Properties

Label 25.1.233...921.1
Degree $25$
Signature $[1, 12]$
Discriminant $2.337\times 10^{49}$
Root discriminant $94.35$
Ramified primes $11, 239$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $D_{25}$ (as 25T4)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 7*x^24 + 49*x^23 - 283*x^22 + 1302*x^21 - 5636*x^20 + 22809*x^19 - 80739*x^18 + 282358*x^17 - 893429*x^16 + 2621604*x^15 - 6788665*x^14 + 15933765*x^13 - 33100241*x^12 + 60008112*x^11 - 95295391*x^10 + 131154926*x^9 - 150037171*x^8 + 139788299*x^7 - 104949621*x^6 + 56076967*x^5 - 27454311*x^4 + 20655635*x^3 - 14146574*x^2 + 4203471*x - 447083)
 
gp: K = bnfinit(x^25 - 7*x^24 + 49*x^23 - 283*x^22 + 1302*x^21 - 5636*x^20 + 22809*x^19 - 80739*x^18 + 282358*x^17 - 893429*x^16 + 2621604*x^15 - 6788665*x^14 + 15933765*x^13 - 33100241*x^12 + 60008112*x^11 - 95295391*x^10 + 131154926*x^9 - 150037171*x^8 + 139788299*x^7 - 104949621*x^6 + 56076967*x^5 - 27454311*x^4 + 20655635*x^3 - 14146574*x^2 + 4203471*x - 447083, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-447083, 4203471, -14146574, 20655635, -27454311, 56076967, -104949621, 139788299, -150037171, 131154926, -95295391, 60008112, -33100241, 15933765, -6788665, 2621604, -893429, 282358, -80739, 22809, -5636, 1302, -283, 49, -7, 1]);
 

\( x^{25} - 7 x^{24} + 49 x^{23} - 283 x^{22} + 1302 x^{21} - 5636 x^{20} + 22809 x^{19} - 80739 x^{18} + 282358 x^{17} - 893429 x^{16} + 2621604 x^{15} - 6788665 x^{14} + 15933765 x^{13} - 33100241 x^{12} + 60008112 x^{11} - 95295391 x^{10} + 131154926 x^{9} - 150037171 x^{8} + 139788299 x^{7} - 104949621 x^{6} + 56076967 x^{5} - 27454311 x^{4} + 20655635 x^{3} - 14146574 x^{2} + 4203471 x - 447083 \)

