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Properties

Label 25.25.7128869711...1601.1
Degree $25$
Signature $[25, 0]$
Discriminant $11^{20}\cdot 71^{20}$
Root discriminant $206.12$
Ramified primes $11, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5^2$ (as 25T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![73350496, 164644368, -1993261592, -7701909436, -5969323408, 11310231959, 21035443090, 3751400425, -15119768150, -10682929821, 2103035116, 4696914774, 1048835196, -688461999, -347862808, 11554131, 35351982, 4077128, -1607358, -328747, 33950, 10686, -272, -164, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 164*x^23 - 272*x^22 + 10686*x^21 + 33950*x^20 - 328747*x^19 - 1607358*x^18 + 4077128*x^17 + 35351982*x^16 + 11554131*x^15 - 347862808*x^14 - 688461999*x^13 + 1048835196*x^12 + 4696914774*x^11 + 2103035116*x^10 - 10682929821*x^9 - 15119768150*x^8 + 3751400425*x^7 + 21035443090*x^6 + 11310231959*x^5 - 5969323408*x^4 - 7701909436*x^3 - 1993261592*x^2 + 164644368*x + 73350496)
 
gp: K = bnfinit(x^25 - 164*x^23 - 272*x^22 + 10686*x^21 + 33950*x^20 - 328747*x^19 - 1607358*x^18 + 4077128*x^17 + 35351982*x^16 + 11554131*x^15 - 347862808*x^14 - 688461999*x^13 + 1048835196*x^12 + 4696914774*x^11 + 2103035116*x^10 - 10682929821*x^9 - 15119768150*x^8 + 3751400425*x^7 + 21035443090*x^6 + 11310231959*x^5 - 5969323408*x^4 - 7701909436*x^3 - 1993261592*x^2 + 164644368*x + 73350496, 1)
 

Normalized defining polynomial

\( x^{25} - 164 x^{23} - 272 x^{22} + 10686 x^{21} + 33950 x^{20} - 328747 x^{19} - 1607358 x^{18} + 4077128 x^{17} + 35351982 x^{16} + 11554131 x^{15} - 347862808 x^{14} - 688461999 x^{13} + 1048835196 x^{12} + 4696914774 x^{11} + 2103035116 x^{10} - 10682929821 x^{9} - 15119768150 x^{8} + 3751400425 x^{7} + 21035443090 x^{6} + 11310231959 x^{5} - 5969323408 x^{4} - 7701909436 x^{3} - 1993261592 x^{2} + 164644368 x + 73350496 \)

