Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 3*x^24 + 4*x^23 - 7*x^22 + 5*x^21 - 6*x^20 + 28*x^19 - 37*x^18 + 24*x^17 + 60*x^15 - 133*x^14 + 74*x^13 + 37*x^12 - 24*x^11 - 112*x^10 + 155*x^9 - 56*x^8 - 45*x^7 + 51*x^6 - 17*x^5 + 3*x^4 - 3*x^3 + 3*x - 1)
gp: K = bnfinit(y^25 - 3*y^24 + 4*y^23 - 7*y^22 + 5*y^21 - 6*y^20 + 28*y^19 - 37*y^18 + 24*y^17 + 60*y^15 - 133*y^14 + 74*y^13 + 37*y^12 - 24*y^11 - 112*y^10 + 155*y^9 - 56*y^8 - 45*y^7 + 51*y^6 - 17*y^5 + 3*y^4 - 3*y^3 + 3*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 - 3*x^24 + 4*x^23 - 7*x^22 + 5*x^21 - 6*x^20 + 28*x^19 - 37*x^18 + 24*x^17 + 60*x^15 - 133*x^14 + 74*x^13 + 37*x^12 - 24*x^11 - 112*x^10 + 155*x^9 - 56*x^8 - 45*x^7 + 51*x^6 - 17*x^5 + 3*x^4 - 3*x^3 + 3*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 3*x^24 + 4*x^23 - 7*x^22 + 5*x^21 - 6*x^20 + 28*x^19 - 37*x^18 + 24*x^17 + 60*x^15 - 133*x^14 + 74*x^13 + 37*x^12 - 24*x^11 - 112*x^10 + 155*x^9 - 56*x^8 - 45*x^7 + 51*x^6 - 17*x^5 + 3*x^4 - 3*x^3 + 3*x - 1)
\( x^{25} - 3 x^{24} + 4 x^{23} - 7 x^{22} + 5 x^{21} - 6 x^{20} + 28 x^{19} - 37 x^{18} + 24 x^{17} + \cdots - 1 \)
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: | $[5, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(722359806654877741268938522624\)
\(\medspace = 2^{30}\cdot 11^{20}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.64\) | 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: | $2^{3/2}11^{4/5}\approx 19.260126783385598$ | ||
| Ramified primes: |
\(2\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $5$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{19}a^{20}+\frac{6}{19}a^{19}-\frac{9}{19}a^{18}-\frac{2}{19}a^{17}+\frac{5}{19}a^{16}+\frac{5}{19}a^{15}+\frac{9}{19}a^{13}+\frac{2}{19}a^{12}+\frac{3}{19}a^{11}+\frac{2}{19}a^{10}-\frac{6}{19}a^{9}+\frac{8}{19}a^{8}-\frac{1}{19}a^{7}+\frac{4}{19}a^{6}-\frac{5}{19}a^{5}-\frac{5}{19}a^{4}+\frac{8}{19}a^{3}+\frac{7}{19}a^{2}-\frac{8}{19}a+\frac{3}{19}$, $\frac{1}{19}a^{21}-\frac{7}{19}a^{19}-\frac{5}{19}a^{18}-\frac{2}{19}a^{17}-\frac{6}{19}a^{16}+\frac{8}{19}a^{15}+\frac{9}{19}a^{14}+\frac{5}{19}a^{13}-\frac{9}{19}a^{12}+\frac{3}{19}a^{11}+\frac{1}{19}a^{10}+\frac{6}{19}a^{9}+\frac{8}{19}a^{8}-\frac{9}{19}a^{7}+\frac{9}{19}a^{6}+\frac{6}{19}a^{5}-\frac{3}{19}a^{3}+\frac{7}{19}a^{2}-\frac{6}{19}a+\frac{1}{19}$, $\frac{1}{19}a^{22}-\frac{1}{19}a^{19}-\frac{8}{19}a^{18}-\frac{1}{19}a^{17}+\frac{5}{19}a^{16}+\frac{6}{19}a^{15}+\frac{5}{19}a^{14}-\frac{3}{19}a^{13}-\frac{2}{19}a^{12}+\frac{3}{19}a^{11}+\frac{1}{19}a^{10}+\frac{4}{19}a^{9}+\frac{9}{19}a^{8}+\frac{2}{19}a^{7}-\frac{4}{19}a^{6}+\frac{3}{19}a^{5}+\frac{6}{19}a^{3}+\frac{5}{19}a^{2}+\frac{2}{19}a+\frac{2}{19}$, $\frac{1}{19}a^{23}-\frac{2}{19}a^{19}+\frac{9}{19}a^{18}+\frac{3}{19}a^{17}-\frac{8}{19}a^{16}-\frac{9}{19}a^{15}-\frac{3}{19}a^{14}+\frac{7}{19}a^{13}+\frac{5}{19}a^{12}+\frac{4}{19}a^{11}+\frac{6}{19}a^{10}+\frac{3}{19}a^{9}-\frac{9}{19}a^{8}-\frac{5}{19}a^{7}+\frac{7}{19}a^{6}-\frac{5}{19}a^{5}+\frac{1}{19}a^{4}-\frac{6}{19}a^{3}+\frac{9}{19}a^{2}-\frac{6}{19}a+\frac{3}{19}$, $\frac{1}{52\!