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

Invariants

Degree:  $25$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(23368375067493109808662942110693103411946589151921\)\(\medspace = 11^{20}\cdot 239^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $94.35$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $11, 239$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} + \frac{5}{17} a^{13} - \frac{7}{17} a^{12} - \frac{1}{17} a^{11} - \frac{1}{17} a^{10} - \frac{5}{17} a^{9} - \frac{1}{17} a^{6} - \frac{5}{17} a^{5} + \frac{7}{17} a^{4} + \frac{1}{17} a^{3} + \frac{1}{17} a^{2} + \frac{5}{17} a$, $\frac{1}{17} a^{15} + \frac{2}{17} a^{13} + \frac{4}{17} a^{11} + \frac{8}{17} a^{9} - \frac{1}{17} a^{7} - \frac{2}{17} a^{5} - \frac{4}{17} a^{3} - \frac{8}{17} a$, $\frac{1}{17} a^{16} + \frac{7}{17} a^{13} + \frac{1}{17} a^{12} + \frac{2}{17} a^{11} - \frac{7}{17} a^{10} - \frac{7}{17} a^{9} - \frac{1}{17} a^{8} - \frac{7}{17} a^{5} - \frac{1}{17} a^{4} - \frac{2}{17} a^{3} + \frac{7}{17} a^{2} + \frac{7}{17} a$, $\frac{1}{119} a^{17} - \frac{3}{119} a^{16} - \frac{3}{119} a^{15} - \frac{2}{119} a^{14} - \frac{54}{119} a^{13} + \frac{11}{119} a^{12} + \frac{18}{119} a^{11} + \frac{57}{119} a^{10} + \frac{58}{119} a^{9} + \frac{3}{119} a^{8} + \frac{20}{119} a^{7} - \frac{15}{119} a^{6} + \frac{54}{119} a^{5} - \frac{45}{119} a^{4} + \frac{50}{119} a^{3} - \frac{6}{119} a^{2} - \frac{25}{119} a$, $\frac{1}{119} a^{18} + \frac{2}{119} a^{16} + \frac{3}{119} a^{15} + \frac{3}{119} a^{14} + \frac{52}{119} a^{13} - \frac{19}{119} a^{12} + \frac{13}{119} a^{11} - \frac{3}{7} a^{10} - \frac{5}{119} a^{9} + \frac{15}{119} a^{8} + \frac{31}{119} a^{7} - \frac{54}{119} a^{6} + \frac{33}{119} a^{5} - \frac{15}{119} a^{4} + \frac{4}{119} a^{3} - \frac{1}{119} a^{2} - \frac{12}{119} a$, $\frac{1}{119} a^{19} + \frac{2}{119} a^{16} + \frac{2}{119} a^{15} - \frac{16}{119} a^{13} + \frac{19}{119} a^{12} + \frac{46}{119} a^{11} - \frac{2}{17} a^{10} + \frac{53}{119} a^{9} + \frac{32}{119} a^{8} + \frac{32}{119} a^{7} - \frac{18}{119} a^{5} - \frac{53}{119} a^{4} + \frac{4}{119} a^{3} + \frac{2}{17} a^{2} + \frac{15}{119} a$, $\frac{1}{29393} a^{20} - \frac{2}{4199} a^{19} - \frac{5}{2261} a^{18} - \frac{10}{29393} a^{17} + \frac{797}{29393} a^{16} + \frac{401}{29393} a^{15} - \frac{565}{29393} a^{14} - \frac{9993}{29393} a^{13} - \frac{10429}{29393} a^{12} + \frac{1179}{29393} a^{11} + \frac{11714}{29393} a^{10} - \frac{6618}{29393} a^{9} + \frac{433}{2261} a^{8} + \frac{708}{2261} a^{7} + \frac{7263}{29393} a^{6} + \frac{12050}{29393} a^{5} - \frac{1240}{4199} a^{4} - \frac{5466}{29393} a^{3} - \frac{10068}{29393} a^{2} + \frac{4146}{29393} a + \frac{6}{19}$, $\frac{1}{499681} a^{21} + \frac{2}{499681} a^{20} + \frac{452}{499681} a^{19} - \frac{2038}{499681} a^{18} + \frac{125}{38437} a^{17} - \frac{97}{71383} a^{16} - \frac{2053}{499681} a^{15} + \frac{1962}{499681} a^{14} - \frac{119188}{499681} a^{13} - \frac{15101}{38437} a^{12} + \frac{97268}{499681} a^{11} - \frac{117076}{499681} a^{10} - \frac{230428}{499681} a^{9} - \frac{4220}{38437} a^{8} - \frac{11120}{26299} a^{7} + \frac{15357}{38437} a^{6} + \frac{10046}{29393} a^{5} + \frac{155018}{499681} a^{4} + \frac{5828}{71383} a^{3} - \frac{28313}{71383} a^{2} - \frac{6231}{29393} a + \frac{9}{19}$, $\frac{1}{499681} a^{22} + \frac{6}{499681} a^{20} - \frac{953}{499681} a^{19} + \frac{839}{499681} a^{18} + \frac{491}{499681} a^{17} + \frac{12344}{499681} a^{16} + \frac{13582}{499681} a^{15} - \frac{3551}{499681} a^{14} - \frac{189324}{499681} a^{13} - \frac{191228}{499681} a^{12} + \frac{17806}{71383} a^{11} + \frac{37095}{499681} a^{10} + \frac{190300}{499681} a^{9} - \frac{24422}{71383} a^{8} + \frac{11599}{71383} a^{7} - \frac{96342}{499681} a^{6} - \frac{79140}{499681} a^{5} + \frac{149334}{499681} a^{4} - \frac{766}{3757} a^{3} - \frac{222707}{499681} a^{2} + \frac{367}{4199} a - \frac{3}{19}$, $\frac{1}{424718356699} a^{23} - \frac{2141}{24983432747} a^{22} - \frac{32388}{32670642823} a^{21} + \frac{523525}{32670642823} a^{20} + \frac{1451038279}{424718356699} a^{19} + \frac{24309496}{32670642823} a^{18} - \frac{365687278}{424718356699} a^{17} - \frac{40721965}{424718356699} a^{16} + \frac{64940826}{4667234689} a^{15} + \frac{175565508}{60674050957} a^{14} + \frac{99014949361}{424718356699} a^{13} - \frac{78858756512}{424718356699} a^{12} - \frac{207562259935}{424718356699} a^{11} + \frac{160139365682}{424718356699} a^{10} + \frac{87133810018}{424718356699} a^{9} - \frac{6546645438}{60674050957} a^{8} - \frac{148144804447}{424718356699} a^{7} - \frac{152884654180}{424718356699} a^{6} - \frac{8777527503}{60674050957} a^{5} + \frac{55550647744}{424718356699} a^{4} + \frac{197603666941}{424718356699} a^{3} + \frac{70599098690}{424718356699} a^{2} - \frac{11444102875}{24983432747} a - \frac{2017682}{16149601}$, $\frac{1}{58043956543233443285184360397601563488696200534346574990140125161} a^{24} - \frac{14566734513647335813323613103488014306623148953991171}{58043956543233443285184360397601563488696200534346574990140125161} a^{23} - \frac{3731760879654017932042393778630320627390805082088516600622}{8291993791890491897883480056800223355528028647763796427162875023} a^{22} + \frac{1724946981466159592435982475185299027884501682026471627408}{4464919734094880252706489261353966422207400041103582691549240397} a^{21} - \frac{15921105605208227998201770246963465338825169729735551535896}{1095168991381763080852535101841538933748984915742388207361134437} a^{20} - \frac{81434828172518905484001264354217698322622901763515490291858201}{58043956543233443285184360397601563488696200534346574990140125161} a^{19} - \frac{207179492209979293503166455908478997901258930199149776953939315}{58043956543233443285184360397601563488696200534346574990140125161} a^{18} - \frac{191035788614123506268004138563604193077055549917447299640768123}{58043956543233443285184360397601563488696200534346574990140125161} a^{17} - \frac{320280867844799838235868910813474174026565993524417387488635660}{58043956543233443285184360397601563488696200534346574990140125161} a^{16} - \frac{539758024848933450161955097236058278030474734107380125965041084}{58043956543233443285184360397601563488696200534346574990140125161} a^{15} - \frac{717307650531773427505374751087734959988833864730252638989739762}{58043956543233443285184360397601563488696200534346574990140125161} a^{14} + \frac{13261197453283385961742168795382157234526686127423367067959693325}{58043956543233443285184360397601563488696200534346574990140125161} a^{13} + \frac{635544890978466901174888664878450603608068368993292157118536199}{8291993791890491897883480056800223355528028647763796427162875023} a^{12} + \frac{9718185329290696692800647112839368326627425549366102078054889767}{58043956543233443285184360397601563488696200534346574990140125161} a^{11} - \frac{3555923268476177513203881104068122034372926496733241505650050566}{8291993791890491897883480056800223355528028647763796427162875023} a^{10} - \frac{155159696032046498534574932277846171340593441787958992914170975}{3054945081222812804483387389347450709931378975491924999481059219} a^{9} + \frac{1727625469265554893783393005213704456265080530148616832943499263}{8291993791890491897883480056800223355528028647763796427162875023} a^{8} + \frac{28013358261651924595647156749028205108427816027935246552946163215}{58043956543233443285184360397601563488696200534346574990140125161} a^{7} - \frac{6737037863713686581317068380603873900227465115400460111131948592}{58043956543233443285184360397601563488696200534346574990140125161} a^{6} - \frac{4216908817732883612986885291139004388444968068511018434451968283}{58043956543233443285184360397601563488696200534346574990140125161} a^{5} - \frac{6076026166491553999511831235919919215679997634882546114442673344}{58043956543233443285184360397601563488696200534346574990140125161} a^{4} + \frac{26474520938618723188135346870402270283403655350378947048981994701}{58043956543233443285184360397601563488696200534346574990140125161} a^{3} - \frac{28836828390787931030888763532029790241492713317667001468305188907}{58043956543233443285184360397601563488696200534346574990140125161} a^{2} - \frac{1288830215862641903273195361176296853196806876412214106576299696}{3414350384896084899128491788094209616982129443196857352361183833} a + \frac{29032500259544950602264032165884740088660396509727200599421}{129828144982550093126297265603034701584932105524805405238267}$

