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Properties

Label 25.25.3454133504...0625.1
Degree $25$
Signature $[25, 0]$
Discriminant $5^{40}\cdot 151^{20}$
Root discriminant $727.00$
Ramified primes $5, 151$
Class number Not computed
Class group Not computed
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![-125065035541807, 486510203387995, 138192470603395, -834575863497100, -138336432441715, 509438422551999, 78900095706790, -151078351731360, -21459822962985, 25320004393065, 3146307382901, -2603477810110, -275695602415, 172402155845, 15285369820, -7542224736, -548593630, 219332280, 12672730, -4180150, -181231, 50015, 1455, -340, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 5*x^24 - 340*x^23 + 1455*x^22 + 50015*x^21 - 181231*x^20 - 4180150*x^19 + 12672730*x^18 + 219332280*x^17 - 548593630*x^16 - 7542224736*x^15 + 15285369820*x^14 + 172402155845*x^13 - 275695602415*x^12 - 2603477810110*x^11 + 3146307382901*x^10 + 25320004393065*x^9 - 21459822962985*x^8 - 151078351731360*x^7 + 78900095706790*x^6 + 509438422551999*x^5 - 138336432441715*x^4 - 834575863497100*x^3 + 138192470603395*x^2 + 486510203387995*x - 125065035541807)
 
gp: K = bnfinit(x^25 - 5*x^24 - 340*x^23 + 1455*x^22 + 50015*x^21 - 181231*x^20 - 4180150*x^19 + 12672730*x^18 + 219332280*x^17 - 548593630*x^16 - 7542224736*x^15 + 15285369820*x^14 + 172402155845*x^13 - 275695602415*x^12 - 2603477810110*x^11 + 3146307382901*x^10 + 25320004393065*x^9 - 21459822962985*x^8 - 151078351731360*x^7 + 78900095706790*x^6 + 509438422551999*x^5 - 138336432441715*x^4 - 834575863497100*x^3 + 138192470603395*x^2 + 486510203387995*x - 125065035541807, 1)
 

Normalized defining polynomial

\( x^{25} - 5 x^{24} - 340 x^{23} + 1455 x^{22} + 50015 x^{21} - 181231 x^{20} - 4180150 x^{19} + 12672730 x^{18} + 219332280 x^{17} - 548593630 x^{16} - 7542224736 x^{15} + 15285369820 x^{14} + 172402155845 x^{13} - 275695602415 x^{12} - 2603477810110 x^{11} + 3146307382901 x^{10} + 25320004393065 x^{9} - 21459822962985 x^{8} - 151078351731360 x^{7} + 78900095706790 x^{6} + 509438422551999 x^{5} - 138336432441715 x^{4} - 834575863497100 x^{3} + 138192470603395 x^{2} + 486510203387995 x - 125065035541807 \)

