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Properties

Label 25.25.1109756376...0625.1
Degree $25$
Signature $[25, 0]$
Discriminant $5^{40}\cdot 101^{20}$
Root discriminant $527.00$
Ramified primes $5, 101$
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![-30329857, 546159595, 9313043145, -38719949575, -107311399140, 151466421774, 451866407240, 145297331040, -327731536960, -299666707460, -30799326749, 64633222815, 26271143660, -2303086930, -3289109505, -368499261, 160854770, 36250180, -3194595, -1338125, 7594, 24890, 530, -240, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 5*x^24 - 240*x^23 + 530*x^22 + 24890*x^21 + 7594*x^20 - 1338125*x^19 - 3194595*x^18 + 36250180*x^17 + 160854770*x^16 - 368499261*x^15 - 3289109505*x^14 - 2303086930*x^13 + 26271143660*x^12 + 64633222815*x^11 - 30799326749*x^10 - 299666707460*x^9 - 327731536960*x^8 + 145297331040*x^7 + 451866407240*x^6 + 151466421774*x^5 - 107311399140*x^4 - 38719949575*x^3 + 9313043145*x^2 + 546159595*x - 30329857)
 
gp: K = bnfinit(x^25 - 5*x^24 - 240*x^23 + 530*x^22 + 24890*x^21 + 7594*x^20 - 1338125*x^19 - 3194595*x^18 + 36250180*x^17 + 160854770*x^16 - 368499261*x^15 - 3289109505*x^14 - 2303086930*x^13 + 26271143660*x^12 + 64633222815*x^11 - 30799326749*x^10 - 299666707460*x^9 - 327731536960*x^8 + 145297331040*x^7 + 451866407240*x^6 + 151466421774*x^5 - 107311399140*x^4 - 38719949575*x^3 + 9313043145*x^2 + 546159595*x - 30329857, 1)
 

Normalized defining polynomial

\( x^{25} - 5 x^{24} - 240 x^{23} + 530 x^{22} + 24890 x^{21} + 7594 x^{20} - 1338125 x^{19} - 3194595 x^{18} + 36250180 x^{17} + 160854770 x^{16} - 368499261 x^{15} - 3289109505 x^{14} - 2303086930 x^{13} + 26271143660 x^{12} + 64633222815 x^{11} - 30799326749 x^{10} - 299666707460 x^{9} - 327731536960 x^{8} + 145297331040 x^{7} + 451866407240 x^{6} + 151466421774 x^{5} - 107311399140 x^{4} - 38719949575 x^{3} + 9313043145 x^{2} + 546159595 x - 30329857 \)

