Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 2*x^24 - 2*x^23 - 10*x^22 + 33*x^21 + 5*x^20 - 6*x^19 - 115*x^18 - 16*x^17 + 168*x^16 + 250*x^15 - 16*x^14 + 291*x^13 - 1042*x^12 + 750*x^11 - 1666*x^10 + 1541*x^9 - 903*x^8 + 418*x^7 + 161*x^6 - 27*x^5 + 92*x^4 + 3*x^3 + 8*x^2 + 4*x - 1)
gp: K = bnfinit(y^25 - 2*y^24 - 2*y^23 - 10*y^22 + 33*y^21 + 5*y^20 - 6*y^19 - 115*y^18 - 16*y^17 + 168*y^16 + 250*y^15 - 16*y^14 + 291*y^13 - 1042*y^12 + 750*y^11 - 1666*y^10 + 1541*y^9 - 903*y^8 + 418*y^7 + 161*y^6 - 27*y^5 + 92*y^4 + 3*y^3 + 8*y^2 + 4*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 - 2*x^24 - 2*x^23 - 10*x^22 + 33*x^21 + 5*x^20 - 6*x^19 - 115*x^18 - 16*x^17 + 168*x^16 + 250*x^15 - 16*x^14 + 291*x^13 - 1042*x^12 + 750*x^11 - 1666*x^10 + 1541*x^9 - 903*x^8 + 418*x^7 + 161*x^6 - 27*x^5 + 92*x^4 + 3*x^3 + 8*x^2 + 4*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 2*x^24 - 2*x^23 - 10*x^22 + 33*x^21 + 5*x^20 - 6*x^19 - 115*x^18 - 16*x^17 + 168*x^16 + 250*x^15 - 16*x^14 + 291*x^13 - 1042*x^12 + 750*x^11 - 1666*x^10 + 1541*x^9 - 903*x^8 + 418*x^7 + 161*x^6 - 27*x^5 + 92*x^4 + 3*x^3 + 8*x^2 + 4*x - 1)
\( x^{25} - 2 x^{24} - 2 x^{23} - 10 x^{22} + 33 x^{21} + 5 x^{20} - 6 x^{19} - 115 x^{18} - 16 x^{17} + \cdots - 1 \)
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: |
\(188919613181312032574569023867244773376\)
\(\medspace = 2^{20}\cdot 41^{20}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(33.97\) | 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: | $2\cdot 41^{4/5}\approx 39.01728815462264$ | ||
| Ramified primes: |
\(2\), \(41\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{9}a^{14}-\frac{1}{9}a^{12}+\frac{1}{9}a^{10}-\frac{1}{9}a^{8}-\frac{4}{9}a^{6}+\frac{4}{9}a^{4}-\frac{4}{9}a^{2}+\frac{4}{9}$, $\frac{1}{9}a^{15}-\frac{1}{9}a^{13}+\frac{1}{9}a^{11}-\frac{1}{9}a^{9}-\frac{4}{9}a^{7}+\frac{4}{9}a^{5}-\frac{4}{9}a^{3}+\frac{4}{9}a$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{8}-\frac{2}{9}$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{9}-\frac{2}{9}a$, $\frac{1}{9}a^{18}+\frac{1}{9}a^{10}-\frac{2}{9}a^{2}$, $\frac{1}{27}a^{19}+\frac{1}{27}a^{17}-\frac{1}{27}a^{16}+\frac{1}{9}a^{13}+\frac{1}{27}a^{11}-\frac{1}{9}a^{10}-\frac{2}{27}a^{9}+\frac{2}{27}a^{8}-\frac{1}{3}a^{6}+\frac{2}{9}a^{5}+\frac{1}{3}a^{4}-\frac{2}{27}a^{3}+\frac{4}{9}a^{2}-\frac{8}{27}a-\frac{10}{27}$, $\frac{1}{27}a^{20}+\frac{1}{27}a^{18}-\frac{1}{27}a^{17}+\frac{4}{27}a^{12}-\frac{1}{9}a^{11}+\frac{4}{27}a^{10}+\frac{2}{27}a^{9}+\frac{1}{9}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{13}{27}a^{4}+\frac{4}{9}a^{3}-\frac{5}{27}a^{2}-\frac{10}{27}a-\frac{4}{9}$, $\frac{1}{27}a^{21}-\frac{1}{27}a^{18}-\frac{1}{27}a^{17}+\frac{1}{27}a^{16}+\frac{1}{27}a^{13}-\frac{1}{9}a^{12}+\frac{1