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Properties

Label 25.1.335...881.1
Degree $25$
Signature $[1, 12]$
Discriminant $3.352\times 10^{41}$
Root discriminant \(45.82\)
Ramified prime $2887$
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 - 5*x^24 + 18*x^23 - 53*x^22 + 184*x^21 - 519*x^20 + 1425*x^19 - 2994*x^18 + 4254*x^17 - 2567*x^16 - 3146*x^15 + 8388*x^14 - 11301*x^13 - 6966*x^12 + 13530*x^11 + 11956*x^10 + 18271*x^9 + 21048*x^8 - 41862*x^7 - 65478*x^6 - 1959*x^5 + 35311*x^4 + 17869*x^3 - 2655*x^2 - 3850*x - 2125)
 
gp: K = bnfinit(y^25 - 5*y^24 + 18*y^23 - 53*y^22 + 184*y^21 - 519*y^20 + 1425*y^19 - 2994*y^18 + 4254*y^17 - 2567*y^16 - 3146*y^15 + 8388*y^14 - 11301*y^13 - 6966*y^12 + 13530*y^11 + 11956*y^10 + 18271*y^9 + 21048*y^8 - 41862*y^7 - 65478*y^6 - 1959*y^5 + 35311*y^4 + 17869*y^3 - 2655*y^2 - 3850*y - 2125, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 - 5*x^24 + 18*x^23 - 53*x^22 + 184*x^21 - 519*x^20 + 1425*x^19 - 2994*x^18 + 4254*x^17 - 2567*x^16 - 3146*x^15 + 8388*x^14 - 11301*x^13 - 6966*x^12 + 13530*x^11 + 11956*x^10 + 18271*x^9 + 21048*x^8 - 41862*x^7 - 65478*x^6 - 1959*x^5 + 35311*x^4 + 17869*x^3 - 2655*x^2 - 3850*x - 2125);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 5*x^24 + 18*x^23 - 53*x^22 + 184*x^21 - 519*x^20 + 1425*x^19 - 2994*x^18 + 4254*x^17 - 2567*x^16 - 3146*x^15 + 8388*x^14 - 11301*x^13 - 6966*x^12 + 13530*x^11 + 11956*x^10 + 18271*x^9 + 21048*x^8 - 41862*x^7 - 65478*x^6 - 1959*x^5 + 35311*x^4 + 17869*x^3 - 2655*x^2 - 3850*x - 2125)
 

\( x^{25} - 5 x^{24} + 18 x^{23} - 53 x^{22} + 184 x^{21} - 519 x^{20} + 1425 x^{19} - 2994 x^{18} + \cdots - 2125 \) Copy content Toggle raw display

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

Invariants

Degree:  $25$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(335244303153752999188341104839710238574881\) \(\medspace = 2887^{12}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(45.82\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2887^{1/2}\approx 53.73081052803875$
Ramified primes:   \(2887\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{9}a^{13}+\frac{1}{9}a^{12}-\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{9}a^{7}+\frac{4}{9}a^{6}+\frac{1}{3}a^{4}+\frac{2}{9}a^{3}-\frac{4}{9}a^{2}+\frac{1}{9}a+\frac{4}{9}$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{12}+\frac{1}{9}a^{10}+\frac{1}{9}a^{8}+\frac{2}{9}a^{6}+\frac{2}{9}a^{4}+\frac{2}{9}a^{2}+\frac{2}{9}$, $\frac{1}{9}a^{15}-\frac{1}{9}a^{12}-\frac{1}{9}a^{11}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{7}-\frac{4}{9}a^{6}+\frac{2}{9}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{4}{9}a^{2}+\frac{1}{9}a-\frac{4}{9}$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{8}-\frac{2}{9}$, $\frac{1}{135}a^{17}+\frac{1}{135}a^{16}-\frac{4}{135}a^{15}-\frac{4}{135}a^{14}-\frac{1}{27}a^{13}-\frac{1}{27}a^{12}+\frac{1}{45}a^{11}+\frac{2}{45}a^{10}-\frac{1}{15}a^{9}-\frac{1}{9}a^{8