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:  \(7128869711569640297994415184144957376207195381442155291601=11^{20}\cdot 71^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $206.12$
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:  $11, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(781=11\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{781}(1,·)$, $\chi_{781}(196,·)$, $\chi_{781}(5,·)$, $\chi_{781}(199,·)$, $\chi_{781}(522,·)$, $\chi_{781}(267,·)$, $\chi_{781}(147,·)$, $\chi_{781}(214,·)$, $\chi_{781}(664,·)$, $\chi_{781}(25,·)$, $\chi_{781}(218,·)$, $\chi_{781}(412,·)$, $\chi_{781}(735,·)$, $\chi_{781}(289,·)$, $\chi_{781}(356,·)$, $\chi_{781}(551,·)$, $\chi_{781}(554,·)$, $\chi_{781}(427,·)$, $\chi_{781}(125,·)$, $\chi_{781}(625,·)$, $\chi_{781}(498,·)$, $\chi_{781}(309,·)$, $\chi_{781}(696,·)$, $\chi_{781}(764,·)$, $\chi_{781}(573,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{138} a^{15} - \frac{3}{46} a^{14} + \frac{2}{69} a^{13} + \frac{4}{23} a^{12} + \frac{1}{46} a^{11} + \frac{21}{46} a^{10} - \frac{13}{138} a^{9} + \frac{5}{46} a^{8} + \frac{2}{69} a^{7} + \frac{59}{138} a^{6} + \frac{26}{69} a^{5} - \frac{32}{69} a^{4} - \frac{21}{46} a^{3} + \frac{9}{46} a^{2} + \frac{2}{23} a + \frac{1}{3}$, $\frac{1}{138} a^{16} - \frac{4}{69} a^{14} - \frac{3}{46} a^{13} + \frac{2}{23} a^{12} + \frac{7}{46} a^{11} - \frac{67}{138} a^{10} + \frac{6}{23} a^{9} + \frac{1}{138} a^{8} - \frac{43}{138} a^{7} + \frac{31}{138} a^{6} - \frac{5}{69} a^{5} - \frac{3}{23} a^{4} + \frac{2}{23} a^{3} + \frac{8}{23} a^{2} + \frac{8}{69} a$, $\frac{1}{138} a^{17} - \frac{2}{23} a^{14} - \frac{25}{138} a^{13} + \frac{1}{23} a^{12} + \frac{13}{69} a^{11} + \frac{19}{46} a^{10} + \frac{35}{138} a^{9} - \frac{61}{138} a^{8} + \frac{21}{46} a^{7} + \frac{8}{23} a^{6} - \frac{8}{69} a^{5} - \frac{17}{138} a^{4} - \frac{7}{23} a^{3} + \frac{25}{138} a^{2} - \frac{7}{23} a - \frac{1}{3}$, $\frac{1}{138} a^{18} + \frac{5}{138} a^{14} - \frac{5}{46} a^{13} - \frac{31}{138} a^{12} + \frac{4}{23} a^{11} + \frac{16}{69} a^{10} - \frac{5}{69} a^{9} - \frac{11}{46} a^{8} - \frac{7}{23} a^{7} + \frac{1}{69} a^{6} + \frac{55}{138} a^{5} + \frac{3}{23} a^{4} + \frac{14}{69} a^{3} + \frac{1}{23} a^{2} + \frac{29}{138} a$, $\frac{1}{138} a^{19} + \frac{5}{23} a^{14} + \frac{3}{23} a^{13} - \frac{9}{46} a^{12} + \frac{17}{138} a^{11} + \frac{10}{69} a^{10} - \frac{37}{138} a^{9} - \frac{8}{23} a^{8} - \frac{3}{23} a^{7} - \frac{11}{46} a^{6} - \frac{35}{138} a^{5} - \frac{11}{23} a^{4} + \frac{15}{46} a^{3} - \frac{37}{138} a^{2} - \frac{10}{23} a + \frac{1}{3}$, $\frac{1}{138} a^{20} + \frac{2}{23} a^{14} - \frac{3}{46} a^{13} - \frac{13}{138} a^{12} - \frac{1}{138} a^{11} + \frac{5}{138} a^{10} + \frac{11}{23} a^{9} + \frac{5}{46} a^{8} - \frac{5}{46} a^{7} + \frac{29}{69} a^{6} - \frac{13}{46} a^{5} + \frac{11}{46} a^{4} - \frac{5}{69} a^{3} + \frac{9}{46} a^{2} + \frac{31}{138} a$, $\frac{1}{138} a^{21} + \frac{5}{23} a^{14} + \frac{4}{69} a^{13} - \frac{13}{138} a^{12} - \frac{31}{138} a^{11} - \frac{1}{2} a^{10} - \frac{6}{23} a^{9} + \frac{2}{23} a^{8} - \frac{59}{138} a^{7} - \frac{19}{46} a^{6} + \frac{5}{23} a^{5} + \frac{34}{69} a^{4} - \frac{15}{46} a^{3} + \frac{26}{69} a^{2} + \frac{21}{46} a$, $\frac{1}{276} a^{22} - \frac{1}{276} a^{16} + \frac{5}{138} a^{14} + \frac{7}{138} a^{13} + \frac{65}{276} a^{12} - \frac{7}{46} a^{11} + \frac{73}{276} a^{10} + \frac{15}{46} a^{9} + \frac{7}{46} a^{8} - \frac{67}{138} a^{7} - \frac{5}{12} a^{6} + \frac{3}{23} a^{5} + \frac{33}{92} a^{4} - \frac{1}{138} a^{3} - \frac{35}{92} a^{2} + \frac{19}{138} a$, $\frac{1}{12696} a^{23} + \frac{1}{1587} a^{22} + \frac{5}{1587} a^{21} - \frac{7}{3174} a^{20} - \frac{19}{6348} a^{19} + \frac{5}{6348} a^{18} - \frac{9}{4232} a^{17} - \frac{19}{6348} a^{16} + \frac{5}{3174} a^{15} + \frac{285}{2116} a^{14} - \frac{781}{12696} a^{13} - \frac{17}{1058} a^{12} + \frac{1397}{12696} a^{11} + \frac{605}{1587} a^{10} - \frac{1187}{6348} a^{9} + \frac{277}{1058} a^{8} + \frac{1749}{4232} a^{7} - \frac{205}{6348} a^{6} - \frac{4279}{12696} a^{5} + \frac{499}{2116} a^{4} - \frac{485}{12696} a^{3} - \frac{515}{1058} a^{2} - \frac{1135}{3174} a + \frac{19}{69}$, $\frac{1}{66742393983874202880378970179419048141740744013178063261937640293458500903596555237776} a^{24} + \frac{352035650962652946403816702828904375128031593549743009689019679989783437390966807}{16685598495968550720094742544854762035435186003294515815484410073364625225899138809444} a^{23} + \frac{8698913680475540269171271288785896922780630053770380844425283093322321601367007627}{5561866165322850240031580848284920678478395334431505271828136691121541741966379603148} a^{22} + \frac{1458093453491264909107597412853201858603954105156607188084770359445478296606510141}{4171399623992137680023685636213690508858796500823628953871102518341156306474784702361} a^{21} + \frac{118046647424216854026816459614521044792870006528531639901735299220986720979737659855}{33371196991937101440189485089709524070870372006589031630968820146729250451798277618888} a^{20} - \frac{35449422381267853394126402120098398211097482437546377798574155160504272293692049633}{33371196991937101440189485089709524070870372006589031630968820146729250451798277618888} a^{19} + \frac{177775809104017628809575991165873613017259926207418908140045618611664437697828765989}{66742393983874202880378970179419048141740744013178063261937640293458500903596555237776} a^{18} + \frac{12051739084269380093985258837250658152226247930596454248962758032000327711823169449}{11123732330645700480063161696569841356956790668863010543656273382243083483932759206296} a^{17} - \frac{4586876348020909211474959715911306579701523518046288393757068448646375609022502466}{1390466541330712560007895212071230169619598833607876317957034172780385435491594900787} a^{16} + \frac{32847389577790934223702864652290391252697507438306762716836694064934460221451503341}{11123732330645700480063161696569841356956790668863010543656273382243083483932759206296} a^{15} - \frac{11537376368117957016760325690938850600693058232129784321596919955490431259669328145877}{66742393983874202880378970179419048141740744013178063261937640293458500903596555237776} a^{14} + \frac{2412714525154310921360728719806645480611282711789505329479638675463448324817897845805}{16685598495968550720094742544854762035435186003294515815484410073364625225899138809444} a^{13} - \frac{13885878735602798744068005955903919081176105616819039361359489859465725738181475479287}{66742393983874202880378970179419048141740744013178063261937640293458500903596555237776} a^{12} - \frac{742026002650636068873871520498752501611311101149351769250812718297568999070790828573}{8342799247984275360047371272427381017717593001647257907742205036682312612949569404722} a^{11} + \frac{4745908813398279289520324228297277848573553789370913290419260362591875204521600482183}{33371196991937101440189485089709524070870372006589031630968820146729250451798277618888} a^{10} + \frac{2803482126605613845027178795680205457794697036383701130054735121106991949735478988575}{16685598495968550720094742544854762035435186003294515815484410073364625225899138809444} a^{9} + \frac{25922660082142166386147346708259136321395713506989574182569244688457710488864879318763}{66742393983874202880378970179419048141740744013178063261937640293458500903596555237776} a^{8} - \frac{3448269216448477815862497972413124967699626102698154916186927799638532764931657797187}{11123732330645700480063161696569841356956790668863010543656273382243083483932759206296} a^{7} - \frac{13239474454634076052734961144195905302985881105856670556377820549300554044882956957879}{66742393983874202880378970179419048141740744013178063261937640293458500903596555237776} a^{6} + \frac{924216058514083771624652321443272848328592465682313123685332977681732192974331227301}{11123732330645700480063161696569841356956790668863010543656273382243083483932759206296} a^{5} + \frac{3559114275383499960603040761009747386578901663794701109228405852072758604781707710093}{22247464661291400960126323393139682713913581337726021087312546764486166967865518412592} a^{4} - \frac{5048892428768341813817523944882294842959947362400578664433801669765011522948697611711}{16685598495968550720094742544854762035435186003294515815484410073364625225899138809444} a^{3} + \frac{6255372597372876873886751511397382958258785692271617577852926903003668229870381645405}{16685598495968550720094742544854762035435186003294515815484410073364625225899138809444} a^{2} - \frac{1369149505747151315036783969825617027653450552637579237676450916591384801651437908069}{2780933082661425120015790424142460339239197667215752635914068345560770870983189801574} a + \frac{422761081551694768271876599964102938379869236219033603128782180689422186805716837}{181365201043136420870595027661464804732991152209722997994395761667006795933686291407}$

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:  \( 1817774689596528400000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2$ (as 25T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 25
The 25 conjugacy class representatives for $C_5^2$
Character table for $C_5^2$ is not computed

Intermediate fields

5.5.25411681.1, 5.5.372052421521.3, 5.5.372052421521.2, 5.5.372052421521.1, 5.5.372052421521.4, \(\Q(\zeta_{11})^+\)

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{5}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{25}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{5}$

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
11Data not computed
71Data not computed