}{9}a^{11}-\frac{4}{27}a^{10}-\frac{4}{27}a^{9}-\frac{2}{27}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{7}{27}a^{5}+\frac{1}{9}a^{4}-\frac{1}{9}a^{3}-\frac{13}{27}a^{2}+\frac{5}{27}a+\frac{1}{27}$, $\frac{1}{27}a^{22}-\frac{1}{27}a^{18}-\frac{1}{27}a^{17}-\frac{1}{27}a^{16}+\frac{1}{27}a^{14}+\frac{1}{9}a^{12}-\frac{1}{9}a^{11}+\frac{2}{27}a^{10}+\frac{2}{27}a^{9}+\frac{2}{27}a^{8}-\frac{1}{3}a^{7}-\frac{2}{27}a^{6}+\frac{1}{3}a^{5}+\frac{2}{9}a^{4}+\frac{4}{9}a^{3}+\frac{8}{27}a^{2}-\frac{10}{27}a+\frac{8}{27}$, $\frac{1}{243}a^{23}-\frac{1}{243}a^{22}-\frac{1}{243}a^{21}-\frac{4}{243}a^{20}-\frac{1}{81}a^{18}+\frac{1}{27}a^{17}-\frac{13}{243}a^{16}-\frac{2}{243}a^{15}+\frac{5}{243}a^{14}+\frac{8}{243}a^{13}+\frac{11}{243}a^{12}+\frac{7}{81}a^{11}-\frac{1}{9}a^{9}-\frac{1}{243}a^{8}+\frac{91}{243}a^{7}-\frac{58}{243}a^{6}-\frac{25}{243}a^{5}+\frac{83}{243}a^{4}+\frac{29}{81}a^{3}+\frac{25}{81}a^{2}-\frac{1}{3}a-\frac{13}{243}$, $\frac{1}{63\!\cdots\!13}a^{24}+\frac{94\!\cdots\!62}{21\!\cdots\!71}a^{23}+\frac{35\!\cdots\!27}{63\!\cdots\!13}a^{22}+\frac{45\!\cdots\!02}{63\!\cdots\!13}a^{21}+\frac{84\!\cdots\!29}{63\!\cdots\!13}a^{20}+\frac{25\!\cdots\!42}{21\!\cdots\!71}a^{19}-\frac{38\!\cdots\!38}{21\!\cdots\!71}a^{18}-\frac{17\!\cdots\!97}{63\!\cdots\!13}a^{17}-\frac{28\!\cdots\!63}{70\!\cdots\!57}a^{16}+\frac{10\!\cdots\!17}{21\!\cdots\!71}a^{15}-\frac{71\!\cdots\!61}{63\!\cdots\!13}a^{14}+\frac{11\!\cdots\!75}{63\!\cdots\!13}a^{13}-\frac{33\!\cdots\!17}{63\!\cdots\!13}a^{12}-\frac{15\!\cdots\!12}{21\!\cdots\!71}a^{11}+\frac{30\!\cdots\!68}{23\!\cdots\!19}a^{10}-\frac{65\!\cdots\!07}{63\!\cdots\!13}a^{9}-\frac{32\!\cdots\!20}{21\!\cdots\!71}a^{8}-\frac{16\!\cdots\!30}{70\!\cdots\!57}a^{7}+\frac{11\!\cdots\!00}{63\!\cdots\!13}a^{6}+\frac{13\!\cdots\!35}{63\!\cdots\!13}a^{5}-\frac{18\!\cdots\!59}{63\!\cdots\!13}a^{4}+\frac{77\!\cdots\!59}{23\!\cdots\!19}a^{3}+\frac{78\!\cdots\!25}{21\!\cdots\!71}a^{2}+\frac{19\!\cdots\!08}{63\!\cdots\!13}a+\frac{12\!\cdots\!28}{63\!\cdots\!13}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
$\frac{60\!\cdots\!78}{63\!\cdots\!13}a^{24}-\frac{11\!\cdots\!98}{21\!\cdots\!71}a^{23}+\frac{10\!\cdots\!79}{63\!\cdots\!13}a^{22}+\frac{16\!\cdots\!55}{63\!\cdots\!13}a^{21}+\frac{46\!\cdots\!44}{63\!\cdots\!13}a^{20}-\frac{16\!\cdots\!66}{21\!\cdots\!71}a^{19}-\frac{25\!\cdots\!73}{21\!\cdots\!71}a^{18}-\frac{87\!\cdots\!91}{63\!\cdots\!13}a^{17}+\frac{30\!\cdots\!83}{70\!\cdots\!57}a^{16}+\frac{12\!\cdots\!88}{21\!\cdots\!71}a^{15}-\frac{14\!\cdots\!35}{63\!\cdots\!13}a^{14}-\frac{98\!\cdots\!49}{63\!\cdots\!13}a^{13}-\frac{34\!\cdots\!96}{63\!\cdots\!13}a^{12}-\frac{36\!\cdots\!60}{21\!\cdots\!71}a^{11}+\frac{87\!\cdots\!27}{23\!\cdots\!19}a^{10}-\frac{67\!\cdots\!94}{63\!