}+\frac{2}{15}a^{7}-\frac{13}{45}a^{6}+\frac{2}{27}a^{5}-\frac{11}{135}a^{4}-\frac{43}{135}a^{3}+\frac{1}{27}a^{2}-\frac{14}{135}a+\frac{5}{27}$, $\frac{1}{135}a^{18}-\frac{1}{27}a^{16}-\frac{1}{135}a^{14}+\frac{8}{135}a^{12}+\frac{1}{45}a^{11}-\frac{1}{9}a^{10}-\frac{2}{45}a^{9}-\frac{4}{45}a^{8}-\frac{4}{45}a^{7}-\frac{41}{135}a^{6}+\frac{8}{45}a^{5}+\frac{13}{135}a^{4}-\frac{14}{45}a^{3}+\frac{26}{135}a^{2}-\frac{17}{45}a+\frac{13}{27}$, $\frac{1}{135}a^{19}+\frac{1}{27}a^{16}-\frac{2}{45}a^{15}-\frac{1}{27}a^{14}-\frac{2}{135}a^{13}-\frac{7}{135}a^{12}+\frac{1}{9}a^{11}-\frac{2}{45}a^{10}+\frac{1}{45}a^{9}+\frac{2}{15}a^{8}-\frac{11}{135}a^{7}+\frac{13}{45}a^{6}+\frac{1}{45}a^{5}-\frac{22}{135}a^{4}+\frac{7}{45}a^{3}-\frac{41}{135}a^{2}+\frac{5}{27}a-\frac{5}{27}$, $\frac{1}{135}a^{20}+\frac{4}{135}a^{16}+\frac{1}{45}a^{14}+\frac{1}{45}a^{13}-\frac{4}{27}a^{12}+\frac{1}{15}a^{11}+\frac{1}{45}a^{10}+\frac{1}{45}a^{9}+\frac{19}{135}a^{8}+\frac{1}{15}a^{7}-\frac{4}{45}a^{6}-\frac{4}{45}a^{5}+\frac{46}{135}a^{4}+\frac{2}{5}a^{3}+\frac{1}{9}a^{2}+\frac{1}{9}a-\frac{10}{27}$, $\frac{1}{1215}a^{21}-\frac{4}{1215}a^{20}-\frac{1}{405}a^{19}-\frac{1}{1215}a^{18}-\frac{2}{81}a^{16}+\frac{52}{1215}a^{15}-\frac{22}{1215}a^{14}-\frac{2}{405}a^{13}+\frac{107}{1215}a^{12}+\frac{8}{135}a^{11}-\frac{11}{81}a^{10}+\frac{29}{243}a^{9}-\frac{94}{1215}a^{8}+\frac{2}{81}a^{7}+\frac{56}{1215}a^{6}+\frac{47}{405}a^{5}-\frac{131}{405}a^{4}+\frac{116}{243}a^{3}-\frac{313}{1215}a^{2}-\frac{67}{135}a-\frac{44}{243}$, $\frac{1}{1215}a^{22}-\frac{1}{1215}a^{20}-\frac{4}{1215}a^{19}-\frac{4}{1215}a^{18}-\frac{1}{405}a^{17}-\frac{59}{1215}a^{16}+\frac{8}{405}a^{15}-\frac{58}{1215}a^{14}-\frac{16}{1215}a^{13}+\frac{77}{1215}a^{12}+\frac{32}{405}a^{11}+\frac{187}{1215}a^{10}-\frac{1}{15}a^{9}+\frac{158}{1215}a^{8}-\frac{17}{243}a^{7}+\frac{122}{1215}a^{6}-\frac{62}{135}a^{5}+\frac{16}{1215}a^{4}+\frac{43}{135}a^{3}-\frac{334}{1215}a^{2}-\frac{98}{243}a+\frac{85}{243}$, $\frac{1}{21130065}a^{23}-\frac{13}{128061}a^{22}+\frac{19}{112995}a^{21}-\frac{61}{414315}a^{20}+\frac{13988}{21130065}a^{19}-\frac{33641}{21130065}a^{18}+\frac{20182}{21130065}a^{17}-\frac{261223}{7043355}a^{16}+\frac{208807}{21130065}a^{15}+\frac{2572}{2347785}a^{14}+\frac{67712}{21130065}a^{13}-\frac{1348631}{21130065}a^{12}+\frac{738277}{21130065}a^{11}-\frac{83026}{782595}a^{10}+\frac{1531012}{21130065}a^{9}+\frac{148085}{1408671}a^{8}+\frac{265186}{4226013}a^{7}+\frac{6728329}{21130065}a^{6}+\frac{2724778}{21130065}a^{5}+\frac{279161}{640305}a^{4}+\frac{6831256}{21130065}a^{3}-\frac{205718}{1408671}a^{2}+\frac{4871678}{21130065}a-\frac{74219}{248589}$, $\frac{1}{49\!\cdots\!75}a^{24}+\frac{41\!\cdots\!33}{33\!\cdots\!05}a^{23}+\frac{11\!\cdots\!53}{45\!\cdots\!25}a^{22}-\frac{17\!\cdots\!58}{97\!\cdots\!25}a^{21}+\frac{10\!\cdots\!49}{49\!\cdots\!75}a^{20}-\frac{14\!\cdots\!34}{49\!\cdots\!75}a^{19}-\frac{59\!\cdots\!91}{99\!\cdots\!15}a^{18}+\frac{43\!