596910223425589931359916237072179551454352504447697813}{766210979607372118004502324099345009224024590117487258827936246728514801809662134} a^{19} - \frac{28846643904979947798551913197788942596425587782996992820609728046925439227846925}{766210979607372118004502324099345009224024590117487258827936246728514801809662134} a^{18} + \frac{116670333705274457775951907838967186001517507039629699005688219746766763922403863}{766210979607372118004502324099345009224024590117487258827936246728514801809662134} a^{17} + \frac{107304578803298511543392787125861088078161611984691758142929718868526605122743045}{766210979607372118004502324099345009224024590117487258827936246728514801809662134} a^{16} + \frac{138154212854277247724083922121238729997593221665141526908222249335634097665650055}{383105489803686059002251162049672504612012295058743629413968123364257400904831067} a^{15} - \frac{28506435448957753747017362024860237754591967952833403614329929496549067881036383}{383105489803686059002251162049672504612012295058743629413968123364257400904831067} a^{14} + \frac{126422137928117895370315546648256624501097438164850483470440184538160774322131181}{766210979607372118004502324099345009224024590117487258827936246728514801809662134} a^{13} + \frac{12515972968736592186120478897707180989232279229690751985161507465776990159969616}{383105489803686059002251162049672504612012295058743629413968123364257400904831067} a^{12} - \frac{120054378209148283890903364835069173813085595178603631708105233120585116412118207}{383105489803686059002251162049672504612012295058743629413968123364257400904831067} a^{11} - \frac{74465409283271287073379880880738165227277771449588893819007520247515584537915110}{383105489803686059002251162049672504612012295058743629413968123364257400904831067} a^{10} + \frac{151165892819492087890229709851974145639794916001793316002792386065833204408061986}{383105489803686059002251162049672504612012295058743629413968123364257400904831067} a^{9} + \frac{197982664961456243472245230688755223638595604789174000612901726317701583486561895}{766210979607372118004502324099345009224024590117487258827936246728514801809662134} a^{8} + \frac{1343136093580717213479067311085752158449768306199820403468492093578532927762847}{766210979607372118004502324099345009224024590117487258827936246728514801809662134} a^{7} + \frac{109027216869155473224314910985632638137563381659808948960029309756332254598125721}{383105489803686059002251162049672504612012295058743629413968123364257400904831067} a^{6} - \frac{25744381701612401856320405753226684342651606104234577958522697513012124856709794}{383105489803686059002251162049672504612012295058743629413968123364257400904831067} a^{5} - \frac{30857520289169316306507581731108199785360636932209911007342517982926969468771238}{383105489803686059002251162049672504612012295058743629413968123364257400904831067} a^{4} + \frac{18430734222904539751719606681634725160200022937395272234500443769672888655120703}{383105489803686059002251162049672504612012295058743629413968123364257400904831067} a^{3} - \frac{259105027865934293463078903696945498293086651039723163347302483742650993127647825}{766210979607372118004502324099345009224024590117487258827936246728514801809662134} a^{2} + \frac{370877839815898753561900465208686789074214701375900954788383075460357958315094669}{766210979607372118004502324099345009224024590117487258827936246728514801809662134} a + \frac{28682812795236212252609111643910185961013485409721389449374196872094500517192895}{383105489803686059002251162049672504612012295058743629413968123364257400904831067}$

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

Class group and class number

$C_{2}\times C_{3218}$, which has order $6436$ (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:  \( 7812223142.924683 \) (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

\(\Q(\sqrt{41}) \), \(\Q(\zeta_{7})^+\), 4.4.68921.1, 6.6.165479321.1, 8.0.194754273881.1, 12.12.1887291702776240766761.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ $24$ ${\href{/LocalNumberField/5.12.0.1}{12} }^{2}$ R $24$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{3}$ $24$ $24$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ $24$ $24$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{8}$

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
7Data not computed
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$