\cdots\!13}a^{9}+\frac{11\!\cdots\!46}{21\!\cdots\!71}a^{8}-\frac{70\!\cdots\!05}{23\!\cdots\!19}a^{7}+\frac{39\!\cdots\!20}{63\!\cdots\!13}a^{6}-\frac{10\!\cdots\!38}{63\!\cdots\!13}a^{5}+\frac{72\!\cdots\!57}{63\!\cdots\!13}a^{4}-\frac{13\!\cdots\!48}{77\!\cdots\!73}a^{3}+\frac{33\!\cdots\!78}{21\!\cdots\!71}a^{2}-\frac{86\!\cdots\!44}{63\!\cdots\!13}a-\frac{64\!\cdots\!87}{63\!\cdots\!13}$, $a$, $\frac{24\!\cdots\!42}{63\!\cdots\!13}a^{24}-\frac{15\!\cdots\!98}{21\!\cdots\!71}a^{23}-\frac{64\!\cdots\!64}{63\!\cdots\!13}a^{22}-\frac{23\!\cdots\!53}{63\!\cdots\!13}a^{21}+\frac{79\!\cdots\!80}{63\!\cdots\!13}a^{20}+\frac{10\!\cdots\!09}{21\!\cdots\!71}a^{19}-\frac{13\!\cdots\!85}{21\!\cdots\!71}a^{18}-\frac{29\!\cdots\!32}{63\!\cdots\!13}a^{17}-\frac{77\!\cdots\!57}{70\!\cdots\!57}a^{16}+\frac{17\!\cdots\!75}{21\!\cdots\!71}a^{15}+\frac{69\!\cdots\!62}{63\!\cdots\!13}a^{14}-\frac{12\!\cdots\!96}{63\!\cdots\!13}a^{13}+\frac{44\!\cdots\!26}{63\!\cdots\!13}a^{12}-\frac{80\!\cdots\!08}{21\!\cdots\!71}a^{11}+\frac{48\!\cdots\!40}{23\!\cdots\!19}a^{10}-\frac{29\!\cdots\!09}{63\!\cdots\!13}a^{9}+\frac{89\!\cdots\!69}{21\!\cdots\!71}a^{8}-\frac{53\!\cdots\!90}{70\!\cdots\!57}a^{7}-\frac{46\!\cdots\!11}{63\!\cdots\!13}a^{6}+\frac{94\!\cdots\!97}{63\!\cdots\!13}a^{5}-\frac{17\!\cdots\!45}{63\!\cdots\!13}a^{4}+\frac{33\!\cdots\!72}{23\!\cdots\!19}a^{3}+\frac{10\!\cdots\!68}{21\!\cdots\!71}a^{2}+\frac{22\!\cdots\!41}{63\!\cdots\!13}a+\frac{14\!\cdots\!87}{63\!\cdots\!13}$, $\frac{32\!\cdots\!95}{63\!\cdots\!13}a^{24}-\frac{25\!\cdots\!80}{21\!\cdots\!71}a^{23}-\frac{41\!\cdots\!49}{63\!\cdots\!13}a^{22}-\frac{31\!\cdots\!01}{63\!\cdots\!13}a^{21}+\frac{11\!\cdots\!76}{63\!\cdots\!13}a^{20}-\frac{65\!\cdots\!86}{21\!\cdots\!71}a^{19}-\frac{32\!\cdots\!37}{21\!\cdots\!71}a^{18}-\frac{37\!\cdots\!37}{63\!\cdots\!13}a^{17}+\frac{66\!\cdots\!41}{70\!\cdots\!57}a^{16}+\frac{17\!\cdots\!46}{21\!\cdots\!71}a^{15}+\frac{66\!\cdots\!12}{63\!\cdots\!13}a^{14}-\frac{23\!\cdots\!91}{63\!\cdots\!13}a^{13}+\frac{10\!\cdots\!14}{63\!\cdots\!13}a^{12}-\frac{12\!\cdots\!14}{21\!\cdots\!71}a^{11}+\frac{13\!\cdots\!94}{23\!\cdots\!19}a^{10}-\frac{66\!\cdots\!47}{63\!\cdots\!13}a^{9}+\frac{23\!\cdots\!24}{21\!\cdots\!71}a^{8}-\frac{59\!\cdots\!48}{70\!\cdots\!57}a^{7}+\frac{31\!\cdots\!67}{63\!\cdots\!13}a^{6}-\frac{62\!\cdots\!98}{63\!\cdots\!13}a^{5}+\frac{22\!\cdots\!21}{63\!\cdots\!13}a^{4}+\frac{93\!\cdots\!44}{23\!\cdots\!19}a^{3}-\frac{10\!\cdots\!53}{21\!\cdots\!71}a^{2}+\frac{64\!\cdots\!55}{63\!\cdots\!13}a+\frac{44\!\cdots\!45}{63\!\cdots\!13}$, $\frac{30\!\cdots\!85}{63\!\cdots\!13}a^{24}-\frac{25\!\cdots\!77}{21\!\cdots\!71}a^{23}-\frac{15\!\cdots\!39}{63\!\cdots\!13}a^{22}-\frac{31\!\cdots\!14}{63\!\cdots\!13}a^{21}+\frac{11\!