\cdots\!73}a^{24}-\frac{11\!\cdots\!48}{52\!\cdots\!73}a^{23}+\frac{438509416742639}{52\!\cdots\!73}a^{22}+\frac{13\!\cdots\!82}{52\!\cdots\!73}a^{21}-\frac{826988159815181}{52\!\cdots\!73}a^{20}-\frac{58\!\cdots\!38}{52\!\cdots\!73}a^{19}-\frac{17\!\cdots\!95}{52\!\cdots\!73}a^{18}+\frac{65\!\cdots\!55}{52\!\cdots\!73}a^{17}+\frac{259134087403138}{22\!\cdots\!51}a^{16}-\frac{608962458370800}{22\!\cdots\!51}a^{15}+\frac{18\!\cdots\!15}{52\!\cdots\!73}a^{14}+\frac{12\!\cdots\!91}{52\!\cdots\!73}a^{13}-\frac{10\!\cdots\!62}{52\!\cdots\!73}a^{12}+\frac{11\!\cdots\!67}{22\!\cdots\!51}a^{11}+\frac{20\!\cdots\!03}{52\!\cdots\!73}a^{10}-\frac{22\!\cdots\!24}{52\!\cdots\!73}a^{9}+\frac{43\!\cdots\!97}{52\!\cdots\!73}a^{8}+\frac{37962263285211}{22\!\cdots\!51}a^{7}-\frac{43407010630088}{52\!\cdots\!73}a^{6}+\frac{16\!\cdots\!20}{52\!\cdots\!73}a^{5}+\frac{11\!\cdots\!12}{52\!\cdots\!73}a^{4}-\frac{21\!\cdots\!04}{52\!\cdots\!73}a^{3}-\frac{12\!\cdots\!96}{52\!\cdots\!73}a^{2}+\frac{23\!\cdots\!54}{52\!\cdots\!73}a+\frac{10\!\cdots\!83}{52\!\cdots\!73}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
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: | $14$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{20\!\cdots\!37}{52\!\cdots\!73}a^{24}-\frac{73\!\cdots\!49}{52\!\cdots\!73}a^{23}+\frac{11\!\cdots\!67}{52\!\cdots\!73}a^{22}-\frac{19\!\cdots\!32}{52\!\cdots\!73}a^{21}+\frac{19\!\cdots\!52}{52\!\cdots\!73}a^{20}-\frac{20\!\cdots\!84}{52\!\cdots\!73}a^{19}+\frac{68\!\cdots\!92}{52\!\cdots\!73}a^{18}-\frac{59\!\cdots\!12}{27\!\cdots\!67}a^{17}+\frac{41\!\cdots\!80}{22\!\cdots\!51}a^{16}-\frac{15\!\cdots\!92}{22\!\cdots\!51}a^{15}+\frac{15\!\cdots\!09}{52\!\cdots\!73}a^{14}-\frac{36\!\cdots\!86}{52\!\cdots\!73}a^{13}+\frac{31\!\cdots\!54}{52\!\cdots\!73}a^{12}-\frac{513266406696436}{120409286217329}a^{11}-\frac{16\!\cdots\!48}{52\!\cdots\!73}a^{10}-\frac{29\!\cdots\!10}{52\!\cdots\!73}a^{9}+\frac{46\!\cdots\!27}{52\!\cdots\!73}a^{8}-\frac{11\!\cdots\!16}{22\!\cdots\!51}a^{7}+\frac{71\!\cdots\!99}{27\!\cdots\!67}a^{6}+\frac{50\!\cdots\!63}{52\!\cdots\!73}a^{5}-\frac{91\!\cdots\!83}{52\!\cdots\!73}a^{4}+\frac{98\!\cdots\!38}{52\!\cdots\!73}a^{3}-\frac{80\!\cdots\!03}{27\!\cdots\!67}a^{2}+\frac{88\!\cdots\!76}{52\!\cdots\!73}a+\frac{12\!\cdots\!60}{52\!\cdots\!73}$, $\frac{15\!\cdots\!50}{52\!\cdots\!73}a^{24}-\frac{13\!\cdots\!41}{52\!\cdots\!73}a^{23}-\frac{39\!\cdots\!05}{52\!\cdots\!73}a^{22}+\frac{35\!\cdots\!37}{52\!\cdots\!73}a^{21}-\frac{18\!\cdots\!06}{52\!\cdots\!73}a^{20}+\frac{12\!\cdots\!95}{52\!\cdots\!73}a^{19}+\frac{16\!\cdots\!02}{52\!\cdots\!73}a^{18}+\frac{42\!\cdots\!39}{52\!\cdots\!73}a^{17}-\frac{42\!\cdots\!69}{22\!\cdots\!51}a^{16}+\frac{46\!\cdots\!73}{22\!\cdots\!51}a^{15}+\frac{50\!\cdots\!63}{52\!\cdots\!73}a^{14}+\frac{21\!\cdots\!87}{52\!\cdots\!73}a^{13}-\frac{35\!\cdots\!93}{52\!\cdots\!73}a^{12}+\frac{16\!\cdots\!62}{22\!\cdots\!51}a^{11}-\frac{42\!\cdots\!13}{52\!\cdots\!73}a^{10}-\frac{15\!\cdots\!04}{52\!\cdots\!73}a^{9}-\frac{15\!\cdots\!49}{52\!\cdots\!73}a^{8}+\frac{19\!\cdots\!06}{22\!\cdots\!51}a^{7}-\frac{36\!