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:  \(345413350438014986350888987668177302412259459742926992475986480712890625=5^{40}\cdot 151^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $727.00$
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:  $5, 151$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3775=5^{2}\cdot 151\)
Dirichlet character group:    $\lbrace$$\chi_{3775}(1216,·)$, $\chi_{3775}(1,·)$, $\chi_{3775}(2626,·)$, $\chi_{3775}(321,·)$, $\chi_{3775}(3341,·)$, $\chi_{3775}(461,·)$, $\chi_{3775}(1876,·)$, $\chi_{3775}(3481,·)$, $\chi_{3775}(2586,·)$, $\chi_{3775}(1871,·)$, $\chi_{3775}(1116,·)$, $\chi_{3775}(2266,·)$, $\chi_{3775}(1121,·)$, $\chi_{3775}(1511,·)$, $\chi_{3775}(2726,·)$, $\chi_{3775}(1831,·)$, $\chi_{3775}(361,·)$, $\chi_{3775}(3021,·)$, $\chi_{3775}(2631,·)$, $\chi_{3775}(366,·)$, $\chi_{3775}(1971,·)$, $\chi_{3775}(1076,·)$, $\chi_{3775}(3381,·)$, $\chi_{3775}(756,·)$, $\chi_{3775}(3386,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{8} a^{15} + \frac{1}{8} a^{14} - \frac{1}{4} a^{13} + \frac{1}{8} a^{12} + \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} + \frac{3}{8} a - \frac{3}{8}$, $\frac{1}{8} a^{16} + \frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{8} a^{8} + \frac{3}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{8} a^{17} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} + \frac{1}{8} a^{12} + \frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{1}{4} a^{5} + \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} + \frac{3}{8}$, $\frac{1}{8} a^{18} + \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{2} a^{9} + \frac{3}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{2} a^{6} + \frac{3}{8} a^{5} - \frac{1}{2} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{8} a^{19} + \frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{2} a^{7} + \frac{3}{8} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{224} a^{20} + \frac{9}{224} a^{19} - \frac{1}{28} a^{18} + \frac{1}{112} a^{17} + \frac{5}{112} a^{16} + \frac{3}{56} a^{15} + \frac{33}{224} a^{14} + \frac{11}{112} a^{13} + \frac{11}{56} a^{12} + \frac{5}{224} a^{11} + \frac{25}{112} a^{10} + \frac{47}{224} a^{9} + \frac{29}{112} a^{8} + \frac{23}{224} a^{7} + \frac{19}{56} a^{6} + \frac{9}{224} a^{5} - \frac{67}{224} a^{4} - \frac{57}{224} a^{3} + \frac{13}{56} a^{2} - \frac{65}{224} a + \frac{95}{224}$, $\frac{1}{672} a^{21} + \frac{23}{672} a^{19} + \frac{3}{112} a^{18} - \frac{1}{84} a^{17} - \frac{11}{336} a^{16} + \frac{37}{672} a^{15} + \frac{5}{672} a^{14} + \frac{5}{48} a^{13} + \frac{1}{672} a^{12} - \frac{163}{672} a^{11} - \frac{67}{672} a^{10} + \frac{307}{672} a^{9} - \frac{275}{672} a^{8} - \frac{81}{224} a^{7} + \frac{53}{672} a^{6} + \frac{5}{168} a^{5} + \frac{5}{16} a^{4} - \frac{107}{672} a^{3} - \frac{85}{672} a^{2} - \frac{5}{21} a + \frac{209}{672}$, $\frac{1}{2016} a^{22} + \frac{1}{2016} a^{21} - \frac{1}{504} a^{20} + \frac{25}{1008} a^{19} - \frac{55}{1008} a^{18} - \frac{1}{24} a^{17} + \frac{9}{224} a^{16} - \frac{19}{336} a^{15} + \frac{23}{168} a^{14} - \frac{439}{2016} a^{13} + \frac{41}{336} a^{12} - \frac{197}{2016} a^{11} + \frac{11}{336} a^{10} + \frac{23}{2016} a^{9} - \frac{31}{63} a^{8} - \frac{811}{2016} a^{7} - \frac{719}{2016} a^{6} + \frac{659}{2016} a^{5} - \frac{215}{504} a^{4} - \frac{167}{672} a^{3} + \frac{115}{2016} a^{2} - \frac{137}{504} a - \frac{211}{504}$, $\frac{1}{6048} a^{23} - \frac{1}{6048} a^{22} + \frac{1}{2016} a^{21} + \frac{13}{6048} a^{20} + \frac{29}{504} a^{19} - \frac{7}{432} a^{18} + \frac{113}{2016} a^{17} + \frac{1}{72} a^{16} + \frac{11}{224} a^{15} - \frac{163}{6048} a^{14} - \frac{61}{1512} a^{13} - \frac{25}{108} a^{12} + \frac{7}{54} a^{11} + \frac{661}{3024} a^{10} + \frac{61}{1008} a^{9} - \frac{79}{252} a^{8} - \frac{605}{2016} a^{7} + \frac{37}{336} a^{6} + \frac{209}{672} a^{5} + \frac{649}{1512} a^{4} - \frac{1565}{6048} a^{3} - \frac{2371}{6048} a^{2} + \frac{277}{672} a - \frac{227}{3024}$, $\frac{1}{359921511975513904638572406300896349234183447568631679754300511058778957682266260407190649693234015447954328881664931420923869149801682811871456} a^{24} + \frac{5766630503224637430863953308843899714876126239879918207284120981556085283939161223813650819010454304302904101504247791567248714062255706341}{119973837325171301546190802100298783078061149189543893251433503686259652560755420135730216564411338482651442960554977140307956383267227603957152} a^{23} + \frac{12534201886450353453780240763191161726255191967024180303651948797645586449700603201264138048208556085575166335046743291605646974620147627393}{89980377993878476159643101575224087308545861892157919938575127764694739420566565101797662423308503861988582220416232855230967287450420702967864} a^{22} - \frac{41493422079335651133711543121084854897161678919559115847335194632695355791035363896855364921863080607409054025823544631959284804780569823257}{359921511975513904638572406300896349234183447568631679754300511058778957682266260407190649693234015447954328881664931420923869149801682811871456} a^{21} - \frac{264922343188165893166916369733911986440562557899147040844786812235046267015057581423777405367300024754838220009675372815575002038320433478227}{179960755987756952319286203150448174617091723784315839877150255529389478841133130203595324846617007723977164440832465710461934574900841405935728} a^{20} - \frac{21194137561672869297795564035640453990925904181265090647958606764269405583931176858113858929582439291625332096618648185177116910774285729676435}{359921511975513904638572406300896349234183447568631679754300511058778957682266260407190649693234015447954328881664931420923869149801682811871456} a^{19} - \frac{7914079042304975309792264136793477943329472696189341212800500822681102874975807626393260782765188674422647446498525909074285816035513268390117}{359921511975513904638572406300896349234183447568631679754300511058778957682266260407190649693234015447954328881664931420923869149801682811871456} a^{18} - \frac{3381508671195132586076590319873106889820111818751987423216802515776352146628536732053711293643939256542433582415495145152180801349522812554}{138858608015244561974757872801271739673681885636046172744714703340578301574948402934872935838439049169735466389531223542023097665818550467543} a^{17} - \frac{211920227655092556643651238920507901808521803114187480890872987398042400691605002124364409447553081889189466740590484933359067778313376873291}{3749182416411603173318462565634336971189410912173246664107296990195614142523606879241569267637854327582857592517343035634623636977100862623661} a^{16} + \frac{69570465316463010523758945485950391964455241380010937587616028204741938315796931758155443544764454206348498464316943114887244645114700102581}{6427169856705605439974507255373149093467561563725565709898223411763909958611897507271261601664893132999184444315445203945069091960744335926276} a^{15} + \frac{39450855891535604448240404656992462388269942315701588371790842204043966818346845143925446647969741460733263358663267711440148414674549930331267}{359921511975513904638572406300896349234183447568631679754300511058778957682266260407190649693234015447954328881664931420923869149801682811871456} a^{14} - \frac{17992432439864526886703876468757033096889667291231821208554977621222409897818392409092809394314707971488623875595994132719743806457102291558023}{119973837325171301546190802100298783078061149189543893251433503686259652560755420135730216564411338482651442960554977140307956383267227603957152} a^{13} + \frac{9811665495293359793055104219496021608681714320674961056020402013127880719004842046741944253645429229308088773941501356824800239799240210052483}{359921511975513904638572406300896349234183447568631679754300511058778957682266260407190649693234015447954328881664931420923869149801682811871456} a^{12} - \frac{523215961265145965955081335703173886839219231118161634265660361070013207194175526791900889883604284618828268749884767129994696542164598117471}{29993459331292825386547700525074695769515287297385973312858375921564913