\cdots\!97}{16\!\cdots\!25}a^{17}-\frac{37\!\cdots\!36}{49\!\cdots\!75}a^{16}+\frac{24\!\cdots\!37}{55\!\cdots\!75}a^{15}-\frac{69\!\cdots\!96}{49\!\cdots\!75}a^{14}+\frac{19\!\cdots\!73}{49\!\cdots\!75}a^{13}+\frac{50\!\cdots\!34}{49\!\cdots\!75}a^{12}-\frac{61\!\cdots\!52}{16\!\cdots\!25}a^{11}+\frac{14\!\cdots\!78}{99\!\cdots\!15}a^{10}+\frac{56\!\cdots\!47}{16\!\cdots\!25}a^{9}-\frac{51\!\cdots\!09}{49\!\cdots\!75}a^{8}+\frac{78\!\cdots\!48}{49\!\cdots\!75}a^{7}+\frac{79\!\cdots\!18}{49\!\cdots\!75}a^{6}-\frac{18\!\cdots\!62}{55\!\cdots\!75}a^{5}-\frac{21\!\cdots\!84}{49\!\cdots\!75}a^{4}-\frac{20\!\cdots\!18}{16\!\cdots\!25}a^{3}+\frac{63\!\cdots\!74}{16\!\cdots\!25}a^{2}-\frac{15\!\cdots\!42}{32\!\cdots\!65}a-\frac{14\!\cdots\!71}{38\!\cdots\!33}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$, $5$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $12$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{81\!\cdots\!89}{49\!\cdots\!75}a^{24}-\frac{12\!\cdots\!73}{19\!\cdots\!83}a^{23}+\frac{87\!\cdots\!42}{45\!\cdots\!25}a^{22}-\frac{13\!\cdots\!51}{29\!\cdots\!75}a^{21}+\frac{85\!\cdots\!06}{49\!\cdots\!75}a^{20}-\frac{19\!\cdots\!96}{49\!\cdots\!75}a^{19}+\frac{34\!\cdots\!12}{33\!\cdots\!05}a^{18}-\frac{56\!\cdots\!56}{49\!\cdots\!75}a^{17}-\frac{73\!\cdots\!59}{49\!\cdots\!75}a^{16}+\frac{55\!\cdots\!27}{49\!\cdots\!75}a^{15}-\frac{11\!\cdots\!89}{49\!\cdots\!75}a^{14}+\frac{12\!\cdots\!92}{49\!\cdots\!75}a^{13}-\frac{70\!\cdots\!76}{55\!\cdots\!75}a^{12}-\frac{21\!\cdots\!99}{49\!\cdots\!75}a^{11}+\frac{39\!\cdots\!38}{99\!\cdots\!15}a^{10}-\frac{11\!\cdots\!71}{49\!\cdots\!75}a^{9}+\frac{97\!\cdots\!69}{49\!\cdots\!75}a^{8}+\frac{35\!\cdots\!72}{49\!\cdots\!75}a^{7}-\frac{59\!\cdots\!56}{16\!\cdots\!25}a^{6}-\frac{51\!\cdots\!32}{49\!\cdots\!75}a^{5}+\frac{94\!\cdots\!07}{29\!\cdots\!75}a^{4}+\frac{38\!\cdots\!34}{49\!\cdots\!75}a^{3}+\frac{53\!\cdots\!41}{49\!\cdots\!75}a^{2}-\frac{11\!\cdots\!67}{99\!\cdots\!15}a-\frac{92\!\cdots\!48}{11\!\cdots\!99}$, $\frac{32\!\cdots\!73}{16\!\cdots\!25}a^{24}-\frac{38\!\cdots\!76}{33\!\cdots\!05}a^{23}+\frac{22\!\cdots\!78}{50\!\cdots\!25}a^{22}-\frac{46\!\cdots\!09}{32\!\cdots\!75}a^{21}+\frac{81\!\cdots\!52}{16\!\cdots\!25}a^{20}-\frac{24\!\cdots\!97}{16\!\cdots\!25}a^{19}+\frac{13\!\cdots\!07}{33\!\cdots\!05}a^{18}-\frac{15\!\cdots\!32}{16\!\cdots\!25}a^{17}+\frac{93\!\cdots\!14}{55\!\cdots\!75}a^{16}-\frac{12\!\cdots\!23}{61\!\cdots\!75}a^{15}+\frac{19\!\cdots\!92}{15\!\cdots\!75}a^{14}+\frac{55\!\cdots\!54}{16\!\cdots\!25}a^{13}-\frac{39\!\cdots\!13}{16\!\cdots\!25}a^{12}+\frac{48\!\cdots\!96}{88\!\cdots\!75}a^{11}+\frac{13\!\cdots\!24}{64\!\cdots\!55}a^{10}+\frac{28\!\cdots\!81}{50\!\cdots\!25}a^{9}+\frac{55\!\cdots\!48}{16\!\cdots\!25}a^{8}+\frac{30\!\cdots\!69}{16\!\cdots\!25}a^{7}-\frac{15\!\cdots\!46}{16\!\cdots\!25}a^{6}-\frac{75\!\cdots\!54}{16\!\cdots\!25}a^{5}+\frac{16\!\cdots\!64}{61\!\cdots\!75}a^{4}+\frac{18\!