\cdots\!15}{52\!\cdots\!73}a^{6}+\frac{93\!\cdots\!05}{52\!\cdots\!73}a^{5}+\frac{50\!\cdots\!44}{52\!\cdots\!73}a^{4}-\frac{36\!\cdots\!54}{52\!\cdots\!73}a^{3}+\frac{20\!\cdots\!73}{52\!\cdots\!73}a^{2}-\frac{21\!\cdots\!08}{52\!\cdots\!73}a+\frac{11\!\cdots\!54}{52\!\cdots\!73}$, $\frac{93\!\cdots\!98}{52\!\cdots\!73}a^{24}-\frac{28\!\cdots\!37}{52\!\cdots\!73}a^{23}+\frac{40\!\cdots\!76}{52\!\cdots\!73}a^{22}-\frac{73\!\cdots\!32}{52\!\cdots\!73}a^{21}+\frac{60\!\cdots\!34}{52\!\cdots\!73}a^{20}-\frac{76\!\cdots\!27}{52\!\cdots\!73}a^{19}+\frac{28\!\cdots\!11}{52\!\cdots\!73}a^{18}-\frac{37\!\cdots\!93}{52\!\cdots\!73}a^{17}+\frac{13\!\cdots\!60}{22\!\cdots\!51}a^{16}-\frac{54\!\cdots\!92}{22\!\cdots\!51}a^{15}+\frac{34\!\cdots\!69}{27\!\cdots\!67}a^{14}-\frac{13\!\cdots\!65}{52\!\cdots\!73}a^{13}+\frac{88\!\cdots\!60}{52\!\cdots\!73}a^{12}-\frac{18\!\cdots\!89}{22\!\cdots\!51}a^{11}+\frac{98\!\cdots\!84}{52\!\cdots\!73}a^{10}-\frac{10\!\cdots\!96}{52\!\cdots\!73}a^{9}+\frac{14\!\cdots\!82}{52\!\cdots\!73}a^{8}-\frac{36\!\cdots\!87}{22\!\cdots\!51}a^{7}+\frac{53\!\cdots\!47}{52\!\cdots\!73}a^{6}+\frac{17\!\cdots\!95}{52\!\cdots\!73}a^{5}-\frac{11\!\cdots\!88}{52\!\cdots\!73}a^{4}+\frac{73\!\cdots\!84}{52\!\cdots\!73}a^{3}-\frac{59\!\cdots\!30}{52\!\cdots\!73}a^{2}+\frac{21\!\cdots\!32}{52\!\cdots\!73}a+\frac{80\!\cdots\!30}{52\!\cdots\!73}$, $\frac{33\!\cdots\!32}{52\!\cdots\!73}a^{24}-\frac{12\!\cdots\!21}{52\!\cdots\!73}a^{23}+\frac{19\!\cdots\!05}{52\!\cdots\!73}a^{22}-\frac{32\!\cdots\!91}{52\!\cdots\!73}a^{21}+\frac{35\!\cdots\!18}{52\!\cdots\!73}a^{20}-\frac{32\!\cdots\!34}{52\!\cdots\!73}a^{19}+\frac{11\!\cdots\!32}{52\!\cdots\!73}a^{18}-\frac{19\!\cdots\!02}{52\!\cdots\!73}a^{17}+\frac{68\!\cdots\!14}{22\!\cdots\!51}a^{16}-\frac{32\!\cdots\!00}{22\!\cdots\!51}a^{15}+\frac{23\!\cdots\!54}{52\!\cdots\!73}a^{14}-\frac{62\!\cdots\!22}{52\!\cdots\!73}a^{13}+\frac{52\!\cdots\!38}{52\!\cdots\!73}a^{12}-\frac{24\!\cdots\!36}{22\!\cdots\!51}a^{11}-\frac{75\!\cdots\!62}{27\!\cdots\!67}a^{10}-\frac{46\!\cdots\!56}{52\!\cdots\!73}a^{9}+\frac{77\!\cdots\!96}{52\!\cdots\!73}a^{8}-\frac{11\!\cdots\!10}{120409286217329}a^{7}+\frac{68\!\cdots\!14}{52\!\cdots\!73}a^{6}+\frac{98\!\cdots\!74}{52\!\cdots\!73}a^{5}-\frac{29\!\cdots\!39}{27\!\cdots\!67}a^{4}+\frac{27\!\cdots\!55}{52\!\cdots\!73}a^{3}-\frac{22\!\cdots\!58}{52\!\cdots\!73}a^{2}+\frac{13\!\cdots\!40}{52\!\cdots\!73}a+\frac{10\!\cdots\!89}{52\!\cdots\!73}$, $\frac{12\!\cdots\!11}{52\!\cdots\!73}a^{24}-\frac{29\!\cdots\!24}{52\!\cdots\!73}a^{23}+\frac{10\!\cdots\!15}{52\!\cdots\!73}a^{22}-\frac{21\!\cdots\!08}{52\!\cdots\!73}a^{21}-\frac{30\!\cdots\!64}{52\!\cdots\!73}a^{20}+\frac{42\!\cdots\!20}{52\!\cdots\!73}a^{19}+\frac{27\!\cdots\!54}{52\!\cdots\!73}a^{18}-\frac{14\!\cdots\!49}{52\!\cdots\!73}a^{17}-\frac{16\!\cdots\!08}{22\!\cdots\!51}a^{16}+\frac{21\!\cdots\!53}{22\!\cdots\!51}a^{15}+\frac{64\!\cdots\!27}{52\!\cdots\!73}a^{14}-\frac{11\!\cdots\!45}{52\!\cdots\!73}a^{13}-\frac{12\!\cdots\!19}{52\!\cdots\!73}a^{12}+\frac{10\!\cdots\!05}{22\!\cdots\!51}a^{11}-\frac{66\!\cdots\!51}{52\!