140188855033932554141102834620662860740138744285076989095816806900989288} a^{11} - \frac{2464599526769171021565356859073620047232845645885494758655967894543128611087848542949185810035134466488469964928689036140289479624916709067403}{359921511975513904638572406300896349234183447568631679754300511058778957682266260407190649693234015447954328881664931420923869149801682811871456} a^{10} - \frac{1249165556071269891585248608286679307126011932217527791848209504007635248055233454965489421740397600588784485394009757034446850003693642386177}{9997819777097608462182566841691565256505095765795324437619458640521637713396285011310851380367611540220953580046248095025663031938935633663096} a^{9} - \frac{27743013801646992772618736278262828435425982341942004824913368200126604070099459992332345754579022999905361795964185159803006517725619173777949}{59986918662585650773095401050149391539030574594771946625716751843129826280377710067865108282205669241325721480277488570153978191633613801978576} a^{8} + \frac{7115288011327107082482979387268655337151215521458930027998802595959047513225580453132718907970599385386323703043818181085959476169349328493171}{59986918662585650773095401050149391539030574594771946625716751843129826280377710067865108282205669241325721480277488570153978191633613801978576} a^{7} - \frac{10268091612603667230069879214436365634490504290103042668901437951614192260563467147735851166351648064342662889796114496244072954272112767286273}{39991279108390433848730267366766261026020383063181297750477834562086550853585140045243405521470446160883814320184992380102652127755742534652384} a^{6} + \frac{9607468498565594719142654756491629079333455719242243969934084562110084036504674215628143440660367477118152157043752576571829740149156762729783}{51417358853644843519796058042985192747740492509804525679185787294111279668895180058170092813319145063993475554523561631560552735685954687410208} a^{5} + \frac{107288771576532816393432266282938248053407911899587961854105467201246739385443405589819281880244883915185202497835236797297207841262334571902987}{359921511975513904638572406300896349234183447568631679754300511058778957682266260407190649693234015447954328881664931420923869149801682811871456} a^{4} + \frac{7211634755847457182281838906404910760208072669642417083627264268620710661504909001855381232628430963121827829253919887242720460271208234171085}{119973837325171301546190802100298783078061149189543893251433503686259652560755420135730216564411338482651442960554977140307956383267227603957152} a^{3} + \frac{105363346079080182714810322997493290547592452698067651733339987663822374104529883881697547238234320662883718531252335938039181596025811745408523}{359921511975513904638572406300896349234183447568631679754300511058778957682266260407190649693234015447954328881664931420923869149801682811871456} a^{2} + \frac{9292707774737356377248690520451894527151172715645861463656152753246508441042773796491449773854159326530552564722999265588258319531296495204181}{44990188996939238079821550787612043654272930946078959969287563882347369710283282550898831211654251930994291110208116427615483643725210351483932} a - \frac{551341043505145319328815946798591399304718754702462572298702276421125899386565569634132208945022584967792377149501584340861674467177358665053}{2415580617285328219050821518797962075397204346098199192981882624555563474377625908773091608679422922469492140145402224301502477515447535650144}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

5.5.390625.1, 5.5.203080312890625.4, 5.5.203080312890625.3, 5.5.203080312890625.2, 5.5.203080312890625.1, 5.5.519885601.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{5}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{5}$

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
5Data not computed
$151$151.5.4.1$x^{5} - 151$$5$$1$$4$$C_5$$[\ ]_{5}$
151.5.4.1$x^{5} - 151$$5$$1$$4$$C_5$$[\ ]_{5}$
151.5.4.1$x^{5} - 151$$5$$1$$4$$C_5$$[\ ]_{5}$
151.5.4.1$x^{5} - 151$$5$$1$$4$$C_5$$[\ ]_{5}$
151.5.4.1$x^{5} - 151$$5$$1$$4$$C_5$$[\ ]_{5}$