Properties

Label 25.1.171...161.1
Degree $25$
Signature $[1, 12]$
Discriminant $1.718\times 10^{46}$
Root discriminant $70.70$
Ramified primes $11, 131$
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 + 18*x^23 - 17*x^22 + 163*x^21 + 778*x^20 + 668*x^19 + 6093*x^18 + 8161*x^17 - 16824*x^16 + 24958*x^15 + 156175*x^14 + 317021*x^13 + 1220510*x^12 + 3228424*x^11 + 5166827*x^10 + 7607147*x^9 + 10396892*x^8 + 8655374*x^7 + 3386053*x^6 + 971637*x^5 + 379214*x^4 - 252700*x^3 + 96379*x^2 - 14488*x + 1088)
 
gp: K = bnfinit(x^25 + 18*x^23 - 17*x^22 + 163*x^21 + 778*x^20 + 668*x^19 + 6093*x^18 + 8161*x^17 - 16824*x^16 + 24958*x^15 + 156175*x^14 + 317021*x^13 + 1220510*x^12 + 3228424*x^11 + 5166827*x^10 + 7607147*x^9 + 10396892*x^8 + 8655374*x^7 + 3386053*x^6 + 971637*x^5 + 379214*x^4 - 252700*x^3 + 96379*x^2 - 14488*x + 1088, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1088, -14488, 96379, -252700, 379214, 971637, 3386053, 8655374, 10396892, 7607147, 5166827, 3228424, 1220510, 317021, 156175, 24958, -16824, 8161, 6093, 668, 778, 163, -17, 18, 0, 1]);
 

\( x^{25} + 18 x^{23} - 17 x^{22} + 163 x^{21} + 778 x^{20} + 668 x^{19} + 6093 x^{18} + 8161 x^{17} - 16824 x^{16} + 24958 x^{15} + 156175 x^{14} + 317021 x^{13} + 1220510 x^{12} + 3228424 x^{11} + 5166827 x^{10} + 7607147 x^{9} + 10396892 x^{8} + 8655374 x^{7} + 3386053 x^{6} + 971637 x^{5} + 379214 x^{4} - 252700 x^{3} + 96379 x^{2} - 14488 x + 1088 \)