\cdots\!73}a^{10}-\frac{20\!\cdots\!48}{52\!\cdots\!73}a^{9}+\frac{47\!\cdots\!41}{52\!\cdots\!73}a^{8}+\frac{95\!\cdots\!10}{22\!\cdots\!51}a^{7}-\frac{19\!\cdots\!59}{52\!\cdots\!73}a^{6}+\frac{16\!\cdots\!08}{52\!\cdots\!73}a^{5}+\frac{52\!\cdots\!63}{52\!\cdots\!73}a^{4}-\frac{95\!\cdots\!27}{52\!\cdots\!73}a^{3}-\frac{18\!\cdots\!16}{52\!\cdots\!73}a^{2}-\frac{88\!\cdots\!63}{52\!\cdots\!73}a+\frac{78\!\cdots\!93}{52\!\cdots\!73}$, $\frac{51\!\cdots\!48}{52\!\cdots\!73}a^{24}-\frac{16\!\cdots\!72}{52\!\cdots\!73}a^{23}+\frac{24\!\cdots\!87}{52\!\cdots\!73}a^{22}-\frac{44\!\cdots\!25}{52\!\cdots\!73}a^{21}+\frac{38\!\cdots\!80}{52\!\cdots\!73}a^{20}-\frac{50\!\cdots\!72}{52\!\cdots\!73}a^{19}+\frac{16\!\cdots\!86}{52\!\cdots\!73}a^{18}-\frac{22\!\cdots\!14}{52\!\cdots\!73}a^{17}+\frac{91\!\cdots\!81}{22\!\cdots\!51}a^{16}-\frac{38\!\cdots\!97}{22\!\cdots\!51}a^{15}+\frac{38\!\cdots\!90}{52\!\cdots\!73}a^{14}-\frac{75\!\cdots\!69}{52\!\cdots\!73}a^{13}+\frac{62\!\cdots\!40}{52\!\cdots\!73}a^{12}-\frac{48\!\cdots\!27}{22\!\cdots\!51}a^{11}+\frac{40\!\cdots\!96}{52\!\cdots\!73}a^{10}-\frac{60\!\cdots\!03}{52\!\cdots\!73}a^{9}+\frac{91\!\cdots\!98}{52\!\cdots\!73}a^{8}-\frac{26\!\cdots\!88}{22\!\cdots\!51}a^{7}+\frac{93\!\cdots\!83}{52\!\cdots\!73}a^{6}+\frac{50\!\cdots\!79}{27\!\cdots\!67}a^{5}-\frac{82\!\cdots\!89}{52\!\cdots\!73}a^{4}+\frac{50\!\cdots\!06}{52\!\cdots\!73}a^{3}-\frac{47\!\cdots\!03}{52\!\cdots\!73}a^{2}+\frac{19\!\cdots\!45}{52\!\cdots\!73}a+\frac{16\!\cdots\!49}{52\!\cdots\!73}$, $\frac{33\!\cdots\!13}{52\!\cdots\!73}a^{24}-\frac{93\!\cdots\!86}{52\!\cdots\!73}a^{23}+\frac{11\!\cdots\!90}{52\!\cdots\!73}a^{22}-\frac{22\!\cdots\!37}{52\!\cdots\!73}a^{21}+\frac{15\!\cdots\!81}{52\!\cdots\!73}a^{20}-\frac{21\!\cdots\!58}{52\!\cdots\!73}a^{19}+\frac{96\!\cdots\!49}{52\!\cdots\!73}a^{18}-\frac{11\!\cdots\!39}{52\!\cdots\!73}a^{17}+\frac{32\!\cdots\!74}{22\!\cdots\!51}a^{16}-\frac{83\!\cdots\!21}{22\!\cdots\!51}a^{15}+\frac{22\!\cdots\!86}{52\!\cdots\!73}a^{14}-\frac{41\!\cdots\!63}{52\!\cdots\!73}a^{13}+\frac{18\!\cdots\!12}{52\!\cdots\!73}a^{12}+\frac{26\!\cdots\!20}{22\!\cdots\!51}a^{11}+\frac{40\!\cdots\!87}{52\!\cdots\!73}a^{10}-\frac{38\!\cdots\!61}{52\!\cdots\!73}a^{9}+\frac{42\!\cdots\!89}{52\!\cdots\!73}a^{8}-\frac{67\!\cdots\!77}{22\!\cdots\!51}a^{7}-\frac{33\!\cdots\!51}{52\!\cdots\!73}a^{6}+\frac{62\!\cdots\!08}{52\!\cdots\!73}a^{5}-\frac{32\!\cdots\!99}{52\!\cdots\!73}a^{4}+\frac{28\!\cdots\!53}{52\!\cdots\!73}a^{3}-\frac{14\!\cdots\!09}{52\!\cdots\!73}a^{2}+\frac{29\!\cdots\!45}{52\!\cdots\!73}a+\frac{46\!\cdots\!53}{52\!\cdots\!73}$, $\frac{38\!\cdots\!58}{52\!\cdots\!73}a^{24}-\frac{11\!\cdots\!96}{52\!\cdots\!73}a^{23}+\frac{16\!\cdots\!04}{52\!\cdots\!73}a^{22}-\frac{30\!\cdots\!64}{52\!\cdots\!73}a^{21}+\frac{24\!\cdots\!12}{52\!\cdots\!73}a^{20}-\frac{32\!\cdots\!53}{52\!\cdots\!73}a^{19}+\frac{11\!\cdots\!30}{52\!\cdots\!73}a^{18}-\frac{15\!\cdots\!79}{52\!\cdots\!73}a^{17}+\frac{56\!\cdots\!21}{22\!\cdots\!51}a^{16}-\frac{21\!\cdots\!92}{22\!\cdots\!51}a^{15}+\frac{26\!\cdots\!22}{52\!\cdots\!73}a^{14}-\frac{53\!