\cdots\!95}{63\!\cdots\!13}a^{20}-\frac{16\!\cdots\!88}{21\!\cdots\!71}a^{19}+\frac{12\!\cdots\!67}{21\!\cdots\!71}a^{18}-\frac{35\!\cdots\!29}{63\!\cdots\!13}a^{17}+\frac{11\!\cdots\!86}{70\!\cdots\!57}a^{16}+\frac{11\!\cdots\!89}{21\!\cdots\!71}a^{15}+\frac{55\!\cdots\!63}{63\!\cdots\!13}a^{14}-\frac{11\!\cdots\!84}{63\!\cdots\!13}a^{13}+\frac{12\!\cdots\!15}{63\!\cdots\!13}a^{12}-\frac{12\!\cdots\!09}{21\!\cdots\!71}a^{11}+\frac{15\!\cdots\!06}{23\!\cdots\!19}a^{10}-\frac{81\!\cdots\!27}{63\!\cdots\!13}a^{9}+\frac{29\!\cdots\!29}{21\!\cdots\!71}a^{8}-\frac{93\!\cdots\!94}{70\!\cdots\!57}a^{7}+\frac{62\!\cdots\!84}{63\!\cdots\!13}a^{6}-\frac{26\!\cdots\!63}{63\!\cdots\!13}a^{5}+\frac{11\!\cdots\!31}{63\!\cdots\!13}a^{4}+\frac{64\!\cdots\!10}{23\!\cdots\!19}a^{3}-\frac{47\!\cdots\!19}{21\!\cdots\!71}a^{2}+\frac{16\!\cdots\!62}{63\!\cdots\!13}a-\frac{27\!\cdots\!12}{63\!\cdots\!13}$, $\frac{54\!\cdots\!61}{70\!\cdots\!57}a^{24}-\frac{23\!\cdots\!28}{70\!\cdots\!57}a^{23}+\frac{78\!\cdots\!56}{70\!\cdots\!57}a^{22}-\frac{86\!\cdots\!97}{70\!\cdots\!57}a^{21}+\frac{33\!\cdots\!02}{77\!\cdots\!73}a^{20}-\frac{11\!\cdots\!42}{23\!\cdots\!19}a^{19}-\frac{14\!\cdots\!69}{25\!\cdots\!91}a^{18}-\frac{35\!\cdots\!80}{70\!\cdots\!57}a^{17}+\frac{16\!\cdots\!55}{70\!\cdots\!57}a^{16}+\frac{18\!\cdots\!45}{70\!\cdots\!57}a^{15}-\frac{17\!\cdots\!26}{70\!\cdots\!57}a^{14}-\frac{51\!\cdots\!36}{70\!\cdots\!57}a^{13}+\frac{52\!\cdots\!85}{23\!\cdots\!19}a^{12}-\frac{63\!\cdots\!18}{77\!\cdots\!73}a^{11}+\frac{18\!\cdots\!24}{77\!\cdots\!73}a^{10}-\frac{99\!\cdots\!05}{70\!\cdots\!57}a^{9}+\frac{16\!\cdots\!92}{70\!\cdots\!57}a^{8}-\frac{13\!\cdots\!87}{70\!\cdots\!57}a^{7}-\frac{68\!\cdots\!71}{70\!\cdots\!57}a^{6}+\frac{75\!\cdots\!49}{70\!\cdots\!57}a^{5}-\frac{14\!\cdots\!81}{23\!\cdots\!19}a^{4}+\frac{59\!\cdots\!71}{23\!\cdots\!19}a^{3}+\frac{45\!\cdots\!87}{77\!\cdots\!73}a^{2}+\frac{14\!\cdots\!98}{70\!\cdots\!57}a+\frac{10\!\cdots\!55}{25\!\cdots\!91}$, $\frac{17\!\cdots\!96}{21\!\cdots\!71}a^{24}-\frac{12\!\cdots\!42}{70\!\cdots\!57}a^{23}-\frac{38\!\cdots\!07}{21\!\cdots\!71}a^{22}-\frac{16\!\cdots\!93}{21\!\cdots\!71}a^{21}+\frac{61\!\cdots\!19}{21\!\cdots\!71}a^{20}+\frac{35\!\cdots\!52}{70\!\cdots\!57}a^{19}-\frac{10\!\cdots\!07}{70\!\cdots\!57}a^{18}-\frac{21\!\cdots\!26}{21\!\cdots\!71}a^{17}-\frac{23\!\cdots\!75}{70\!\cdots\!57}a^{16}+\frac{41\!\cdots\!89}{23\!\cdots\!19}a^{15}+\frac{45\!\cdots\!63}{21\!\cdots\!71}a^{14}-\frac{17\!\cdots\!30}{21\!\cdots\!71}a^{13}+\frac{30\!\cdots\!55}{21\!\cdots\!71}a^{12}-\frac{64\!\cdots\!68}{70\!\cdots\!57}a^{11}+\frac{16\!\cdots\!93}{25\!\cdots\!91}a^{10}-\frac{24\!\cdots\!94}{21\!\cdots\!71}a^{9}+\frac{93\!\cdots\!48}{70\!\cdots\!57}a^{8}-\frac{33\!\cdots\!41}{70\!