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:  \(17183405982116876392097404040231169905747048161\)\(\medspace = 11^{20}\cdot 131^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $70.70$
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:  $11, 131$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{7} - \frac{1}{8} a^{4} + \frac{1}{8} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{8} a^{5} + \frac{1}{8} a^{2}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{1}{16} a^{7} - \frac{3}{16} a^{6} - \frac{3}{16} a^{5} - \frac{3}{16} a^{4} - \frac{3}{16} a^{3} - \frac{3}{16} a^{2} - \frac{3}{8} a$, $\frac{1}{32} a^{14} + \frac{1}{16} a^{8} - \frac{3}{32} a^{2}$, $\frac{1}{32} a^{15} + \frac{1}{16} a^{9} - \frac{3}{32} a^{3}$, $\frac{1}{64} a^{16} - \frac{1}{64} a^{15} - \frac{1}{64} a^{14} - \frac{1}{32} a^{13} - \frac{1}{32} a^{12} - \frac{1}{32} a^{11} + \frac{1}{16} a^{9} + \frac{1}{16} a^{8} + \frac{1}{32} a^{7} - \frac{3}{32} a^{6} - \frac{3}{32} a^{5} - \frac{9}{64} a^{4} - \frac{11}{64} a^{3} - \frac{11}{64} a^{2} - \frac{1}{8} a$, $\frac{1}{64} a^{17} - \frac{1}{64} a^{14} + \frac{1}{32} a^{11} - \frac{1}{32} a^{8} - \frac{3}{64} a^{5} + \frac{3}{64} a^{2}$, $\frac{1}{64} a^{18} - \frac{1}{64} a^{15} + \frac{1}{32} a^{12} - \frac{1}{32} a^{9} - \frac{3}{64} a^{6} + \frac{3}{64} a^{3}$, $\frac{1}{128} a^{19} - \frac{1}{128} a^{18} - \frac{1}{128} a^{17} - \frac{1}{128} a^{16} + \frac{1}{128} a^{15} + \frac{1}{128} a^{14} - \frac{1}{64} a^{13} + \frac{1}{64} a^{12} + \frac{1}{64} a^{11} + \frac{1}{64} a^{10} + \frac{3}{64} a^{9} + \frac{3}{64} a^{8} - \frac{7}{128} a^{7} - \frac{17}{128} a^{6} - \frac{17}{128} a^{5} - \frac{17}{128} a^{4} + \frac{41}{128} a^{3} - \frac{23}{128} a^{2} - \frac{1}{16} a - \frac{1}{2}$, $\frac{1}{128} a^{20} + \frac{1}{128} a^{14} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{13}{128} a^{8} - \frac{1}{16} a^{7} - \frac{3}{16} a^{6} - \frac{3}{16} a^{5} - \frac{3}{16} a^{4} + \frac{5}{16} a^{3} - \frac{21}{128} a^{2} - \frac{3}{16} a - \frac{1}{2}$, $\frac{1}{512} a^{21} - \frac{1}{512} a^{19} + \frac{1}{512} a^{18} + \frac{3}{512} a^{17} - \frac{3}{512} a^{16} - \frac{7}{512} a^{14} - \frac{3}{256} a^{13} + \frac{3}{256} a^{12} - \frac{11}{256} a^{11} + \frac{15}{256} a^{10} - \frac{35}{512} a^{9} - \frac{21}{256} a^{8} - \frac{1}{512} a^{7} + \frac{41}{512} a^{6} + \frac{3}{512} a^{5} - \frac{43}{512} a^{4} - \frac{119}{256} a^{3} + \frac{33}{512} a^{2} - \frac{1}{64} a + \frac{3}{8}$, $\frac{1}{512} a^{22} - \frac{1}{512} a^{20} + \frac{1}{512} a^{19} + \frac{3}{512} a^{18} - \frac{3}{512} a^{17} - \frac{7}{512} a^{15} - \frac{3}{256} a^{14} + \frac{3}{256} a^{13} - \frac{11}{256} a^{12} + \frac{15}{256} a^{11} + \frac{29}{512} a^{10} - \frac{21}{256} a^{9} - \frac{1}{512} a^{8} - \frac{23}{512} a^{7} + \frac{3}{512} a^{6} - \frac{43}{512} a^{5} - \frac{23}{256} a^{4} + \frac{33}{512} a^{3} - \frac{1}{64} a^{2}$, $\frac{1}{98311168} a^{23} - \frac{87863}{98311168} a^{22} + \frac{4107}{12288896} a^{21} + \frac{8619}{6144448} a^{20} - \frac{177013}{98311168} a^{19} + \frac{119809}{98311168} a^{18} + \frac{8805}{12288896} a^{17} + \frac{58391}{49155584} a^{16} + \frac{1031315}{98311168} a^{15} + \frac{505785}{98311168} a^{14} + \frac{13253}{49155584} a^{13} + \frac{2062479}{49155584} a^{12} + \frac{5088517}{98311168} a^{11} - \frac{4510119}{98311168} a^{10} - \frac{155597}{1585664} a^{9} + \frac{4544351}{49155584} a^{8} + \frac{11676163}{98311168} a^{7} + \frac{3163441}{98311168} a^{6} - \frac{4233359}{49155584} a^{5} - \frac{9707}{80848} a^{4} - \frac{34225717}{98311168} a^{3} + \frac{29441513}{98311168} a^{2} + \frac{3135647}{12288896} a - \frac{570141}{1536112}$, $\frac{1}{266562372380805553492987920756701635514090859430827252736} a^{24} + \frac{1054762804269807390187529661884604925387024529045}{266562372380805553492987920756701635514090859430827252736} a^{23} + \frac{103546041795546060622399625455800788096417852500948939}{133281186190402776746493960378350817757045429715413626368} a^{22} + \frac{22358991056657140819579762389746903700053907966739785}{66640593095201388373246980189175408878522714857706813184} a^{21} - \frac{494069392432063120432583449171311058218774546582085439}{266562372380805553492987920756701635514090859430827252736} a^{20} - \frac{362813383858702594068489155696890326702071284455900981}{266562372380805553492987920756701635514090859430827252736} a^{19} + \frac{7396684575876040484579278204263186734709163773315825}{1017413635041242570583923361666800135549965112331401728} a^{18} - \frac{310330338157811383437969090264896620427369874716626761}{66640593095201388373246980189175408878522714857706813184} a^{17} - \frac{2081744362336256383546331936308556071002782653097179609}{266562372380805553492987920756701635514090859430827252736} a^{16} + \frac{3207544678785630102715136847444775527924720733177837319}{266562372380805553492987920756701635514090859430827252736} a^{15} + \frac{407748484305696477860740188734546035365534154321953879}{133281186190402776746493960378350817757045429715413626368} a^{14} - \frac{1294692773612998967303578295340158134694028633540432275}{133281186190402776746493960378350817757045429715413626368} a^{13} + \frac{4889869022563511996432964786530743412352400478418618985}{266562372380805553492987920756701635514090859430827252736} a^{12} - \frac{11591914550166495959102522123996766137739339197826710903}{266562372380805553492987920756701635514090859430827252736} a^{11} + \frac{29955771429928449647401754408211077238796459277242455}{33320296547600694186623490094587704439261357428853406592} a^{10} - \frac{1186232937773623989579545609727646390238409917677996085}{133281186190402776746493960378350817757045429715413626368} a^{9} + \frac{1031117477335768345741009265675911062844182450820475759}{11589668364382850151869040032900071109308298236122924032} a^{8} + \frac{21355943761999485134454870663962271582416906333229630091}{266562372380805553492987920756701635514090859430827252736} a^{7} - \frac{26091416958519995327178284782188494039687655467203803}{362177136386964067245907501028127222165884319878841376} a^{6} + \frac{1230909344980294127473415189323956038864335012520123321}{7014799273179093512973366335702674618791864721863875072} a^{5} + \frac{16835221380069483202036576043209997382522839104820211963}{266562372380805553492987920756701635514090859430827252736} a^{4} - \frac{24265222199480489158765828217485964414580977310472759377}{266562372380805553492987920756701635514090859430827252736} a^{3} + \frac{6866093934654085863923467960738753415582210828505434401}{33320296547600694186623490094587704439261357428853406592} a^{2} + \frac{1546391898573609291506111370131652378230862223550240085}{4165037068450086773327936261823463054907669678606675824} a + \frac{188406420672918675975699533198258532479232916547334}{805928225319289236325065066142310962636932987346493}$