\cdots\!55}{52\!\cdots\!73}a^{13}+\frac{37\!\cdots\!40}{52\!\cdots\!73}a^{12}-\frac{16\!\cdots\!92}{22\!\cdots\!51}a^{11}+\frac{11\!\cdots\!23}{52\!\cdots\!73}a^{10}-\frac{43\!\cdots\!93}{52\!\cdots\!73}a^{9}+\frac{62\!\cdots\!92}{52\!\cdots\!73}a^{8}-\frac{16\!\cdots\!05}{22\!\cdots\!51}a^{7}+\frac{14\!\cdots\!34}{52\!\cdots\!73}a^{6}+\frac{94\!\cdots\!22}{52\!\cdots\!73}a^{5}-\frac{44\!\cdots\!87}{52\!\cdots\!73}a^{4}+\frac{23\!\cdots\!04}{52\!\cdots\!73}a^{3}-\frac{22\!\cdots\!98}{52\!\cdots\!73}a^{2}+\frac{13\!\cdots\!07}{52\!\cdots\!73}a+\frac{30\!\cdots\!38}{52\!\cdots\!73}$, $\frac{42\!\cdots\!11}{52\!\cdots\!73}a^{24}-\frac{10\!\cdots\!74}{52\!\cdots\!73}a^{23}+\frac{10\!\cdots\!80}{52\!\cdots\!73}a^{22}-\frac{22\!\cdots\!00}{52\!\cdots\!73}a^{21}+\frac{35\!\cdots\!61}{27\!\cdots\!67}a^{20}-\frac{18\!\cdots\!57}{52\!\cdots\!73}a^{19}+\frac{10\!\cdots\!70}{52\!\cdots\!73}a^{18}-\frac{92\!\cdots\!21}{52\!\cdots\!73}a^{17}+\frac{740188362127014}{120409286217329}a^{16}+\frac{14\!\cdots\!28}{22\!\cdots\!51}a^{15}+\frac{26\!\cdots\!19}{52\!\cdots\!73}a^{14}-\frac{41\!\cdots\!79}{52\!\cdots\!73}a^{13}+\frac{40\!\cdots\!81}{52\!\cdots\!73}a^{12}+\frac{10\!\cdots\!95}{22\!\cdots\!51}a^{11}+\frac{37\!\cdots\!53}{52\!\cdots\!73}a^{10}-\frac{49\!\cdots\!98}{52\!\cdots\!73}a^{9}+\frac{37\!\cdots\!91}{52\!\cdots\!73}a^{8}+\frac{19\!\cdots\!45}{22\!\cdots\!51}a^{7}-\frac{22\!\cdots\!36}{52\!\cdots\!73}a^{6}+\frac{74\!\cdots\!64}{52\!\cdots\!73}a^{5}+\frac{15\!\cdots\!69}{52\!\cdots\!73}a^{4}-\frac{29\!\cdots\!35}{52\!\cdots\!73}a^{3}-\frac{11\!\cdots\!48}{52\!\cdots\!73}a^{2}-\frac{68\!\cdots\!52}{52\!\cdots\!73}a+\frac{96\!\cdots\!32}{52\!\cdots\!73}$, $\frac{308155242237218}{52\!\cdots\!73}a^{24}+\frac{12\!\cdots\!04}{52\!\cdots\!73}a^{23}-\frac{31\!\cdots\!57}{52\!\cdots\!73}a^{22}+\frac{27\!\cdots\!35}{52\!\cdots\!73}a^{21}-\frac{66\!\cdots\!13}{52\!\cdots\!73}a^{20}+\frac{14\!\cdots\!78}{52\!\cdots\!73}a^{19}-\frac{41\!\cdots\!01}{52\!\cdots\!73}a^{18}+\frac{33\!\cdots\!91}{52\!\cdots\!73}a^{17}-\frac{11\!\cdots\!92}{22\!\cdots\!51}a^{16}+\frac{16\!\cdots\!35}{22\!\cdots\!51}a^{15}+\frac{12\!\cdots\!26}{52\!\cdots\!73}a^{14}+\frac{84\!\cdots\!17}{52\!\cdots\!73}a^{13}-\frac{63\!\cdots\!23}{27\!\cdots\!67}a^{12}-\frac{47\!\cdots\!99}{22\!\cdots\!51}a^{11}+\frac{81\!\cdots\!03}{52\!\cdots\!73}a^{10}+\frac{26\!\cdots\!00}{52\!\cdots\!73}a^{9}-\frac{13\!\cdots\!92}{52\!\cdots\!73}a^{8}+\frac{35\!\cdots\!62}{22\!\cdots\!51}a^{7}+\frac{25\!\cdots\!87}{52\!\cdots\!73}a^{6}-\frac{53\!\cdots\!97}{52\!\cdots\!73}a^{5}+\frac{31\!\cdots\!68}{52\!\cdots\!73}a^{4}-\frac{14\!\cdots\!66}{52\!\cdots\!73}a^{3}+\frac{35\!\cdots\!66}{52\!\cdots\!73}a^{2}+\frac{13\!\cdots\!58}{52\!\cdots\!73}a-\frac{38\!\cdots\!21}{52\!\cdots\!73}$, $\frac{89\!\cdots\!55}{52\!\cdots\!73}a^{24}-\frac{19\!\cdots\!95}{52\!\cdots\!73}a^{23}+\frac{24\!\cdots\!28}{52\!\cdots\!73}a^{22}-\frac{64\!\cdots\!28}{52\!\cdots\!73}a^{21}+\frac{34\!\cdots\!92}{52\!\cdots\!73}a^{20}-\frac{98\!\cdots\!18}{52\!\cdots\!73}a^{19}+\frac{26\!\cdots\!48}{52\!\cdots\!73}a^{18}-\frac{19\!\cdots\!92}{52\!