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

Class group and class number

$C_{5}$, which has order $5$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $12$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 291702564786496.44 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{12}\cdot 291702564786496.44 \cdot 5}{2\sqrt{23368375067493109808662942110693103411946589151921}}\approx 1.14223224229078$ (assuming GRH)

Galois group

$D_{25}$ (as 25T4):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 50
The 14 conjugacy class representatives for $D_{25}$
Character table for $D_{25}$

Intermediate fields

5.1.57121.1

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 $25$ $25$ $25$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{25}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $25$ $25$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
11Data not computed
239Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.239.2t1.a.a$1$ $ 239 $ $x^{2} - x + 60$ $C_2$ (as 2T1) $1$ $-1$
* 2.239.5t2.a.b$2$ $ 239 $ $x^{5} - 2 x^{4} + 4 x^{3} - 5 x^{2} + 2 x + 1$ $D_{5}$ (as 5T2) $1$ $0$
* 2.239.5t2.a.a$2$ $ 239 $ $x^{5} - 2 x^{4} + 4 x^{3} - 5 x^{2} + 2 x + 1$ $D_{5}$ (as 5T2) $1$ $0$
* 2.28919.25t4.a.g$2$ $ 11^{2} \cdot 239 $ $x^{25} - 7 x^{24} + 49 x^{23} - 283 x^{22} + 1302 x^{21} - 5636 x^{20} + 22809 x^{19} - 80739 x^{18} + 282358 x^{17} - 893429 x^{16} + 2621604 x^{15} - 6788665 x^{14} + 15933765 x^{13} - 33100241 x^{12} + 60008112 x^{11} - 95295391 x^{10} + 131154926 x^{9} - 150037171 x^{8} + 139788299 x^{7} - 104949621 x^{6} + 56076967 x^{5} - 27454311 x^{4} + 20655635 x^{3} - 14146574 x^{2} + 4203471 x - 447083$ $D_{25}$ (as 25T4) $1$ $0$
* 2.28919.25t4.a.h$2$ $ 11^{2} \cdot 239 $ $x^{25} - 7 x^{24} + 49 x^{23} - 283 x^{22} + 1302 x^{21} - 5636 x^{20} + 22809 x^{19} - 80739 x^{18} + 282358 x^{17} - 893429 x^{16} + 2621604 x^{15} - 6788665 x^{14} + 15933765 x^{13} - 33100241 x^{12} + 60008112 x^{11} - 95295391 x^{10} + 131154926 x^{9} - 150037171 x^{8} + 139788299 x^{7} - 104949621 x^{6} + 56076967 x^{5} - 27454311 x^{4} + 20655635 x^{3} - 14146574 x^{2} + 4203471 x - 447083$ $D_{25}$ (as 25T4) $1$ $0$
* 2.28919.25t4.a.i$2$ $ 11^{2} \cdot 239 $ $x^{25} - 7 x^{24} + 49 x^{23} - 283 x^{22} + 1302 x^{21} - 5636 x^{20} + 22809 x^{19} - 80739 x^{18} + 282358 x^{17} - 893429 x^{16} + 2621604 x^{15} - 6788665 x^{14} + 15933765 x^{13} - 33100241 x^{12} + 60008112 x^{11} - 95295391 x^{10} + 131154926 x^{9} - 150037171 x^{8} + 139788299 x^{7} - 104949621 x^{6} + 56076967 x^{5} - 27454311 x^{4} + 20655635 x^{3} - 14146574 x^{2} + 4203471 x - 447083$ $D_{25}$ (as 25T4) $1$ $0$
* 2.28919.25t4.a.f$2$ $ 11^{2} \cdot 239 $ $x^{25} - 7 x^{24} + 49 x^{23} - 283 x^{22} + 1302 x^{21} - 5636 x^{20} + 22809 x^{19} - 80739 x^{18} + 282358 x^{17} - 893429 x^{16} + 2621604 x^{15} - 6788665 x^{14} + 15933765 x^{13} - 33100241 x^{12} + 60008112 x^{11} - 95295391 x^{10} + 131154926 x^{9} - 150037171 x^{8} + 139788299 x^{7} - 104949621 x^{6} + 56076967 x^{5} - 27454311 x^{4} + 20655635 x^{3} - 14146574 x^{2} + 4203471 x - 447083$ $D_{25}$ (as 25T4) $1$ $0$