\cdots\!71}{55\!\cdots\!75}a^{3}-\frac{24\!\cdots\!93}{16\!\cdots\!25}a^{2}-\frac{82\!\cdots\!57}{66\!\cdots\!61}a-\frac{40\!\cdots\!00}{12\!\cdots\!11}$, $\frac{22\!\cdots\!66}{33\!\cdots\!05}a^{24}-\frac{37\!\cdots\!39}{99\!\cdots\!15}a^{23}+\frac{43\!\cdots\!93}{30\!\cdots\!55}a^{22}-\frac{26\!\cdots\!54}{58\!\cdots\!95}a^{21}+\frac{34\!\cdots\!13}{22\!\cdots\!87}a^{20}-\frac{14\!\cdots\!33}{32\!\cdots\!65}a^{19}+\frac{12\!\cdots\!64}{99\!\cdots\!15}a^{18}-\frac{29\!\cdots\!07}{99\!\cdots\!15}a^{17}+\frac{34\!\cdots\!21}{66\!\cdots\!61}a^{16}-\frac{10\!\cdots\!12}{18\!\cdots\!53}a^{15}+\frac{12\!\cdots\!94}{33\!\cdots\!05}a^{14}+\frac{11\!\cdots\!26}{99\!\cdots\!15}a^{13}-\frac{71\!\cdots\!09}{99\!\cdots\!15}a^{12}+\frac{55\!\cdots\!82}{99\!\cdots\!15}a^{11}+\frac{18\!\cdots\!68}{33\!\cdots\!05}a^{10}+\frac{32\!\cdots\!98}{99\!\cdots\!15}a^{9}+\frac{47\!\cdots\!87}{36\!\cdots\!45}a^{8}+\frac{86\!\cdots\!36}{90\!\cdots\!65}a^{7}-\frac{28\!\cdots\!06}{99\!\cdots\!15}a^{6}-\frac{20\!\cdots\!18}{99\!\cdots\!15}a^{5}+\frac{12\!\cdots\!56}{35\!\cdots\!85}a^{4}+\frac{11\!\cdots\!53}{99\!\cdots\!15}a^{3}+\frac{31\!\cdots\!41}{19\!\cdots\!65}a^{2}+\frac{25\!\cdots\!24}{99\!\cdots\!15}a-\frac{73\!\cdots\!79}{11\!\cdots\!99}$, $\frac{17\!\cdots\!29}{18\!\cdots\!75}a^{24}-\frac{26\!\cdots\!72}{43\!\cdots\!39}a^{23}+\frac{41\!\cdots\!32}{16\!\cdots\!25}a^{22}-\frac{88\!\cdots\!11}{10\!\cdots\!75}a^{21}+\frac{50\!\cdots\!71}{18\!\cdots\!75}a^{20}-\frac{51\!\cdots\!22}{60\!\cdots\!25}a^{19}+\frac{86\!\cdots\!53}{36\!\cdots\!15}a^{18}-\frac{10\!\cdots\!16}{18\!\cdots\!75}a^{17}+\frac{19\!\cdots\!56}{18\!\cdots\!75}a^{16}-\frac{78\!\cdots\!88}{53\!\cdots\!75}a^{15}+\frac{12\!\cdots\!43}{10\!\cdots\!75}a^{14}-\frac{97\!\cdots\!41}{60\!\cdots\!25}a^{13}-\frac{23\!\cdots\!79}{18\!\cdots\!75}a^{12}+\frac{18\!\cdots\!06}{18\!\cdots\!75}a^{11}+\frac{33\!\cdots\!84}{36\!\cdots\!15}a^{10}-\frac{95\!\cdots\!46}{18\!\cdots\!75}a^{9}+\frac{24\!\cdots\!74}{18\!\cdots\!75}a^{8}-\frac{28\!\cdots\!91}{55\!\cdots\!75}a^{7}-\frac{10\!\cdots\!83}{18\!\cdots\!75}a^{6}-\frac{55\!\cdots\!42}{18\!\cdots\!75}a^{5}+\frac{51\!\cdots\!74}{18\!\cdots\!75}a^{4}+\frac{33\!\cdots\!49}{18\!\cdots\!75}a^{3}-\frac{77\!\cdots\!59}{18\!\cdots\!75}a^{2}-\frac{89\!\cdots\!27}{40\!\cdots\!35}a-\frac{43\!\cdots\!88}{14\!\cdots\!13}$, $\frac{46\!\cdots\!81}{55\!\cdots\!75}a^{24}-\frac{48\!\cdots\!38}{99\!\cdots\!15}a^{23}+\frac{27\!\cdots\!64}{15\!\cdots\!75}a^{22}-\frac{16\!\cdots\!26}{29\!\cdots\!75}a^{21}+\frac{10\!\cdots\!99}{55\!\cdots\!75}a^{20}-\frac{27\!\cdots\!11}{49\!\cdots\!75}a^{19}+\frac{31\!\cdots\!14}{19\!\cdots\!83}a^{18}-\frac{17\!\cdots\!21}{49\!\cdots\!75}a^{17}+\frac{95\!\cdots\!02}{16\!\cdots\!25}a^{16}-\frac{27\!\cdots\!73}{49\!\cdots\!75}a^{15}+\frac{13\!\cdots\!32}{15\!\cdots\!75}a^{14}+\frac{38\!\cdots\!52}{49\!\cdots\!75}a^{13}-\frac{68\!\cdots\!34}{49\!\cdots\!75}a^{12}+\frac{70\!\cdots\!71}{45\!\cdots\!25}a^{11}+\frac{21\!