\cdots\!73}a^{17}+\frac{11\!\cdots\!08}{22\!\cdots\!51}a^{16}-\frac{98\!\cdots\!78}{22\!\cdots\!51}a^{15}+\frac{82\!\cdots\!97}{52\!\cdots\!73}a^{14}-\frac{82\!\cdots\!73}{52\!\cdots\!73}a^{13}+\frac{44\!\cdots\!59}{52\!\cdots\!73}a^{12}-\frac{24\!\cdots\!67}{22\!\cdots\!51}a^{11}+\frac{89\!\cdots\!99}{52\!\cdots\!73}a^{10}-\frac{10\!\cdots\!03}{52\!\cdots\!73}a^{9}+\frac{70\!\cdots\!66}{52\!\cdots\!73}a^{8}-\frac{25\!\cdots\!00}{22\!\cdots\!51}a^{7}+\frac{72\!\cdots\!50}{52\!\cdots\!73}a^{6}-\frac{63\!\cdots\!75}{52\!\cdots\!73}a^{5}+\frac{16\!\cdots\!48}{52\!\cdots\!73}a^{4}+\frac{11\!\cdots\!51}{52\!\cdots\!73}a^{3}-\frac{16\!\cdots\!36}{52\!\cdots\!73}a^{2}+\frac{61\!\cdots\!20}{52\!\cdots\!73}a-\frac{51\!\cdots\!60}{52\!\cdots\!73}$, $\frac{18\!\cdots\!31}{52\!\cdots\!73}a^{24}-\frac{63\!\cdots\!50}{52\!\cdots\!73}a^{23}+\frac{11\!\cdots\!83}{52\!\cdots\!73}a^{22}-\frac{21\!\cdots\!32}{52\!\cdots\!73}a^{21}+\frac{22\!\cdots\!79}{52\!\cdots\!73}a^{20}-\frac{15\!\cdots\!80}{27\!\cdots\!67}a^{19}+\frac{68\!\cdots\!16}{52\!\cdots\!73}a^{18}-\frac{10\!\cdots\!68}{52\!\cdots\!73}a^{17}+\frac{55\!\cdots\!72}{22\!\cdots\!51}a^{16}-\frac{38\!\cdots\!85}{22\!\cdots\!51}a^{15}+\frac{17\!\cdots\!17}{52\!\cdots\!73}a^{14}-\frac{32\!\cdots\!43}{52\!\cdots\!73}a^{13}+\frac{36\!\cdots\!98}{52\!\cdots\!73}a^{12}-\frac{98\!\cdots\!25}{22\!\cdots\!51}a^{11}+\frac{10\!\cdots\!19}{52\!\cdots\!73}a^{10}-\frac{22\!\cdots\!20}{52\!\cdots\!73}a^{9}+\frac{41\!\cdots\!18}{52\!\cdots\!73}a^{8}-\frac{18\!\cdots\!01}{22\!\cdots\!51}a^{7}+\frac{23\!\cdots\!06}{52\!\cdots\!73}a^{6}-\frac{26\!\cdots\!48}{27\!\cdots\!67}a^{5}-\frac{30\!\cdots\!45}{52\!\cdots\!73}a^{4}+\frac{37\!\cdots\!74}{52\!\cdots\!73}a^{3}-\frac{34\!\cdots\!49}{52\!\cdots\!73}a^{2}+\frac{24\!\cdots\!29}{52\!\cdots\!73}a-\frac{11\!\cdots\!52}{52\!\cdots\!73}$, $\frac{17\!\cdots\!43}{52\!\cdots\!73}a^{24}-\frac{21\!\cdots\!38}{27\!\cdots\!67}a^{23}+\frac{27\!\cdots\!16}{52\!\cdots\!73}a^{22}-\frac{50\!\cdots\!12}{52\!\cdots\!73}a^{21}-\frac{34\!\cdots\!17}{52\!\cdots\!73}a^{20}+\frac{26\!\cdots\!00}{52\!\cdots\!73}a^{19}+\frac{34\!\cdots\!66}{52\!\cdots\!73}a^{18}-\frac{23\!\cdots\!74}{52\!\cdots\!73}a^{17}-\frac{10\!\cdots\!54}{22\!\cdots\!51}a^{16}+\frac{29\!\cdots\!37}{22\!\cdots\!51}a^{15}+\frac{64\!\cdots\!71}{52\!\cdots\!73}a^{14}-\frac{74\!\cdots\!18}{27\!\cdots\!67}a^{13}-\frac{44\!\cdots\!70}{27\!\cdots\!67}a^{12}+\frac{11\!\cdots\!33}{22\!\cdots\!51}a^{11}-\frac{12\!\cdots\!18}{52\!\cdots\!73}a^{10}-\frac{18\!\cdots\!56}{52\!\cdots\!73}a^{9}+\frac{12\!\cdots\!53}{52\!\cdots\!73}a^{8}+\frac{73\!\cdots\!73}{22\!\cdots\!51}a^{7}-\frac{29\!\cdots\!98}{52\!\cdots\!73}a^{6}+\frac{16\!\cdots\!61}{52\!\cdots\!73}a^{5}-\frac{13\!\cdots\!99}{52\!\cdots\!73}a^{4}-\frac{21\!\cdots\!33}{52\!\cdots\!73}a^{3}+\frac{17\!\cdots\!78}{52\!\cdots\!73}a^{2}-\frac{13\!\cdots\!44}{52\!\cdots\!73}a+\frac{11\!\cdots\!53}{52\!\cdots\!73}$, $\frac{47\!\cdots\!71}{52\!\cdots\!73}a^{24}-\frac{13\!\cdots\!93}{52\!\cdots\!73}a^{23}+\frac{20\!\cdots\!19}{52\!\cdots\!73}a^{22}-\frac{36\!\cdots\!20}{52\!\cdots\!73}a^{21}+\frac{27\!