\cdots\!57}a^{7}+\frac{23\!\cdots\!11}{21\!\cdots\!71}a^{6}+\frac{34\!\cdots\!60}{21\!\cdots\!71}a^{5}-\frac{17\!\cdots\!82}{21\!\cdots\!71}a^{4}+\frac{46\!\cdots\!31}{23\!\cdots\!19}a^{3}-\frac{97\!\cdots\!94}{70\!\cdots\!57}a^{2}-\frac{70\!\cdots\!64}{21\!\cdots\!71}a+\frac{52\!\cdots\!15}{21\!\cdots\!71}$, $\frac{36\!\cdots\!96}{63\!\cdots\!13}a^{24}-\frac{29\!\cdots\!21}{21\!\cdots\!71}a^{23}-\frac{39\!\cdots\!54}{63\!\cdots\!13}a^{22}-\frac{34\!\cdots\!94}{63\!\cdots\!13}a^{21}+\frac{13\!\cdots\!19}{63\!\cdots\!13}a^{20}-\frac{11\!\cdots\!69}{21\!\cdots\!71}a^{19}-\frac{53\!\cdots\!07}{21\!\cdots\!71}a^{18}-\frac{40\!\cdots\!08}{63\!\cdots\!13}a^{17}+\frac{12\!\cdots\!26}{70\!\cdots\!57}a^{16}+\frac{19\!\cdots\!23}{21\!\cdots\!71}a^{15}+\frac{64\!\cdots\!29}{63\!\cdots\!13}a^{14}-\frac{37\!\cdots\!18}{63\!\cdots\!13}a^{13}+\frac{11\!\cdots\!35}{63\!\cdots\!13}a^{12}-\frac{13\!\cdots\!21}{21\!\cdots\!71}a^{11}+\frac{16\!\cdots\!77}{23\!\cdots\!19}a^{10}-\frac{76\!\cdots\!65}{63\!\cdots\!13}a^{9}+\frac{27\!\cdots\!14}{21\!\cdots\!71}a^{8}-\frac{26\!\cdots\!48}{25\!\cdots\!91}a^{7}+\frac{35\!\cdots\!26}{63\!\cdots\!13}a^{6}-\frac{64\!\cdots\!50}{63\!\cdots\!13}a^{5}+\frac{13\!\cdots\!22}{63\!\cdots\!13}a^{4}+\frac{11\!\cdots\!85}{25\!\cdots\!91}a^{3}-\frac{13\!\cdots\!67}{21\!\cdots\!71}a^{2}+\frac{68\!\cdots\!18}{63\!\cdots\!13}a+\frac{41\!\cdots\!59}{63\!\cdots\!13}$, $\frac{77\!\cdots\!57}{63\!\cdots\!13}a^{24}-\frac{45\!\cdots\!03}{21\!\cdots\!71}a^{23}-\frac{20\!\cdots\!05}{63\!\cdots\!13}a^{22}-\frac{79\!\cdots\!00}{63\!\cdots\!13}a^{21}+\frac{23\!\cdots\!35}{63\!\cdots\!13}a^{20}+\frac{38\!\cdots\!53}{21\!\cdots\!71}a^{19}-\frac{24\!\cdots\!65}{21\!\cdots\!71}a^{18}-\frac{90\!\cdots\!95}{63\!\cdots\!13}a^{17}-\frac{12\!\cdots\!86}{23\!\cdots\!19}a^{16}+\frac{46\!\cdots\!24}{21\!\cdots\!71}a^{15}+\frac{22\!\cdots\!16}{63\!\cdots\!13}a^{14}+\frac{17\!\cdots\!01}{63\!\cdots\!13}a^{13}+\frac{19\!\cdots\!53}{63\!\cdots\!13}a^{12}-\frac{24\!\cdots\!20}{21\!\cdots\!71}a^{11}+\frac{13\!\cdots\!19}{23\!\cdots\!19}a^{10}-\frac{10\!\cdots\!26}{63\!\cdots\!13}a^{9}+\frac{25\!\cdots\!93}{21\!\cdots\!71}a^{8}-\frac{24\!\cdots\!29}{70\!\cdots\!57}a^{7}-\frac{56\!\cdots\!19}{63\!\cdots\!13}a^{6}+\frac{32\!\cdots\!47}{63\!\cdots\!13}a^{5}-\frac{57\!\cdots\!47}{63\!\cdots\!13}a^{4}+\frac{27\!\cdots\!82}{23\!\cdots\!19}a^{3}+\frac{62\!\cdots\!98}{21\!\cdots\!71}a^{2}+\frac{48\!\cdots\!44}{63\!\cdots\!13}a+\frac{73\!\cdots\!75}{63\!\cdots\!13}$, $\frac{23\!\cdots\!68}{63\!\cdots\!13}a^{24}-\frac{15\!\cdots\!70}{21\!\cdots\!71}a^{23}-\frac{45\!\cdots\!18}{63\!\cdots\!13}a^{22}-\frac{22\!\cdots\!34}{63\!\cdots\!13}a^{21}+\frac{76\!\cdots\!09}{63\!\cdots\!13}a^{20}+\frac{34\!\cdots\!69}{21\!\cdots\!71}a^{19}-\frac{52\!