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Properties

Label 25.25.2988580393...9601.1
Degree $25$
Signature $[25, 0]$
Discriminant $401^{24}$
Root discriminant $315.51$
Ramified prime $401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{25}$ (as 25T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6307554361, 6292934527, -73057285978, -45833333710, 171412082812, 110262430876, -155205497967, -103704806969, 71255379842, 49846712475, -18425686096, -13790113019, 2777496960, 2318100658, -241493765, -242203703, 11305013, 15781195, -221401, -632791, -1306, 15030, 107, -192, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - x^24 - 192*x^23 + 107*x^22 + 15030*x^21 - 1306*x^20 - 632791*x^19 - 221401*x^18 + 15781195*x^17 + 11305013*x^16 - 242203703*x^15 - 241493765*x^14 + 2318100658*x^13 + 2777496960*x^12 - 13790113019*x^11 - 18425686096*x^10 + 49846712475*x^9 + 71255379842*x^8 - 103704806969*x^7 - 155205497967*x^6 + 110262430876*x^5 + 171412082812*x^4 - 45833333710*x^3 - 73057285978*x^2 + 6292934527*x + 6307554361)
 
gp: K = bnfinit(x^25 - x^24 - 192*x^23 + 107*x^22 + 15030*x^21 - 1306*x^20 - 632791*x^19 - 221401*x^18 + 15781195*x^17 + 11305013*x^16 - 242203703*x^15 - 241493765*x^14 + 2318100658*x^13 + 2777496960*x^12 - 13790113019*x^11 - 18425686096*x^10 + 49846712475*x^9 + 71255379842*x^8 - 103704806969*x^7 - 155205497967*x^6 + 110262430876*x^5 + 171412082812*x^4 - 45833333710*x^3 - 73057285978*x^2 + 6292934527*x + 6307554361, 1)
 

Normalized defining polynomial

\( x^{25} - x^{24} - 192 x^{23} + 107 x^{22} + 15030 x^{21} - 1306 x^{20} - 632791 x^{19} - 221401 x^{18} + 15781195 x^{17} + 11305013 x^{16} - 242203703 x^{15} - 241493765 x^{14} + 2318100658 x^{13} + 2777496960 x^{12} - 13790113019 x^{11} - 18425686096 x^{10} + 49846712475 x^{9} + 71255379842 x^{8} - 103704806969 x^{7} - 155205497967 x^{6} + 110262430876 x^{5} + 171412082812 x^{4} - 45833333710 x^{3} - 73057285978 x^{2} + 6292934527 x + 6307554361 \)