* 2.28919.25t4.a.e$2$ $ 11^{2} \cdot 239 $ $x^{25} - 7 x^{24} + 49 x^{23} - 283 x^{22} + 1302 x^{21} - 5636 x^{20} + 22809 x^{19} - 80739 x^{18} + 282358 x^{17} - 893429 x^{16} + 2621604 x^{15} - 6788665 x^{14} + 15933765 x^{13} - 33100241 x^{12} + 60008112 x^{11} - 95295391 x^{10} + 131154926 x^{9} - 150037171 x^{8} + 139788299 x^{7} - 104949621 x^{6} + 56076967 x^{5} - 27454311 x^{4} + 20655635 x^{3} - 14146574 x^{2} + 4203471 x - 447083$ $D_{25}$ (as 25T4) $1$ $0$
* 2.28919.25t4.a.b$2$ $ 11^{2} \cdot 239 $ $x^{25} - 7 x^{24} + 49 x^{23} - 283 x^{22} + 1302 x^{21} - 5636 x^{20} + 22809 x^{19} - 80739 x^{18} + 282358 x^{17} - 893429 x^{16} + 2621604 x^{15} - 6788665 x^{14} + 15933765 x^{13} - 33100241 x^{12} + 60008112 x^{11} - 95295391 x^{10} + 131154926 x^{9} - 150037171 x^{8} + 139788299 x^{7} - 104949621 x^{6} + 56076967 x^{5} - 27454311 x^{4} + 20655635 x^{3} - 14146574 x^{2} + 4203471 x - 447083$ $D_{25}$ (as 25T4) $1$ $0$
* 2.28919.25t4.a.d$2$ $ 11^{2} \cdot 239 $ $x^{25} - 7 x^{24} + 49 x^{23} - 283 x^{22} + 1302 x^{21} - 5636 x^{20} + 22809 x^{19} - 80739 x^{18} + 282358 x^{17} - 893429 x^{16} + 2621604 x^{15} - 6788665 x^{14} + 15933765 x^{13} - 33100241 x^{12} + 60008112 x^{11} - 95295391 x^{10} + 131154926 x^{9} - 150037171 x^{8} + 139788299 x^{7} - 104949621 x^{6} + 56076967 x^{5} - 27454311 x^{4} + 20655635 x^{3} - 14146574 x^{2} + 4203471 x - 447083$ $D_{25}$ (as 25T4) $1$ $0$
* 2.28919.25t4.a.c$2$ $ 11^{2} \cdot 239 $ $x^{25} - 7 x^{24} + 49 x^{23} - 283 x^{22} + 1302 x^{21} - 5636 x^{20} + 22809 x^{19} - 80739 x^{18} + 282358 x^{17} - 893429 x^{16} + 2621604 x^{15} - 6788665 x^{14} + 15933765 x^{13} - 33100241 x^{12} + 60008112 x^{11} - 95295391 x^{10} + 131154926 x^{9} - 150037171 x^{8} + 139788299 x^{7} - 104949621 x^{6} + 56076967 x^{5} - 27454311 x^{4} + 20655635 x^{3} - 14146574 x^{2} + 4203471 x - 447083$ $D_{25}$ (as 25T4) $1$ $0$
* 2.28919.25t4.a.a$2$ $ 11^{2} \cdot 239 $ $x^{25} - 7 x^{24} + 49 x^{23} - 283 x^{22} + 1302 x^{21} - 5636 x^{20} + 22809 x^{19} - 80739 x^{18} + 282358 x^{17} - 893429 x^{16} + 2621604 x^{15} - 6788665 x^{14} + 15933765 x^{13} - 33100241 x^{12} + 60008112 x^{11} - 95295391 x^{10} + 131154926 x^{9} - 150037171 x^{8} + 139788299 x^{7} - 104949621 x^{6} + 56076967 x^{5} - 27454311 x^{4} + 20655635 x^{3} - 14146574 x^{2} + 4203471 x - 447083$ $D_{25}$ (as 25T4) $1$ $0$
* 2.28919.25t4.a.j$2$ $ 11^{2} \cdot 239 $ $x^{25} - 7 x^{24} + 49 x^{23} - 283 x^{22} + 1302 x^{21} - 5636 x^{20} + 22809 x^{19} - 80739 x^{18} + 282358 x^{17} - 893429 x^{16} + 2621604 x^{15} - 6788665 x^{14} + 15933765 x^{13} - 33100241 x^{12} + 60008112 x^{11} - 95295391 x^{10} + 131154926 x^{9} - 150037171 x^{8} + 139788299 x^{7} - 104949621 x^{6} + 56076967 x^{5} - 27454311 x^{4} + 20655635 x^{3} - 14146574 x^{2} + 4203471 x - 447083$ $D_{25}$ (as 25T4) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.