Properties

Label 25.1.106...201.1
Degree $25$
Signature $[1, 12]$
Discriminant $1.064\times 10^{43}$
Root discriminant $52.61$
Ramified prime $3851$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{25}$ (as 25T4)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 10*x^24 + 41*x^23 - 105*x^22 + 177*x^21 - 437*x^20 + 1470*x^19 - 3743*x^18 + 4598*x^17 - 1850*x^16 + 3734*x^15 - 33702*x^14 + 37801*x^13 + 70620*x^12 - 146863*x^11 - 115089*x^10 + 114649*x^9 + 185307*x^8 - 74232*x^7 - 239939*x^6 - 516736*x^5 - 486684*x^4 - 187312*x^3 - 162320*x^2 + 32064*x - 22784)
 
gp: K = bnfinit(x^25 - 10*x^24 + 41*x^23 - 105*x^22 + 177*x^21 - 437*x^20 + 1470*x^19 - 3743*x^18 + 4598*x^17 - 1850*x^16 + 3734*x^15 - 33702*x^14 + 37801*x^13 + 70620*x^12 - 146863*x^11 - 115089*x^10 + 114649*x^9 + 185307*x^8 - 74232*x^7 - 239939*x^6 - 516736*x^5 - 486684*x^4 - 187312*x^3 - 162320*x^2 + 32064*x - 22784, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-22784, 32064, -162320, -187312, -486684, -516736, -239939, -74232, 185307, 114649, -115089, -146863, 70620, 37801, -33702, 3734, -1850, 4598, -3743, 1470, -437, 177, -105, 41, -10, 1]);
 

\( x^{25} - 10 x^{24} + 41 x^{23} - 105 x^{22} + 177 x^{21} - 437 x^{20} + 1470 x^{19} - 3743 x^{18} + 4598 x^{17} - 1850 x^{16} + 3734 x^{15} - 33702 x^{14} + 37801 x^{13} + 70620 x^{12} - 146863 x^{11} - 115089 x^{10} + 114649 x^{9} + 185307 x^{8} - 74232 x^{7} - 239939 x^{6} - 516736 x^{5} - 486684 x^{4} - 187312 x^{3} - 162320 x^{2} + 32064 x - 22784 \)

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:  \(10638544719000572788610677956885758729581201\)\(\medspace = 3851^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $52.61$
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:  $3851$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{10} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{3}{16} a^{6} + \frac{1}{8} a^{4} + \frac{3}{16} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{13} - \frac{1}{8} a^{11} + \frac{1}{16} a^{10} - \frac{1}{8} a^{9} + \frac{3}{16} a^{7} + \frac{1}{8} a^{5} + \frac{3}{16} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{14} + \frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{16} a^{8} + \frac{3}{16} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{15} - \frac{1}{32} a^{14} - \frac{1}{32} a^{13} - \frac{1}{32} a^{12} + \frac{3}{32} a^{11} - \frac{1}{32} a^{10} + \frac{1}{32} a^{9} + \frac{1}{32} a^{8} - \frac{3}{32} a^{7} - \frac{3}{32} a^{6} - \frac{7}{32} a^{5} - \frac{3}{32} a^{4} + \frac{5}{16} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{16} + \frac{1}{16} a^{10} - \frac{1}{4} a^{7} - \frac{3}{32} a^{4} + \frac{1}{4} a$, $\frac{1}{448} a^{17} - \frac{1}{224} a^{16} + \frac{1}{224} a^{15} - \frac{5}{224} a^{14} - \frac{1}{32} a^{12} - \frac{3}{112} a^{11} - \frac{3}{112} a^{10} - \frac{23}{224} a^{9} + \frac{25}{224} a^{8} - \frac{1}{4} a^{7} + \frac{43}{224} a^{6} - \frac{109}{448} a^{5} + \frac{47}{224} a^{4} + \frac{7}{16} a^{3} - \frac{5}{28} a^{2} + \frac{5}{14} a - \frac{3}{7}$, $\frac{1}{448} a^{18} - \frac{1}{224} a^{16} - \frac{3}{224} a^{15} + \frac{1}{56} a^{14} - \frac{1}{32} a^{13} - \frac{3}{112} a^{12} - \frac{1}{56} a^{11} + \frac{3}{32} a^{10} + \frac{3}{32} a^{9} + \frac{1}{28} a^{8} - \frac{13}{224} a^{7} - \frac{11}{64} a^{6} - \frac{5}{56} a^{5} + \frac{3}{28} a^{4} - \frac{27}{112} a^{3} - \frac{3}{8} a^{2} - \frac{13}{28} a + \frac{1}{7}$, $\frac{1}{448} a^{19} + \frac{1}{112} a^{16} - \frac{1}{224} a^{15} + \frac{1}{56} a^{14} + \frac{1}{224} a^{13} + \frac{3}{224} a^{12} + \frac{1}{112} a^{11} - \frac{13}{112} a^{10} - \frac{3}{224} a^{9} - \frac{3}{56} a^{8} - \frac{5}{64} a^{7} + \frac{17}{224} a^{6} + \frac{3}{112} a^{5} - \frac{1}{14} a^{4} + \frac{1}{4} a^{3} - \frac{9}{28} a^{2} + \frac{5}{14} a + \frac{1}{7}$, $\frac{1}{19712} a^{20} + \frac{1}{2816} a^{19} - \frac{13}{19712} a^{18} + \frac{9}{9856} a^{17} + \frac{19}{9856} a^{16} + \frac{15}{9856} a^{15} - \frac{113}{4928} a^{14} + \frac{5}{616} a^{13} + \frac{145}{4928} a^{12} + \frac{19}{9856} a^{11} + \frac{571}{9856} a^{10} + \frac{93}{9856} a^{9} - \frac{1013}{19712} a^{8} + \frac{275}{1792} a^{7} - \frac{3215}{19712} a^{6} + \frac{73}{616} a^{5} - \frac{625}{4928} a^{4} + \frac{15}{176} a^{3} - \frac{43}{112} a^{2} + \frac{3}{308} a - \frac{26}{77}$, $\frac{1}{78848} a^{21} + \frac{13}{39424} a^{19} - \frac{23}{78848} a^{18} + \frac{1}{1792} a^{17} + \frac{29}{19712} a^{16} + \frac{109}{39424} a^{15} - \frac{73}{2816} a^{14} - \frac{35}{2816} a^{13} - \frac{31}{39424} a^{12} + \frac{245}{2816} a^{11} + \frac{949}{9856} a^{10} + \frac{6485}{78848} a^{9} - \frac{265}{19712} a^{8} - \frac{133}{5632} a^{7} + \frac{11157}{78848} a^{6} + \frac{179}{9856} a^{5} - \frac{3631}{19712} a^{4} + \frac{1531}{4928} a^{3} + \frac{2179}{4928} a^{2} + \frac{23}{176} a - \frac{5}{308}$, $\frac{1}{78848} a^{22} - \frac{1}{39424} a^{20} - \frac{43}{78848} a^{19} + \frac{1}{1408} a^{18} - \frac{9}{19712} a^{17} - \frac{71}{39424} a^{16} - \frac{15}{2816} a^{15} + \frac{65}{2816} a^{14} + \frac{369}{39424} a^{13} + \frac{207}{19712} a^{12} + \frac{1}{88} a^{11} - \frac{4371}{78848} a^{10} + \frac{281}{19712} a^{9} - \frac{125}{39424} a^{8} + \frac{8297}{78848} a^{7} - \frac{455}{2816} a^{6} + \frac{345}{19712} a^{5} + \frac{73}{352} a^{4} + \frac{2231}{4928} a^{3} - \frac{111}{308} a^{2} + \frac{32}{77} a + \frac{4}{11}$, $\frac{1}{6938624} a^{23} + \frac{1}{6938624} a^{22} + \frac{15}{6938624} a^{21} + \frac{23}{6938624} a^{20} - \frac{125}{6938624} a^{19} - \frac{2663}{6938624} a^{18} - \frac{3855}{3469312} a^{17} - \frac{2755}{3469312} a^{16} + \frac{6419}{495616} a^{15} + \frac{22217}{3469312} a^{14} - \frac{6859}{3469312} a^{13} - \frac{537}{3469312} a^{12} - \frac{74233}{991232} a^{11} - \frac{773327}{6938624} a^{10} + \frac{103217}{991232} a^{9} + \frac{234775}{6938624} a^{8} + \frac{713795}{6938624} a^{7} + \frac{868201}{6938624} a^{6} + \frac{393431}{1734656} a^{5} - \frac{42111}{1734656} a^{4} - \frac{106257}{216832} a^{3} + \frac{38767}{433664} a^{2} + \frac{49221}{108416} a - \frac{8165}{27104}$, $\frac{1}{5092290188686268460548565900541724450816} a^{24} - \frac{2013627016732062895433300524001}{90933753369397651081224391081102222336} a^{23} + \frac{856668070279931247266983870092853}{363735013477590604324897564324408889344} a^{22} - \frac{111853166073574280962782564778373}{57866933962343959778960976142519596032} a^{21} - \frac{21781735854044412575442944136771959}{1273072547171567115137141475135431112704} a^{20} - \frac{84509605691158802867964227364655387}{231467735849375839115843904570078384128} a^{19} + \frac{2614171757507926350283361761015964857}{5092290188686268460548565900541724450816} a^{18} + \frac{69338705184080061261521087375850087}{90933753369397651081224391081102222336} a^{17} + \frac{221020880297612508856092233222201809}{22733438342349412770306097770275555584} a^{16} - \frac{122353093654900060251037708151743739}{636536273585783557568570737567715556352} a^{15} - \frac{6772787589020313532614146792845137047}{636536273585783557568570737567715556352} a^{14} - \frac{6861755056947336192680602248842520403}{1273072547171567115137141475135431112704} a^{13} + \frac{4987100930749561781029421050390482053}{727470026955181208649795128648817778688} a^{12} - \frac{59196723798960178237371658324681559619}{636536273585783557568570737567715556352} a^{11} + \frac{115050200605345465430177550133897691359}{2546145094343134230274282950270862225408} a^{10} + \frac{711767481942723232073577797979820929}{22733438342349412770306097770275555584} a^{9} + \frac{921659058340805858114326425545793463}{181867506738795302162448782162204444672} a^{8} - \frac{3496201657539520288685371161563305091}{231467735849375839115843904570078384128} a^{7} - \frac{396831720444815271185354683687801920317}{5092290188686268460548565900541724450816} a^{6} - \frac{21887699556205585526293216932476885769}{90933753369397651081224391081102222336} a^{5} - \frac{657688958311408296308613493975477885}{16533409703526845651131707469291313152} a^{4} + \frac{17844335995373314234363362485242364137}{318268136792891778784285368783857778176} a^{3} + \frac{2966216239468777111956959248046742869}{318268136792891778784285368783857778176} a^{2} - \frac{200034417725420165573506865228182587}{7233366745292994972370122017814949504} a - \frac{3923331358749167917115153060263077067}{19891758549555736174017835548991111136}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 11838033464233.441 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{12}\cdot 11838033464233.441 \cdot 1}{2\sqrt{10638544719000572788610677956885758729581201}}\approx 13.7403207814964$ (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

5.1.14830201.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ $25$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $25$ $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $25$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $25$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
3851Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.3851.2t1.a.a$1$ $ 3851 $ $x^{2} - x + 963$ $C_2$ (as 2T1) $1$ $-1$
* 2.3851.5t2.a.a$2$ $ 3851 $ $x^{5} - 8 x^{3} - 53 x^{2} - 90 x - 100$ $D_{5}$ (as 5T2) $1$ $0$
* 2.3851.5t2.a.b$2$ $ 3851 $ $x^{5} - 8 x^{3} - 53 x^{2} - 90 x - 100$ $D_{5}$ (as 5T2) $1$ $0$
* 2.3851.25t4.a.j$2$ $ 3851 $ $x^{25} - 10 x^{24} + 41 x^{23} - 105 x^{22} + 177 x^{21} - 437 x^{20} + 1470 x^{19} - 3743 x^{18} + 4598 x^{17} - 1850 x^{16} + 3734 x^{15} - 33702 x^{14} + 37801 x^{13} + 70620 x^{12} - 146863 x^{11} - 115089 x^{10} + 114649 x^{9} + 185307 x^{8} - 74232 x^{7} - 239939 x^{6} - 516736 x^{5} - 486684 x^{4} - 187312 x^{3} - 162320 x^{2} + 32064 x - 22784$ $D_{25}$ (as 25T4) $1$ $0$
* 2.3851.25t4.a.e$2$ $ 3851 $ $x^{25} - 10 x^{24} + 41 x^{23} - 105 x^{22} + 177 x^{21} - 437 x^{20} + 1470 x^{19} - 3743 x^{18} + 4598 x^{17} - 1850 x^{16} + 3734 x^{15} - 33702 x^{14} + 37801 x^{13} + 70620 x^{12} - 146863 x^{11} - 115089 x^{10} + 114649 x^{9} + 185307 x^{8} - 74232 x^{7} - 239939 x^{6} - 516736 x^{5} - 486684 x^{4} - 187312 x^{3} - 162320 x^{2} + 32064 x - 22784$ $D_{25}$ (as 25T4) $1$ $0$
* 2.3851.25t4.a.a$2$ $ 3851 $ $x^{25} - 10 x^{24} + 41 x^{23} - 105 x^{22} + 177 x^{21} - 437 x^{20} + 1470 x^{19} - 3743 x^{18} + 4598 x^{17} - 1850 x^{16} + 3734 x^{15} - 33702 x^{14} + 37801 x^{13} + 70620 x^{12} - 146863 x^{11} - 115089 x^{10} + 114649 x^{9} + 185307 x^{8} - 74232 x^{7} - 239939 x^{6} - 516736 x^{5} - 486684 x^{4} - 187312 x^{3} - 162320 x^{2} + 32064 x - 22784$ $D_{25}$ (as 25T4) $1$ $0$
* 2.3851.25t4.a.b$2$ $ 3851 $ $x^{25} - 10 x^{24} + 41 x^{23} - 105 x^{22} + 177 x^{21} - 437 x^{20} + 1470 x^{19} - 3743 x^{18} + 4598 x^{17} - 1850 x^{16} + 3734 x^{15} - 33702 x^{14} + 37801 x^{13} + 70620 x^{12} - 146863 x^{11} - 115089 x^{10} + 114649 x^{9} + 185307 x^{8} - 74232 x^{7} - 239939 x^{6} - 516736 x^{5} - 486684 x^{4} - 187312 x^{3} - 162320 x^{2} + 32064 x - 22784$ $D_{25}$ (as 25T4) $1$ $0$