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:  \(110975637648877348008932048428738579805221888818778097629547119140625=5^{40}\cdot 101^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $527.00$
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:  $5, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2525=5^{2}\cdot 101\)
Dirichlet character group:    $\lbrace$$\chi_{2525}(1,·)$, $\chi_{2525}(2306,·)$, $\chi_{2525}(196,·)$, $\chi_{2525}(2006,·)$, $\chi_{2525}(2056,·)$, $\chi_{2525}(1801,·)$, $\chi_{2525}(1551,·)$, $\chi_{2525}(1296,·)$, $\chi_{2525}(1046,·)$, $\chi_{2525}(791,·)$, $\chi_{2525}(996,·)$, $\chi_{2525}(1501,·)$, $\chi_{2525}(2511,·)$, $\chi_{2525}(541,·)$, $\chi_{2525}(286,·)$, $\chi_{2525}(36,·)$, $\chi_{2525}(2021,·)$, $\chi_{2525}(2216,·)$, $\chi_{2525}(491,·)$, $\chi_{2525}(1516,·)$, $\chi_{2525}(1711,·)$, $\chi_{2525}(1011,·)$, $\chi_{2525}(1206,·)$, $\chi_{2525}(506,·)$, $\chi_{2525}(701,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{19} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{476} a^{20} + \frac{37}{476} a^{19} + \frac{37}{476} a^{18} - \frac{15}{238} a^{17} - \frac{27}{476} a^{16} - \frac{43}{476} a^{15} - \frac{1}{476} a^{14} + \frac{3}{68} a^{13} - \frac{9}{476} a^{12} - \frac{15}{238} a^{11} - \frac{81}{476} a^{10} - \frac{8}{119} a^{9} + \frac{29}{238} a^{8} - \frac{67}{476} a^{7} + \frac{111}{476} a^{6} + \frac{159}{476} a^{5} + \frac{41}{476} a^{4} + \frac{15}{34} a^{3} - \frac{73}{238} a^{2} - \frac{111}{476} a + \frac{45}{238}$, $\frac{1}{476} a^{21} - \frac{23}{476} a^{19} + \frac{29}{476} a^{18} + \frac{3}{119} a^{17} + \frac{1}{119} a^{16} + \frac{43}{476} a^{15} + \frac{29}{238} a^{14} + \frac{47}{476} a^{13} - \frac{27}{238} a^{12} - \frac{3}{34} a^{11} - \frac{129}{476} a^{10} - \frac{67}{476} a^{9} - \frac{71}{476} a^{8} + \frac{13}{68} a^{7} + \frac{31}{68} a^{6} + \frac{27}{119} a^{5} + \frac{121}{476} a^{4} - \frac{181}{476} a^{3} - \frac{183}{476} a^{2} + \frac{151}{476} a - \frac{59}{119}$, $\frac{1}{476} a^{22} + \frac{47}{476} a^{19} + \frac{15}{238} a^{18} + \frac{1}{17} a^{17} + \frac{1}{28} a^{16} + \frac{3}{68} a^{15} + \frac{6}{119} a^{14} - \frac{47}{476} a^{13} - \frac{11}{476} a^{12} - \frac{15}{68} a^{11} + \frac{53}{119} a^{10} - \frac{53}{119} a^{9} - \frac{61}{238} a^{8} + \frac{223}{476} a^{7} - \frac{195}{476} a^{6} + \frac{52}{119} a^{5} + \frac{167}{476} a^{4} + \frac{125}{476} a^{3} - \frac{113}{476} a^{2} + \frac{93}{238} a - \frac{18}{119}$, $\frac{1}{952} a^{23} - \frac{1}{952} a^{22} - \frac{1}{952} a^{21} - \frac{1}{952} a^{20} + \frac{15}{952} a^{19} + \frac{97}{952} a^{18} + \frac{27}{238} a^{17} + \frac{53}{476} a^{16} - \frac{59}{476} a^{15} + \frac{19}{476} a^{14} - \frac{67}{952} a^{13} + \frac{5}{136} a^{12} + \frac{19}{136} a^{11} - \frac{453}{952} a^{10} + \frac{48}{119} a^{9} - \frac{113}{476} a^{8} - \frac{67}{238} a^{7} - \frac{191}{476} a^{6} + \frac{215}{476} a^{5} + \frac{65}{476} a^{4} + \frac{27}{119} a^{3} - \frac{13}{34} a^{2} - \frac{317}{952} a + \frac{9}{56}$, $\frac{1}{5159898649226105669802084204752246168649401119334573727038531158523173390829267425433663285505389939349230896} a^{24} + \frac{57571995438625241089937529708782225011463376880520985803554006463515949967838718733547409930831216665353}{322493665576631604362630262797015385540587569958410857939908197407698336926829214089603955344086871209326931} a^{23} - \frac{552336174209629836733746366624765837536498193209950357644770385955299598217350512358718174265750466519041}{644987331153263208725260525594030771081175139916821715879816394815396673853658428179207910688173742418653862} a^{22} - \frac{1135224921040819650582337486752047455610657771266204707003134570989877463464010101952637509053113408351483}{2579949324613052834901042102376123084324700559667286863519265579261586695414633712716831642752694969674615448} a^{21} - \frac{166713405580486187967230008302656184479840296460503441990802284415304079201470112853705581451669286704459}{184282094615218059635788721598294506023192897119091918822804684232970478243902408051202260196621069262472532} a^{20} - \frac{28633506307436407075944479871462790232938859247141392947310144471503426588568064066997037643231879847065117}{2579949324613052834901042102376123084324700559667286863519265579261586695414633712716831642752694969674615448} a^{19} - \frac{1905608114968251391207647741811070220067980743573774442676214199220019985095419127717031190564528709740557}{43360492850639543443714993317245766123104211086845157370071690407757759586800566600282884752146133944111184} a^{18} + \frac{209195957792436226975802553398257842142774371090741497829728404398296155741986037140392747416267114892