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

Invariants

Degree:  $25$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[25, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(298858039306015483445329296147756574767800059094329115180169601=401^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $315.51$
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:  $401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(401\)
Dirichlet character group:    $\lbrace$$\chi_{401}(256,·)$, $\chi_{401}(1,·)$, $\chi_{401}(387,·)$, $\chi_{401}(196,·)$, $\chi_{401}(5,·)$, $\chi_{401}(385,·)$, $\chi_{401}(72,·)$, $\chi_{401}(331,·)$, $\chi_{401}(77,·)$, $\chi_{401}(195,·)$, $\chi_{401}(88,·)$, $\chi_{401}(25,·)$, $\chi_{401}(224,·)$, $\chi_{401}(39,·)$, $\chi_{401}(360,·)$, $\chi_{401}(173,·)$, $\chi_{401}(321,·)$, $\chi_{401}(255,·)$, $\chi_{401}(178,·)$, $\chi_{401}(51,·)$, $\chi_{401}(372,·)$, $\chi_{401}(315,·)$, $\chi_{401}(125,·)$, $\chi_{401}(318,·)$, $\chi_{401}(63,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{83779} a^{22} - \frac{25785}{83779} a^{21} + \frac{11041}{83779} a^{20} - \frac{7118}{83779} a^{19} + \frac{1770}{83779} a^{18} + \frac{22139}{83779} a^{17} - \frac{39363}{83779} a^{16} + \frac{15394}{83779} a^{15} - \frac{23355}{83779} a^{14} - \frac{29210}{83779} a^{13} - \frac{32951}{83779} a^{12} + \frac{17362}{83779} a^{11} - \frac{3543}{83779} a^{10} + \frac{12707}{83779} a^{9} - \frac{6361}{83779} a^{8} - \frac{24383}{83779} a^{7} + \frac{17298}{83779} a^{6} - \frac{18139}{83779} a^{5} + \frac{12742}{83779} a^{4} + \frac{1678}{83779} a^{3} + \frac{15339}{83779} a^{2} - \frac{11958}{83779} a - \frac{23019}{83779}$, $\frac{1}{63420703} a^{23} - \frac{1}{318697} a^{22} - \frac{30208563}{63420703} a^{21} + \frac{14562666}{63420703} a^{20} - \frac{1994528}{63420703} a^{19} - \frac{2193334}{63420703} a^{18} - \frac{8398628}{63420703} a^{17} - \frac{13842500}{63420703} a^{16} + \frac{25722603}{63420703} a^{15} - \frac{1335097}{63420703} a^{14} + \frac{18591386}{63420703} a^{13} + \frac{2430744}{63420703} a^{12} - \frac{7934674}{63420703} a^{11} + \frac{27071004}{63420703} a^{10} - \frac{10252396}{63420703} a^{9} - \frac{10131591}{63420703} a^{8} + \frac{28457154}{63420703} a^{7} - \frac{16540431}{63420703} a^{6} + \frac{5489583}{63420703} a^{5} - \frac{31131387}{63420703} a^{4} + \frac{9437047}{63420703} a^{3} + \frac{27259035}{63420703} a^{2} - \frac{1359963}{63420703} a + \frac{19438964}{63420703}$, $\frac{1}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{24} - \frac{524969040480937357290430408358836888220926610225961839955029092876812203743758130239549517066191811099615}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{23} - \frac{2795131841844358061977760229438070551230810796977399540383317902968024289876651948386457803635828783934425578}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{22} - \frac{88908854969178587229220806372564858452490354102180052225306705469341760106527411576415590928547177034628709926591}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{21} + \frac{230095599188016452779107373154195882206158181600071160773606135550409653069858948296197489027244391433091603755057}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{20} + \frac{201918908507430037315062012354125577614096733217981884589356577045389803434110632741532879109648928874918750617220}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{19} - \frac{58612243105713060162431273374881226261254298436267864591822071814644009904838285542085979779951936441628755613993}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{18} + \frac{253489264902815193916622672241298159229723237290415840922200947955582913988542311507839515305065514475666478129220}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{17} - \frac{203835677171688603604887338080935591890886957704095328762497238958841572208799519100159151732190742499966315628221}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{16} + \frac{232143710812571315324397343905888992722693283443985982813216101899995115287249615265228019472556158743155651030819}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{15} + \frac{255621473596664062466901202152800563650356753933980112231743305529421243555917067417779048219359597512745211158575}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{14} - \frac{16985770058303648432093330290029490538098186259546528396794253067637712771380468098816949889007943545526549618830}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{13} + \frac{26573881876568359106904413736759741500173670398634158198824939852064364289048647630826730476395098059892975224462}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{12} + \frac{191870268698428605407188362028587747271094930999089474663122430890437585031024571803236064643639077323863902155375}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{11} - \frac{144986532116680497316989928999223145789684495286275336750229220397323541416461761468040884558736617382101485145698}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{10} + \frac{253650915687392258902217199250535183610580030562278605275975922921900446293776405608293849554757066297627583503922}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{9} - \frac{228861219078091304535313614973897936969944095822794157060308997489578271029517355626443163796252563932547038214300}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{8} + \frac{82296863506649485736351325267266586572335458788528374986993446557214381663893991740286199960111126342791563932467}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{7} + \frac{178082420195237902078186165579621428073495147442974417163236070389218449809593753791313789684652665053538552570272}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{6} + \frac{50778874397041455192933224328605043105828809328404073438278065238093440424069153117526668536135069860027209733737}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{5} + \frac{27276038560595778937576492581190864424826984402594926394445651099076611196271984463266621071798450616294334159475}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{4} - \frac{114516301344100481429607135871815757958592051371767148772784179655699714034905855601421135776782059305536560811984}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{3} + \frac{25952091085684422254815940195470274607846097761087554495357457375275034689737875147247896837610062915552292414712}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a^{2} + \frac{187640211982386579179282330457794300936092075472737947036101317827867340282005294607677722007456638985228857562122}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893} a + \frac{250896246161887602699724089127185943377761960650393933541728597887099676612958195752236822392924952377785384510554}{519288430613656478699572007953413859357664060195809546356931846898041089723726579883295122696456805152193467146893}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $24$
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:  \( 119518165408622930000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{25}$ (as 25T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 25
The 25 conjugacy class representatives for $C_{25}$
Character table for $C_{25}$ is not computed

Intermediate fields

5.5.25856961601.1

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 $25$ $25$ $25$ $25$ $25$ $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{5}$ $25$ $25$ $25$ $25$ $25$ $25$ $25$

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
401Data not computed