\cdots\!96}{21\!\cdots\!71}a^{18}-\frac{26\!\cdots\!09}{63\!\cdots\!13}a^{17}-\frac{32\!\cdots\!83}{70\!\cdots\!57}a^{16}+\frac{13\!\cdots\!87}{21\!\cdots\!71}a^{15}+\frac{56\!\cdots\!03}{63\!\cdots\!13}a^{14}-\frac{59\!\cdots\!34}{63\!\cdots\!13}a^{13}+\frac{68\!\cdots\!47}{63\!\cdots\!13}a^{12}-\frac{79\!\cdots\!32}{21\!\cdots\!71}a^{11}+\frac{65\!\cdots\!75}{23\!\cdots\!19}a^{10}-\frac{38\!\cdots\!47}{63\!\cdots\!13}a^{9}+\frac{11\!\cdots\!82}{21\!\cdots\!71}a^{8}-\frac{23\!\cdots\!28}{70\!\cdots\!57}a^{7}+\frac{94\!\cdots\!94}{63\!\cdots\!13}a^{6}+\frac{39\!\cdots\!92}{63\!\cdots\!13}a^{5}+\frac{33\!\cdots\!25}{63\!\cdots\!13}a^{4}+\frac{46\!\cdots\!46}{23\!\cdots\!19}a^{3}+\frac{12\!\cdots\!05}{21\!\cdots\!71}a^{2}+\frac{27\!\cdots\!91}{63\!\cdots\!13}a+\frac{23\!\cdots\!61}{63\!\cdots\!13}$, $\frac{55\!\cdots\!98}{21\!\cdots\!71}a^{24}-\frac{44\!\cdots\!38}{70\!\cdots\!57}a^{23}-\frac{56\!\cdots\!92}{21\!\cdots\!71}a^{22}-\frac{52\!\cdots\!69}{21\!\cdots\!71}a^{21}+\frac{20\!\cdots\!42}{21\!\cdots\!71}a^{20}-\frac{18\!\cdots\!75}{70\!\cdots\!57}a^{19}-\frac{47\!\cdots\!10}{70\!\cdots\!57}a^{18}-\frac{62\!\cdots\!56}{21\!\cdots\!71}a^{17}+\frac{57\!\cdots\!20}{70\!\cdots\!57}a^{16}+\frac{96\!\cdots\!63}{23\!\cdots\!19}a^{15}+\frac{10\!\cdots\!22}{21\!\cdots\!71}a^{14}-\frac{52\!\cdots\!26}{21\!\cdots\!71}a^{13}+\frac{18\!\cdots\!88}{21\!\cdots\!71}a^{12}-\frac{21\!\cdots\!08}{70\!\cdots\!57}a^{11}+\frac{25\!\cdots\!70}{77\!\cdots\!73}a^{10}-\frac{11\!\cdots\!05}{21\!\cdots\!71}a^{9}+\frac{44\!\cdots\!30}{70\!\cdots\!57}a^{8}-\frac{34\!\cdots\!43}{70\!\cdots\!57}a^{7}+\frac{63\!\cdots\!65}{21\!\cdots\!71}a^{6}-\frac{15\!\cdots\!64}{21\!\cdots\!71}a^{5}+\frac{41\!\cdots\!29}{21\!\cdots\!71}a^{4}+\frac{36\!\cdots\!36}{23\!\cdots\!19}a^{3}-\frac{39\!\cdots\!78}{70\!\cdots\!57}a^{2}+\frac{88\!\cdots\!06}{21\!\cdots\!71}a-\frac{16\!\cdots\!66}{21\!\cdots\!71}$, $\frac{52\!\cdots\!32}{23\!\cdots\!19}a^{24}-\frac{12\!\cdots\!79}{70\!\cdots\!57}a^{23}-\frac{60\!\cdots\!25}{70\!\cdots\!57}a^{22}-\frac{21\!\cdots\!71}{70\!\cdots\!57}a^{21}+\frac{30\!\cdots\!00}{70\!\cdots\!57}a^{20}+\frac{77\!\cdots\!77}{86\!\cdots\!97}a^{19}+\frac{11\!\cdots\!96}{23\!\cdots\!19}a^{18}-\frac{62\!\cdots\!61}{23\!\cdots\!19}a^{17}-\frac{26\!\cdots\!35}{70\!\cdots\!57}a^{16}+\frac{11\!\cdots\!26}{70\!\cdots\!57}a^{15}+\frac{71\!\cdots\!34}{70\!\cdots\!57}a^{14}+\frac{66\!\cdots\!18}{70\!\cdots\!57}a^{13}+\frac{67\!\cdots\!67}{70\!\cdots\!57}a^{12}-\frac{39\!\cdots\!74}{23\!\cdots\!19}a^{11}-\frac{76\!\cdots\!77}{77\!\cdots\!73}a^{10}-\frac{74\!\cdots\!56}{23\!\cdots\!19}a^{9}-\frac{54\!\cdots\!60}{70\!\cdots\!57}a^{8}+\frac{28\!\cdots\!24}{70\!\cdots\!57}a^{7}+\frac{44\!\cdots\!40}{70\!\cdots\!57}a^{6}+\frac{54\!\cdots\!86}{70\!