253039}{2579949324613052834901042102376123084324700559667286863519265579261586695414633712716831642752694969674615448} a^{17} + \frac{117690586366419022962823237498679766581780695899376249454560431677288940539941050239151799671627216767746489}{2579949324613052834901042102376123084324700559667286863519265579261586695414633712716831642752694969674615448} a^{16} - \frac{133234161488552421601438871197498491300241370364474028858914479469010392396529092406337634439776354864048141}{1289974662306526417450521051188061542162350279833643431759632789630793347707316856358415821376347484837307724} a^{15} - \frac{348844092920876249780495833451752506519468795793792945634182502231064590054197557949198619804762387259868461}{5159898649226105669802084204752246168649401119334573727038531158523173390829267425433663285505389939349230896} a^{14} - \frac{18975648234288168740534534302618763189505799884319114486547266065895446038406782784985524660341467265827471}{368564189230436119271577443196589012046385794238183837645609368465940956487804816102404520393242138524945064} a^{13} + \frac{90307956642145469061663313538027241515721609990510663266276991689598739028692769420267131078845415856830437}{1289974662306526417450521051188061542162350279833643431759632789630793347707316856358415821376347484837307724} a^{12} + \frac{16811306196929290336252686753191497802847374657627168131120059686103637714017030668120644595643553894060507}{184282094615218059635788721598294506023192897119091918822804684232970478243902408051202260196621069262472532} a^{11} - \frac{1645157705797226261713683347710639543994499524209906343868481533321953433541794079888136498047927337735866269}{5159898649226105669802084204752246168649401119334573727038531158523173390829267425433663285505389939349230896} a^{10} + \frac{617133734700998958011898488832177848371151281831481237473919519120575570884629067801992466488312029715571725}{2579949324613052834901042102376123084324700559667286863519265579261586695414633712716831642752694969674615448} a^{9} + \frac{378318887169181016044307791649006165106715532311537377545075375430737952640352211466258574158210060027599705}{2579949324613052834901042102376123084324700559667286863519265579261586695414633712716831642752694969674615448} a^{8} + \frac{19897876139111335451701081210046338263921991821276746320758359199938369495248248686340270799476818294290611}{151761724977238402053002476610360181430864738803958050795250916427152158553801983100990096632511468804389144} a^{7} + \frac{1165124606236092065309696074332815317072841214330283414776068281082913779150288577320599753061772535141087949}{2579949324613052834901042102376123084324700559667286863519265579261586695414633712716831642752694969674615448} a^{6} + \frac{75450707678870490017995466822416011559356248390361059451850992202860599732225350733074908583365305186762459}{151761724977238402053002476610360181430864738803958050795250916427152158553801983100990096632511468804389144} a^{5} - \frac{530211179929536869295212077820490839404560762648821742663366249665785827923007521339133861351561307815532423}{1289974662306526417450521051188061542162350279833643431759632789630793347707316856358415821376347484837307724} a^{4} + \frac{47402590125915636779463614635416135486302819844293900283097215789555799380155232342045377935729363527755861}{322493665576631604362630262797015385540587569958410857939908197407698336926829214089603955344086871209326931} a^{3} + \frac{39625617977511721726135199904647053024647232661720664981263116885986943030273734555759846146990279460316799}{737128378460872238543154886393178024092771588476367675291218736931881912975609632204809040786484277049890128} a^{2} + \frac{621745211225496631395590814851502536122437094397842157767279804175586911592290077417062165728458617955259471}{2579949324613052834901042102376123084324700559667286863519265579261586695414633712716831642752694969674615448} a - \frac{121641307554600115742144430128248532169957628064165534981216818918022250186237526209148060127645516739965391}{303523449954476804106004953220720362861729477607916101590501832854304317107603966201980193265022937608778288}$

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:  \( 113526971950914610000000000 \) (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

5.5.390625.1, 5.5.40648594140625.2, 5.5.40648594140625.5, 5.5.40648594140625.3, 5.5.40648594140625.4, 5.5.104060401.1

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
$101$101.5.4.1$x^{5} - 101$$5$$1$$4$$C_5$$[\ ]_{5}$
101.5.4.1$x^{5} - 101$$5$$1$$4$$C_5$$[\ ]_{5}$
101.5.4.1$x^{5} - 101$$5$$1$$4$$C_5$$[\ ]_{5}$
101.5.4.1$x^{5} - 101$$5$$1$$4$$C_5$$[\ ]_{5}$
101.5.4.1$x^{5} - 101$$5$$1$$4$$C_5$$[\ ]_{5}$