\cdots\!73}{12\!\cdots\!15}a^{10}-\frac{17\!\cdots\!66}{45\!\cdots\!25}a^{9}+\frac{79\!\cdots\!36}{55\!\cdots\!75}a^{8}+\frac{48\!\cdots\!27}{49\!\cdots\!75}a^{7}-\frac{14\!\cdots\!64}{29\!\cdots\!75}a^{6}-\frac{11\!\cdots\!47}{49\!\cdots\!75}a^{5}+\frac{51\!\cdots\!98}{16\!\cdots\!25}a^{4}+\frac{85\!\cdots\!74}{49\!\cdots\!75}a^{3}-\frac{77\!\cdots\!68}{16\!\cdots\!25}a^{2}-\frac{35\!\cdots\!21}{19\!\cdots\!83}a-\frac{31\!\cdots\!16}{11\!\cdots\!99}$, $\frac{55\!\cdots\!87}{49\!\cdots\!75}a^{24}-\frac{51\!\cdots\!92}{99\!\cdots\!15}a^{23}+\frac{26\!\cdots\!26}{14\!\cdots\!75}a^{22}-\frac{15\!\cdots\!03}{29\!\cdots\!75}a^{21}+\frac{94\!\cdots\!03}{49\!\cdots\!75}a^{20}-\frac{25\!\cdots\!73}{49\!\cdots\!75}a^{19}+\frac{53\!\cdots\!99}{36\!\cdots\!45}a^{18}-\frac{14\!\cdots\!03}{49\!\cdots\!75}a^{17}+\frac{65\!\cdots\!73}{16\!\cdots\!25}a^{16}-\frac{11\!\cdots\!94}{49\!\cdots\!75}a^{15}-\frac{12\!\cdots\!42}{49\!\cdots\!75}a^{14}+\frac{31\!\cdots\!06}{49\!\cdots\!75}a^{13}-\frac{15\!\cdots\!54}{16\!\cdots\!25}a^{12}-\frac{50\!\cdots\!57}{49\!\cdots\!75}a^{11}+\frac{65\!\cdots\!71}{99\!\cdots\!15}a^{10}+\frac{78\!\cdots\!57}{49\!\cdots\!75}a^{9}+\frac{15\!\cdots\!07}{49\!\cdots\!75}a^{8}+\frac{62\!\cdots\!51}{16\!\cdots\!25}a^{7}-\frac{37\!\cdots\!88}{16\!\cdots\!25}a^{6}-\frac{38\!\cdots\!91}{49\!\cdots\!75}a^{5}-\frac{24\!\cdots\!48}{49\!\cdots\!75}a^{4}+\frac{23\!\cdots\!32}{49\!\cdots\!75}a^{3}+\frac{97\!\cdots\!38}{49\!\cdots\!75}a^{2}+\frac{87\!\cdots\!97}{90\!\cdots\!65}a+\frac{39\!\cdots\!82}{11\!\cdots\!99}$, $\frac{33\!\cdots\!09}{45\!\cdots\!25}a^{24}-\frac{19\!\cdots\!82}{66\!\cdots\!61}a^{23}+\frac{46\!\cdots\!77}{45\!\cdots\!25}a^{22}-\frac{24\!\cdots\!47}{88\!\cdots\!75}a^{21}+\frac{51\!\cdots\!71}{49\!\cdots\!75}a^{20}-\frac{13\!\cdots\!21}{49\!\cdots\!75}a^{19}+\frac{14\!\cdots\!58}{19\!\cdots\!83}a^{18}-\frac{21\!\cdots\!27}{16\!\cdots\!25}a^{17}+\frac{71\!\cdots\!91}{49\!\cdots\!75}a^{16}+\frac{34\!\cdots\!63}{55\!\cdots\!75}a^{15}-\frac{20\!\cdots\!89}{49\!\cdots\!75}a^{14}+\frac{30\!\cdots\!12}{49\!\cdots\!75}a^{13}-\frac{18\!\cdots\!17}{29\!\cdots\!75}a^{12}-\frac{56\!\cdots\!16}{55\!\cdots\!75}a^{11}+\frac{27\!\cdots\!56}{99\!\cdots\!15}a^{10}+\frac{16\!\cdots\!98}{16\!\cdots\!25}a^{9}+\frac{70\!\cdots\!84}{49\!\cdots\!75}a^{8}+\frac{16\!\cdots\!27}{49\!\cdots\!75}a^{7}+\frac{24\!\cdots\!62}{49\!\cdots\!75}a^{6}-\frac{40\!\cdots\!64}{16\!\cdots\!25}a^{5}-\frac{40\!\cdots\!91}{45\!\cdots\!25}a^{4}+\frac{40\!\cdots\!58}{16\!\cdots\!25}a^{3}+\frac{46\!\cdots\!66}{49\!\cdots\!75}a^{2}+\frac{98\!\cdots\!27}{99\!\cdots\!15}a+\frac{18\!\cdots\!97}{48\!\cdots\!93}$, $\frac{12\!\cdots\!34}{49\!\cdots\!75}a^{24}-\frac{48\!\cdots\!86}{33\!\cdots\!05}a^{23}+\frac{26\!\cdots\!82}{45\!\cdots\!25}a^{22}-\frac{17\!\cdots\!42}{97\!\cdots\!25}a^{21}+\frac{30\!\cdots\!86}{49\!\cdots\!75}a^{20}-\frac{53\!\cdots\!38}{29\!\cdots\!75}a^{19}+\frac{46\!\cdots\!03}{90\!\cdots\!65}a^{18}-\frac{20\!