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:  \( 79672132715035.66 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{12}\cdot 79672132715035.66 \cdot 5}{2\sqrt{17183405982116876392097404040231169905747048161}}\approx 11.5048309581894$ (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

5.1.17161.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ $25$ $25$ $25$ R $25$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $25$ $25$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{5}$ $25$

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
$131$$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.131.2t1.a.a$1$ $ 131 $ $x^{2} - x + 33$ $C_2$ (as 2T1) $1$ $-1$
* 2.131.5t2.a.b$2$ $ 131 $ $x^{5} - x^{4} + 2 x^{3} - x^{2} + x + 2$ $D_{5}$ (as 5T2) $1$ $0$
* 2.131.5t2.a.a$2$ $ 131 $ $x^{5} - x^{4} + 2 x^{3} - x^{2} + x + 2$ $D_{5}$ (as 5T2) $1$ $0$
* 2.15851.25t4.a.j$2$ $ 11^{2} \cdot 131 $ $x^{25} + 18 x^{23} - 17 x^{22} + 163 x^{21} + 778 x^{20} + 668 x^{19} + 6093 x^{18} + 8161 x^{17} - 16824 x^{16} + 24958 x^{15} + 156175 x^{14} + 317021 x^{13} + 1220510 x^{12} + 3228424 x^{11} + 5166827 x^{10} + 7607147 x^{9} + 10396892 x^{8} + 8655374 x^{7} + 3386053 x^{6} + 971637 x^{5} + 379214 x^{4} - 252700 x^{3} + 96379 x^{2} - 14488 x + 1088$ $D_{25}$ (as 25T4) $1$ $0$
* 2.15851.25t4.a.e$2$ $ 11^{2} \cdot 131 $ $x^{25} + 18 x^{23} - 17 x^{22} + 163 x^{21} + 778 x^{20} + 668 x^{19} + 6093 x^{18} + 8161 x^{17} - 16824 x^{16} + 24958 x^{15} + 156175 x^{14} + 317021 x^{13} + 1220510 x^{12} + 3228424 x^{11} + 5166827 x^{10} + 7607147 x^{9} + 10396892 x^{8} + 8655374 x^{7} + 3386053 x^{6} + 971637 x^{5} + 379214 x^{4} - 252700 x^{3} + 96379 x^{2} - 14488 x + 1088$ $D_{25}$ (as 25T4) $1$ $0$
* 2.15851.25t4.a.a$2$ $ 11^{2} \cdot 131 $ $x^{25} + 18 x^{23} - 17 x^{22} + 163 x^{21} + 778 x^{20} + 668 x^{19} + 6093 x^{18} + 8161 x^{17} - 16824 x^{16} + 24958 x^{15} + 156175 x^{14} + 317021 x^{13} + 1220510 x^{12} + 3228424 x^{11} + 5166827 x^{10} + 7607147 x^{9} + 10396892 x^{8} + 8655374 x^{7} + 3386053 x^{6} + 971637 x^{5} + 379214 x^{4} - 252700 x^{3} + 96379 x^{2} - 14488 x + 1088$ $D_{25}$ (as 25T4) $1$ $0$
* 2.15851.25t4.a.b$2$ $ 11^{2} \cdot 131 $ $x^{25} + 18 x^{23} - 17 x^{22} + 163 x^{21} + 778 x^{20} + 668 x^{19} + 6093 x^{18} + 8161 x^{17} - 16824 x^{16} + 24958 x^{15} + 156175 x^{14} + 317021 x^{13} + 1220510 x^{12} + 3228424 x^{11} + 5166827 x^{10} + 7607147 x^{9} + 10396892 x^{8} + 8655374 x^{7} + 3386053 x^{6} + 971637 x^{5} + 379214 x^{4} - 252700 x^{3} + 96379 x^{2} - 14488 x + 1088$ $D_{25}$ (as 25T4) $1$ $0$
* 2.15851.25t4.a.d$2$ $ 11^{2} \cdot 131 $ $x^{25} + 18 x^{23} - 17 x^{22} + 163 x^{21} + 778 x^{20} + 668 x^{19} + 6093 x^{18} + 8161 x^{17} - 16824 x^{16} + 24958 x^{15} + 156175 x^{14} + 317021 x^{13} + 1220510 x^{12} + 3228424 x^{11} + 5166827 x^{10} + 7607147 x^{9} + 10396892 x^{8} + 8655374 x^{7} + 3386053 x^{6} + 971637 x^{5} + 379214 x^{4} - 252700 x^{3} + 96379 x^{2} - 14488 x + 1088$ $D_{25}$ (as 25T4) $1$ $0$
* 2.15851.25t4.a.c$2$ $ 11^{2} \cdot 131 $ $x^{25} + 18 x^{23} - 17 x^{22} + 163 x^{21} + 778 x^{20} + 668 x^{19} + 6093 x^{18} + 8161 x^{17} - 16824 x^{16} + 24958 x^{15} + 156175 x^{14} + 317021 x^{13} + 1220510 x^{12} + 3228424 x^{11} + 5166827 x^{10} + 7607147 x^{9} + 10396892 x^{8} + 8655374 x^{7} + 3386053 x^{6} + 971637 x^{5} + 379214 x^{4} - 252700 x^{3} + 96379 x^{2} - 14488 x + 1088$ $D_{25}$ (as 25T4) $1$ $0$
* 2.15851.25t4.a.i$2$ $ 11^{2} \cdot 131 $ $x^{25} + 18 x^{23} - 17 x^{22} + 163 x^{21} + 778 x^{20} + 668 x^{19} + 6093 x^{18} + 8161 x^{17} - 16824 x^{16} + 24958 x^{15} + 156175 x^{14} + 317021 x^{13} + 1220510 x^{12} + 3228424 x^{11} + 5166827 x^{10} + 7607147 x^{9} + 10396892 x^{8} + 8655374 x^{7} + 3386053 x^{6} + 971637 x^{5} + 379214 x^{4} - 252700 x^{3} + 96379 x^{2} - 14488 x + 1088$ $D_{25}$ (as 25T4) $1$ $0$
* 2.15851.25t4.a.g$2$ $ 11^{2} \cdot 131 $ $x^{25} + 18 x^{23} - 17 x^{22} + 163 x^{21} + 778 x^{20} + 668 x^{19} + 6093 x^{18} + 8161 x^{17} - 16824 x^{16} + 24958 x^{15} + 156175 x^{14} + 317021 x^{13} + 1220510 x^{12} + 3228424 x^{11} + 5166827 x^{10} + 7607147 x^{9} + 10396892 x^{8} + 8655374 x^{7} + 3386053 x^{6} + 971637 x^{5} + 379214 x^{4} - 252700 x^{3} + 96379 x^{2} - 14488 x + 1088$ $D_{25}$ (as 25T4) $1$ $0$
* 2.15851.25t4.a.h$2$ $ 11^{2} \cdot 131 $ $x^{25} + 18 x^{23} - 17 x^{22} + 163 x^{21} + 778 x^{20} + 668 x^{19} + 6093 x^{18} + 8161 x^{17} - 16824 x^{16} + 24958 x^{15} + 156175 x^{14} + 317021 x^{13} + 1220510 x^{12} + 3228424 x^{11} + 5166827 x^{10} + 7607147 x^{9} + 10396892 x^{8} + 8655374 x^{7} + 3386053 x^{6} + 971637 x^{5} + 379214 x^{4} - 252700 x^{3} + 96379 x^{2} - 14488 x + 1088$ $D_{25}$ (as 25T4) $1$ $0$
* 2.15851.25t4.a.f$2$ $ 11^{2} \cdot 131 $ $x^{25} + 18 x^{23} - 17 x^{22} + 163 x^{21} + 778 x^{20} + 668 x^{19} + 6093 x^{18} + 8161 x^{17} - 16824 x^{16} + 24958 x^{15} + 156175 x^{14} + 317021 x^{13} + 1220510 x^{12} + 3228424 x^{11} + 5166827 x^{10} + 7607147 x^{9} + 10396892 x^{8} + 8655374 x^{7} + 3386053 x^{6} + 971637 x^{5} + 379214 x^{4} - 252700 x^{3} + 96379 x^{2} - 14488 x + 1088$ $D_{25}$ (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 *.