\cdots\!57}a^{5}+\frac{39\!\cdots\!69}{70\!\cdots\!57}a^{4}+\frac{79\!\cdots\!04}{23\!\cdots\!19}a^{3}+\frac{36\!\cdots\!79}{23\!\cdots\!19}a^{2}+\frac{10\!\cdots\!70}{23\!\cdots\!19}a+\frac{40\!\cdots\!76}{70\!\cdots\!57}$, $\frac{22\!\cdots\!39}{21\!\cdots\!71}a^{24}-\frac{12\!\cdots\!21}{70\!\cdots\!57}a^{23}-\frac{60\!\cdots\!08}{21\!\cdots\!71}a^{22}-\frac{23\!\cdots\!85}{21\!\cdots\!71}a^{21}+\frac{65\!\cdots\!40}{21\!\cdots\!71}a^{20}+\frac{12\!\cdots\!19}{70\!\cdots\!57}a^{19}-\frac{39\!\cdots\!49}{70\!\cdots\!57}a^{18}-\frac{26\!\cdots\!46}{21\!\cdots\!71}a^{17}-\frac{14\!\cdots\!66}{23\!\cdots\!19}a^{16}+\frac{12\!\cdots\!28}{70\!\cdots\!57}a^{15}+\frac{72\!\cdots\!59}{21\!\cdots\!71}a^{14}+\frac{14\!\cdots\!09}{21\!\cdots\!71}a^{13}+\frac{52\!\cdots\!00}{21\!\cdots\!71}a^{12}-\frac{72\!\cdots\!97}{70\!\cdots\!57}a^{11}+\frac{29\!\cdots\!98}{77\!\cdots\!73}a^{10}-\frac{29\!\cdots\!58}{21\!\cdots\!71}a^{9}+\frac{77\!\cdots\!01}{70\!\cdots\!57}a^{8}-\frac{69\!\cdots\!23}{23\!\cdots\!19}a^{7}+\frac{37\!\cdots\!00}{21\!\cdots\!71}a^{6}+\frac{49\!\cdots\!04}{21\!\cdots\!71}a^{5}+\frac{50\!\cdots\!02}{21\!\cdots\!71}a^{4}+\frac{12\!\cdots\!02}{25\!\cdots\!91}a^{3}+\frac{28\!\cdots\!50}{70\!\cdots\!57}a^{2}-\frac{55\!\cdots\!70}{21\!\cdots\!71}a+\frac{17\!\cdots\!63}{21\!\cdots\!71}$, $\frac{13\!\cdots\!58}{21\!\cdots\!71}a^{24}+\frac{23\!\cdots\!95}{70\!\cdots\!57}a^{23}-\frac{15\!\cdots\!46}{21\!\cdots\!71}a^{22}-\frac{46\!\cdots\!31}{21\!\cdots\!71}a^{21}-\frac{65\!\cdots\!04}{21\!\cdots\!71}a^{20}+\frac{90\!\cdots\!91}{70\!\cdots\!57}a^{19}+\frac{91\!\cdots\!10}{70\!\cdots\!57}a^{18}-\frac{20\!\cdots\!45}{21\!\cdots\!71}a^{17}-\frac{42\!\cdots\!66}{70\!\cdots\!57}a^{16}-\frac{78\!\cdots\!33}{23\!\cdots\!19}a^{15}+\frac{20\!\cdots\!39}{21\!\cdots\!71}a^{14}+\frac{38\!\cdots\!93}{21\!\cdots\!71}a^{13}+\frac{16\!\cdots\!45}{21\!\cdots\!71}a^{12}+\frac{77\!\cdots\!35}{70\!\cdots\!57}a^{11}-\frac{30\!\cdots\!43}{77\!\cdots\!73}a^{10}-\frac{21\!\cdots\!30}{21\!\cdots\!71}a^{9}-\frac{26\!\cdots\!71}{70\!\cdots\!57}a^{8}+\frac{17\!\cdots\!16}{70\!\cdots\!57}a^{7}+\frac{33\!\cdots\!26}{21\!\cdots\!71}a^{6}-\frac{31\!\cdots\!79}{21\!\cdots\!71}a^{5}+\frac{21\!\cdots\!57}{21\!\cdots\!71}a^{4}+\frac{66\!\cdots\!53}{23\!\cdots\!19}a^{3}+\frac{69\!\cdots\!51}{70\!\cdots\!57}a^{2}+\frac{53\!\cdots\!20}{21\!\cdots\!71}a-\frac{19\!\cdots\!79}{21\!\cdots\!71}$