\cdots\!87}{16\!\cdots\!25}a^{17}+\frac{10\!\cdots\!46}{49\!\cdots\!75}a^{16}-\frac{43\!\cdots\!76}{16\!\cdots\!25}a^{15}+\frac{79\!\cdots\!51}{49\!\cdots\!75}a^{14}+\frac{30\!\cdots\!47}{49\!\cdots\!75}a^{13}-\frac{16\!\cdots\!79}{49\!\cdots\!75}a^{12}+\frac{22\!\cdots\!92}{16\!\cdots\!25}a^{11}+\frac{20\!\cdots\!94}{90\!\cdots\!65}a^{10}+\frac{13\!\cdots\!83}{16\!\cdots\!25}a^{9}+\frac{16\!\cdots\!24}{45\!\cdots\!25}a^{8}+\frac{96\!\cdots\!57}{49\!\cdots\!75}a^{7}-\frac{60\!\cdots\!83}{49\!\cdots\!75}a^{6}-\frac{89\!\cdots\!16}{18\!\cdots\!25}a^{5}+\frac{19\!\cdots\!14}{49\!\cdots\!75}a^{4}+\frac{25\!\cdots\!86}{55\!\cdots\!75}a^{3}-\frac{51\!\cdots\!84}{16\!\cdots\!25}a^{2}-\frac{80\!\cdots\!99}{32\!\cdots\!65}a-\frac{68\!\cdots\!67}{12\!\cdots\!11}$, $\frac{10\!\cdots\!94}{49\!\cdots\!75}a^{24}+\frac{10\!\cdots\!12}{99\!\cdots\!15}a^{23}-\frac{44\!\cdots\!28}{45\!\cdots\!25}a^{22}+\frac{13\!\cdots\!79}{29\!\cdots\!75}a^{21}-\frac{70\!\cdots\!24}{49\!\cdots\!75}a^{20}+\frac{24\!\cdots\!69}{49\!\cdots\!75}a^{19}-\frac{50\!\cdots\!07}{33\!\cdots\!05}a^{18}+\frac{22\!\cdots\!94}{49\!\cdots\!75}a^{17}-\frac{56\!\cdots\!49}{49\!\cdots\!75}a^{16}+\frac{10\!\cdots\!32}{49\!\cdots\!75}a^{15}-\frac{14\!\cdots\!44}{49\!\cdots\!75}a^{14}+\frac{94\!\cdots\!57}{49\!\cdots\!75}a^{13}+\frac{18\!\cdots\!43}{18\!\cdots\!25}a^{12}-\frac{25\!\cdots\!09}{49\!\cdots\!75}a^{11}+\frac{45\!\cdots\!07}{99\!\cdots\!15}a^{10}+\frac{12\!\cdots\!49}{16\!\cdots\!25}a^{9}-\frac{10\!\cdots\!46}{16\!\cdots\!25}a^{8}+\frac{25\!\cdots\!57}{49\!\cdots\!75}a^{7}-\frac{32\!\cdots\!36}{16\!\cdots\!25}a^{6}-\frac{60\!\cdots\!32}{49\!\cdots\!75}a^{5}+\frac{61\!\cdots\!59}{49\!\cdots\!75}a^{4}+\frac{31\!\cdots\!79}{49\!\cdots\!75}a^{3}+\frac{69\!\cdots\!71}{49\!\cdots\!75}a^{2}+\frac{20\!\cdots\!76}{18\!\cdots\!53}a-\frac{89\!\cdots\!05}{11\!\cdots\!99}$, $\frac{40\!\cdots\!77}{90\!\cdots\!65}a^{24}-\frac{24\!\cdots\!47}{99\!\cdots\!15}a^{23}+\frac{86\!\cdots\!83}{90\!\cdots\!65}a^{22}-\frac{15\!\cdots\!28}{53\!\cdots\!45}a^{21}+\frac{99\!\cdots\!88}{99\!\cdots\!15}a^{20}-\frac{96\!\cdots\!03}{33\!\cdots\!05}a^{19}+\frac{80\!\cdots\!31}{99\!\cdots\!15}a^{18}-\frac{36\!\cdots\!46}{19\!\cdots\!83}a^{17}+\frac{30\!\cdots\!17}{99\!\cdots\!15}a^{16}-\frac{30\!\cdots\!41}{99\!\cdots\!15}a^{15}+\frac{55\!\cdots\!76}{99\!\cdots\!15}a^{14}+\frac{65\!\cdots\!94}{19\!\cdots\!65}a^{13}-\frac{23\!\cdots\!37}{32\!\cdots\!65}a^{12}+\frac{14\!\cdots\!32}{99\!\cdots\!15}a^{11}+\frac{54\!\cdots\!92}{99\!\cdots\!15}a^{10}-\frac{11\!\cdots\!44}{99\!\cdots\!15}a^{9}+\frac{14\!\cdots\!92}{99\!\cdots\!15}a^{8}+\frac{78\!\cdots\!13}{33\!\cdots\!05}a^{7}-\frac{17\!\cdots\!91}{99\!\cdots\!15}a^{6}-\frac{18\!\cdots\!69}{11\!\cdots\!99}a^{5}-\frac{41\!\cdots\!37}{90\!\cdots\!65}a^{4}+\frac{81\!\cdots\!86}{99\!\cdots\!15}a^{3}+\frac{24\!\cdots\!34}{99\!\cdots\!15}a^{2}+\frac{46\!\cdots\!57}{33\!\cdots\!05}a-\frac{34\!\cdots\!66}{11\!\cdots\!99}$, $\frac{65\!