\cdots\!25}{52\!\cdots\!73}a^{20}-\frac{40\!\cdots\!76}{52\!\cdots\!73}a^{19}+\frac{13\!\cdots\!35}{52\!\cdots\!73}a^{18}-\frac{18\!\cdots\!53}{52\!\cdots\!73}a^{17}+\frac{68\!\cdots\!39}{22\!\cdots\!51}a^{16}-\frac{19\!\cdots\!96}{22\!\cdots\!51}a^{15}+\frac{32\!\cdots\!35}{52\!\cdots\!73}a^{14}-\frac{33\!\cdots\!34}{27\!\cdots\!67}a^{13}+\frac{45\!\cdots\!46}{52\!\cdots\!73}a^{12}-\frac{24\!\cdots\!92}{22\!\cdots\!51}a^{11}-\frac{13\!\cdots\!03}{52\!\cdots\!73}a^{10}-\frac{56\!\cdots\!69}{52\!\cdots\!73}a^{9}+\frac{77\!\cdots\!52}{52\!\cdots\!73}a^{8}-\frac{17\!\cdots\!81}{22\!\cdots\!51}a^{7}-\frac{37\!\cdots\!02}{52\!\cdots\!73}a^{6}+\frac{11\!\cdots\!17}{52\!\cdots\!73}a^{5}-\frac{29\!\cdots\!26}{52\!\cdots\!73}a^{4}+\frac{13\!\cdots\!26}{52\!\cdots\!73}a^{3}-\frac{32\!\cdots\!54}{52\!\cdots\!73}a^{2}+\frac{13\!\cdots\!25}{52\!\cdots\!73}a+\frac{54\!\cdots\!45}{52\!\cdots\!73}$
| 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: | \( 87435.46969808154 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{10}\cdot 87435.46969808154 \cdot 1}{2\cdot\sqrt{722359806654877741268938522624}}\cr\approx \mathstrut & 0.157844526650959 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^25 - 3*x^24 + 4*x^23 - 7*x^22 + 5*x^21 - 6*x^20 + 28*x^19 - 37*x^18 + 24*x^17 + 60*x^15 - 133*x^14 + 74*x^13 + 37*x^12 - 24*x^11 - 112*x^10 + 155*x^9 - 56*x^8 - 45*x^7 + 51*x^6 - 17*x^5 + 3*x^4 - 3*x^3 + 3*x - 1)
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 - 3*x^24 + 4*x^23 - 7*x^22 + 5*x^21 - 6*x^20 + 28*x^19 - 37*x^18 + 24*x^17 + 60*x^15 - 133*x^14 + 74*x^13 + 37*x^12 - 24*x^11 - 112*x^10 + 155*x^9 - 56*x^8 - 45*x^7 + 51*x^6 - 17*x^5 + 3*x^4 - 3*x^3 + 3*x - 1, 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 - 3*x^24 + 4*x^23 - 7*x^22 + 5*x^21 - 6*x^20 + 28*x^19 - 37*x^18 + 24*x^17 + 60*x^15 - 133*x^14 + 74*x^13 + 37*x^12 - 24*x^11 - 112*x^10 + 155*x^9 - 56*x^8 - 45*x^7 + 51*x^6 - 17*x^5 + 3*x^4 - 3*x^3 + 3*x - 1);
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 - 3*x^24 + 4*x^23 - 7*x^22 + 5*x^21 - 6*x^20 + 28*x^19 - 37*x^18 + 24*x^17 + 60*x^15 - 133*x^14 + 74*x^13 + 37*x^12 - 24*x^11 - 112*x^10 + 155*x^9 - 56*x^8 - 45*x^7 + 51*x^6 - 17*x^5 + 3*x^4 - 3*x^3 + 3*x - 1);
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
$C_5\times D_5$ (as 25T3):
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 20 conjugacy class representatives for $C_5\times D_5$ |
| Character table for $C_5\times D_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 5.1.937024.1 |
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]
Sibling fields
| Degree 10 siblings: | data not computed |