* 2.3851.25t4.a.d$2$ $ 3851 $ $x^{25} - 10 x^{24} + 41 x^{23} - 105 x^{22} + 177 x^{21} - 437 x^{20} + 1470 x^{19} - 3743 x^{18} + 4598 x^{17} - 1850 x^{16} + 3734 x^{15} - 33702 x^{14} + 37801 x^{13} + 70620 x^{12} - 146863 x^{11} - 115089 x^{10} + 114649 x^{9} + 185307 x^{8} - 74232 x^{7} - 239939 x^{6} - 516736 x^{5} - 486684 x^{4} - 187312 x^{3} - 162320 x^{2} + 32064 x - 22784$ $D_{25}$ (as 25T4) $1$ $0$
* 2.3851.25t4.a.c$2$ $ 3851 $ $x^{25} - 10 x^{24} + 41 x^{23} - 105 x^{22} + 177 x^{21} - 437 x^{20} + 1470 x^{19} - 3743 x^{18} + 4598 x^{17} - 1850 x^{16} + 3734 x^{15} - 33702 x^{14} + 37801 x^{13} + 70620 x^{12} - 146863 x^{11} - 115089 x^{10} + 114649 x^{9} + 185307 x^{8} - 74232 x^{7} - 239939 x^{6} - 516736 x^{5} - 486684 x^{4} - 187312 x^{3} - 162320 x^{2} + 32064 x - 22784$ $D_{25}$ (as 25T4) $1$ $0$
* 2.3851.25t4.a.i$2$ $ 3851 $ $x^{25} - 10 x^{24} + 41 x^{23} - 105 x^{22} + 177 x^{21} - 437 x^{20} + 1470 x^{19} - 3743 x^{18} + 4598 x^{17} - 1850 x^{16} + 3734 x^{15} - 33702 x^{14} + 37801 x^{13} + 70620 x^{12} - 146863 x^{11} - 115089 x^{10} + 114649 x^{9} + 185307 x^{8} - 74232 x^{7} - 239939 x^{6} - 516736 x^{5} - 486684 x^{4} - 187312 x^{3} - 162320 x^{2} + 32064 x - 22784$ $D_{25}$ (as 25T4) $1$ $0$
* 2.3851.25t4.a.g$2$ $ 3851 $ $x^{25} - 10 x^{24} + 41 x^{23} - 105 x^{22} + 177 x^{21} - 437 x^{20} + 1470 x^{19} - 3743 x^{18} + 4598 x^{17} - 1850 x^{16} + 3734 x^{15} - 33702 x^{14} + 37801 x^{13} + 70620 x^{12} - 146863 x^{11} - 115089 x^{10} + 114649 x^{9} + 185307 x^{8} - 74232 x^{7} - 239939 x^{6} - 516736 x^{5} - 486684 x^{4} - 187312 x^{3} - 162320 x^{2} + 32064 x - 22784$ $D_{25}$ (as 25T4) $1$ $0$
* 2.3851.25t4.a.h$2$ $ 3851 $ $x^{25} - 10 x^{24} + 41 x^{23} - 105 x^{22} + 177 x^{21} - 437 x^{20} + 1470 x^{19} - 3743 x^{18} + 4598 x^{17} - 1850 x^{16} + 3734 x^{15} - 33702 x^{14} + 37801 x^{13} + 70620 x^{12} - 146863 x^{11} - 115089 x^{10} + 114649 x^{9} + 185307 x^{8} - 74232 x^{7} - 239939 x^{6} - 516736 x^{5} - 486684 x^{4} - 187312 x^{3} - 162320 x^{2} + 32064 x - 22784$ $D_{25}$ (as 25T4) $1$ $0$
* 2.3851.25t4.a.f$2$ $ 3851 $ $x^{25} - 10 x^{24} + 41 x^{23} - 105 x^{22} + 177 x^{21} - 437 x^{20} + 1470 x^{19} - 3743 x^{18} + 4598 x^{17} - 1850 x^{16} + 3734 x^{15} - 33702 x^{14} + 37801 x^{13} + 70620 x^{12} - 146863 x^{11} - 115089 x^{10} + 114649 x^{9} + 185307 x^{8} - 74232 x^{7} - 239939 x^{6} - 516736 x^{5} - 486684 x^{4} - 187312 x^{3} - 162320 x^{2} + 32064 x - 22784$ $D_{25}$ (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 *.