\cdots\!08}{55\!\cdots\!75}a^{24}-\frac{22\!\cdots\!69}{32\!\cdots\!65}a^{23}+\frac{41\!\cdots\!42}{15\!\cdots\!75}a^{22}-\frac{26\!\cdots\!48}{29\!\cdots\!75}a^{21}+\frac{49\!\cdots\!41}{16\!\cdots\!25}a^{20}-\frac{44\!\cdots\!78}{49\!\cdots\!75}a^{19}+\frac{22\!\cdots\!07}{90\!\cdots\!65}a^{18}-\frac{29\!\cdots\!13}{49\!\cdots\!75}a^{17}+\frac{17\!\cdots\!96}{16\!\cdots\!25}a^{16}-\frac{65\!\cdots\!74}{49\!\cdots\!75}a^{15}+\frac{49\!\cdots\!62}{55\!\cdots\!75}a^{14}+\frac{44\!\cdots\!81}{49\!\cdots\!75}a^{13}-\frac{67\!\cdots\!42}{49\!\cdots\!75}a^{12}+\frac{20\!\cdots\!33}{49\!\cdots\!75}a^{11}+\frac{36\!\cdots\!64}{30\!\cdots\!55}a^{10}+\frac{19\!\cdots\!87}{49\!\cdots\!75}a^{9}+\frac{26\!\cdots\!24}{15\!\cdots\!75}a^{8}+\frac{34\!\cdots\!41}{49\!\cdots\!75}a^{7}-\frac{16\!\cdots\!27}{29\!\cdots\!75}a^{6}-\frac{12\!\cdots\!16}{49\!\cdots\!75}a^{5}+\frac{11\!\cdots\!23}{55\!\cdots\!75}a^{4}+\frac{12\!\cdots\!42}{49\!\cdots\!75}a^{3}-\frac{33\!\cdots\!79}{16\!\cdots\!25}a^{2}-\frac{60\!\cdots\!15}{19\!\cdots\!83}a-\frac{24\!\cdots\!64}{11\!\cdots\!99}$, $\frac{14\!\cdots\!92}{18\!\cdots\!25}a^{24}-\frac{16\!\cdots\!98}{33\!\cdots\!05}a^{23}+\frac{31\!\cdots\!59}{15\!\cdots\!75}a^{22}-\frac{68\!\cdots\!47}{97\!\cdots\!25}a^{21}+\frac{39\!\cdots\!57}{16\!\cdots\!25}a^{20}-\frac{39\!\cdots\!29}{55\!\cdots\!75}a^{19}+\frac{67\!\cdots\!39}{33\!\cdots\!05}a^{18}-\frac{82\!\cdots\!02}{16\!\cdots\!25}a^{17}+\frac{15\!\cdots\!07}{16\!\cdots\!25}a^{16}-\frac{22\!\cdots\!41}{16\!\cdots\!25}a^{15}+\frac{20\!\cdots\!57}{16\!\cdots\!25}a^{14}-\frac{89\!\cdots\!59}{18\!\cdots\!25}a^{13}-\frac{13\!\cdots\!23}{16\!\cdots\!25}a^{12}+\frac{13\!\cdots\!52}{16\!\cdots\!25}a^{11}+\frac{63\!\cdots\!97}{33\!\cdots\!05}a^{10}+\frac{70\!\cdots\!89}{97\!\cdots\!25}a^{9}+\frac{11\!\cdots\!14}{97\!\cdots\!25}a^{8}+\frac{17\!\cdots\!99}{32\!\cdots\!75}a^{7}-\frac{54\!\cdots\!66}{16\!\cdots\!25}a^{6}+\frac{77\!\cdots\!91}{16\!\cdots\!25}a^{5}+\frac{19\!\cdots\!08}{16\!\cdots\!25}a^{4}+\frac{87\!\cdots\!23}{16\!\cdots\!25}a^{3}-\frac{13\!\cdots\!48}{16\!\cdots\!25}a^{2}-\frac{27\!\cdots\!86}{10\!\cdots\!85}a-\frac{10\!\cdots\!06}{38\!\cdots\!33}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 305458559273.9288 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 305458559273.9288 \cdot 1}{2\cdot\sqrt{335244303153752999188341104839710238574881}}\cr\approx \mathstrut & 1.99723889942240 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^25 - 5*x^24 + 18*x^23 - 53*x^22 + 184*x^21 - 519*x^20 + 1425*x^19 - 2994*x^18 + 4254*x^17 - 2567*x^16 - 3146*x^15 + 8388*x^14 - 11301*x^13 - 6966*x^12 + 13530*x^11 + 11956*x^10 + 18271*x^9 + 21048*x^8 - 41862*x^7 - 65478*x^6 - 1959*x^5 + 35311*x^4 + 17869*x^3 - 2655*x^2 - 3850*x - 2125)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^25 - 5*x^24 + 18*x^23 - 53*x^22 + 184*x^21 - 519*x^20 + 1425*x^19 - 2994*x^18 + 4254*x^17 - 2567*x^16 - 3146*x^15 + 8388*x^14 - 11301*x^13 - 