| Minimal sibling: | 10.0.479756288.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{5}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.10.0.1}{10} }^{2}{,}\,{\href{/padicField/7.5.0.1}{5} }$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.5.0.1}{5} }$ | ${\href{/padicField/17.5.0.1}{5} }^{5}$ | ${\href{/padicField/19.5.0.1}{5} }^{5}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}{,}\,{\href{/padicField/23.1.0.1}{1} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.5.0.1}{5} }$ | ${\href{/padicField/37.10.0.1}{10} }^{2}{,}\,{\href{/padicField/37.5.0.1}{5} }$ | ${\href{/padicField/41.5.0.1}{5} }^{5}$ | ${\href{/padicField/43.5.0.1}{5} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.10.0.1}{10} }^{2}{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.5.0.1}{5} }^{5}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# 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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 2.10.15.9 | $x^{10} + 20 x^{9} + 206 x^{8} + 1376 x^{7} + 6504 x^{6} + 22496 x^{5} + 57328 x^{4} + 105856 x^{3} + 135504 x^{2} + 108864 x + 9568$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ | |
| 2.10.15.9 | $x^{10} + 20 x^{9} + 206 x^{8} + 1376 x^{7} + 6504 x^{6} + 22496 x^{5} + 57328 x^{4} + 105856 x^{3} + 135504 x^{2} + 108864 x + 9568$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ | |
|
\(11\)
| Deg $25$ | $5$ | $5$ | $20$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.11.5t1.a.b | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.11.5t1.a.a | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| 1.88.10t1.a.d | $1$ | $ 2^{3} \cdot 11 $ | 10.0.7024111812608.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| * | 1.11.5t1.a.c | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| 1.88.10t1.a.c | $1$ | $ 2^{3} \cdot 11 $ | 10.0.7024111812608.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.88.10t1.a.b | $1$ | $ 2^{3} \cdot 11 $ | 10.0.7024111812608.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| * | 1.11.5t1.a.d | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| 1.88.10t1.a.a | $1$ | $ 2^{3} \cdot 11 $ | 10.0.7024111812608.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| * | 2.968.10t6.a.c | $2$ | $ 2^{3} \cdot 11^{2}$ | 25.5.722359806654877741268938522624.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.88.10t6.a.d | $2$ | $ 2^{3} \cdot 11 $ | 25.5.722359806654877741268938522624.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.88.10t6.a.c | $2$ | $ 2^{3} \cdot 11 $ | 25.5.722359806654877741268938522624.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.968.5t2.a.b | $2$ | $ 2^{3} \cdot 11^{2}$ | 5.1.937024.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.968.5t2.a.a | $2$ | $ 2^{3} \cdot 11^{2}$ | 5.1.937024.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.88.10t6.a.a | $2$ | $ 2^{3} \cdot 11 $ | 25.5.722359806654877741268938522624.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.968.10t6.a.d | $2$ | $ 2^{3} \cdot 11^{2}$ | 25.5.722359806654877741268938522624.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.968.10t6.a.a | $2$ | $ 2^{3} \cdot 11^{2}$ | 25.5.722359806654877741268938522624.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.968.10t6.a.b | $2$ | $ 2^{3} \cdot 11^{2}$ | 25.5.722359806654877741268938522624.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.88.10t6.a.b | $2$ | $ 2^{3} \cdot 11 $ | 25.5.722359806654877741268938522624.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
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 *.