6966*x^12 + 13530*x^11 + 11956*x^10 + 18271*x^9 + 21048*x^8 - 41862*x^7 - 65478*x^6 - 1959*x^5 + 35311*x^4 + 17869*x^3 - 2655*x^2 - 3850*x - 2125, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^25 - 5*x^24 + 18*x^23 - 53*x^22 + 184*x^21 - 519*x^20 + 1425*x^19 - 2994*x^18 + 4254*x^17 - 2567*x^16 - 3146*x^15 + 8388*x^14 - 11301*x^13 - 6966*x^12 + 13530*x^11 + 11956*x^10 + 18271*x^9 + 21048*x^8 - 41862*x^7 - 65478*x^6 - 1959*x^5 + 35311*x^4 + 17869*x^3 - 2655*x^2 - 3850*x - 2125);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 5*x^24 + 18*x^23 - 53*x^22 + 184*x^21 - 519*x^20 + 1425*x^19 - 2994*x^18 + 4254*x^17 - 2567*x^16 - 3146*x^15 + 8388*x^14 - 11301*x^13 - 6966*x^12 + 13530*x^11 + 11956*x^10 + 18271*x^9 + 21048*x^8 - 41862*x^7 - 65478*x^6 - 1959*x^5 + 35311*x^4 + 17869*x^3 - 2655*x^2 - 3850*x - 2125);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

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

Intermediate fields

5.1.8334769.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $25$ ${\href{/padicField/3.2.0.1}{2} }^{12}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.2.0.1}{2} }^{12}{,}\,{\href{/padicField/5.1.0.1}{1} }$ $25$ ${\href{/padicField/11.2.0.1}{2} }^{12}{,}\,{\href{/padicField/11.1.0.1}{1} }$ $25$ ${\href{/padicField/17.2.0.1}{2} }^{12}{,}\,{\href{/padicField/17.1.0.1}{1} }$ $25$ ${\href{/padicField/23.2.0.1}{2} }^{12}{,}\,{\href{/padicField/23.1.0.1}{1} }$ $25$ ${\href{/padicField/31.2.0.1}{2} }^{12}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.5.0.1}{5} }^{5}$ ${\href{/padicField/41.2.0.1}{2} }^{12}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.2.0.1}{2} }^{12}{,}\,{\href{/padicField/43.1.0.1}{1} }$ $25$ ${\href{/padicField/53.5.0.1}{5} }^{5}$ $25$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2887\) Copy content Toggle raw display $\Q_{2887}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2887.2t1.a.a$1$ $ 2887 $ \(\Q(\sqrt{-2887}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.2887.5t2.a.a$2$ $ 2887 $ 5.1.8334769.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.2887.5t2.a.b$2$ $ 2887 $ 5.1.8334769.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.2887.25t4.a.f$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.2887.25t4.a.d$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.2887.25t4.a.h$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.2887.25t4.a.c$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.2887.25t4.a.i$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.2887.25t4.a.g$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.2887.25t4.a.a$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.2887.25t4.a.j$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.2887.25t4.a.e$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (as 25T4) $1$ $0$
* 2.2887.25t4.a.b$2$ $ 2887 $ 25.1.335244303153752999188341104839710238574881.1 $D_{25}$ (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 *.