| 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: | \( 28022710779.842506 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{10}\cdot 28022710779.842506 \cdot 1}{2\cdot\sqrt{188919613181312032574569023867244773376}}\cr\approx \mathstrut & 3.12816956924097 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^25 - 2*x^24 - 2*x^23 - 10*x^22 + 33*x^21 + 5*x^20 - 6*x^19 - 115*x^18 - 16*x^17 + 168*x^16 + 250*x^15 - 16*x^14 + 291*x^13 - 1042*x^12 + 750*x^11 - 1666*x^10 + 1541*x^9 - 903*x^8 + 418*x^7 + 161*x^6 - 27*x^5 + 92*x^4 + 3*x^3 + 8*x^2 + 4*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 - 2*x^24 - 2*x^23 - 10*x^22 + 33*x^21 + 5*x^20 - 6*x^19 - 115*x^18 - 16*x^17 + 168*x^16 + 250*x^15 - 16*x^14 + 291*x^13 - 1042*x^12 + 750*x^11 - 1666*x^10 + 1541*x^9 - 903*x^8 + 418*x^7 + 161*x^6 - 27*x^5 + 92*x^4 + 3*x^3 + 8*x^2 + 4*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 - 2*x^24 - 2*x^23 - 10*x^22 + 33*x^21 + 5*x^20 - 6*x^19 - 115*x^18 - 16*x^17 + 168*x^16 + 250*x^15 - 16*x^14 + 291*x^13 - 1042*x^12 + 750*x^11 - 1666*x^10 + 1541*x^9 - 903*x^8 + 418*x^7 + 161*x^6 - 27*x^5 + 92*x^4 + 3*x^3 + 8*x^2 + 4*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 - 2*x^24 - 2*x^23 - 10*x^22 + 33*x^21 + 5*x^20 - 6*x^19 - 115*x^18 - 16*x^17 + 168*x^16 + 250*x^15 - 16*x^14 + 291*x^13 - 1042*x^12 + 750*x^11 - 1666*x^10 + 1541*x^9 - 903*x^8 + 418*x^7 + 161*x^6 - 27*x^5 + 92*x^4 + 3*x^3 + 8*x^2 + 4*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
| 5.5.2825761.1, 5.1.45212176.1 |
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: | 10.0.2893579264.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.2.0.1}{2} }^{10}{,}\,{\href{/padicField/3.1.0.1}{1} }^{5}$ | ${\href{/padicField/5.5.0.1}{5} }^{5}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}{,}\,{\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.10.0.1}{10} }^{2}{,}\,{\href{/padicField/11.5.0.1}{5} }$ | ${\href{/padicField/13.5.0.1}{5} }^{5}$ | ${\href{/padicField/17.5.0.1}{5} }^{5}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.10.0.1}{10} }^{2}{,}\,{\href{/padicField/23.5.0.1}{5} }$ | ${\href{/padicField/29.5.0.1}{5} }^{5}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.5.0.1}{5} }$ | ${\href{/padicField/37.5.0.1}{5} }^{5}$ | R | ${\href{/padicField/43.10.0.1}{10} }^{2}{,}\,{\href{/padicField/43.5.0.1}{5} }$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.5.0.1}{5} }^{5}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 2.10.10.7 | $x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 592 x^{5} + 656 x^{4} + 384 x^{3} - 112 x^{2} - 352 x - 1248$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
| 2.10.10.7 | $x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 592 x^{5} + 656 x^{4} + 384 x^{3} - 112 x^{2} - 352 x - 1248$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
|
\(41\)
| Deg $25$ | $5$ | $5$ | $20$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.41.5t1.a.b | $1$ | $ 41 $ | 5.5.2825761.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.41.5t1.a.c | $1$ | $ 41 $ | 5.5.2825761.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.41.5t1.a.a | $1$ | $ 41 $ | 5.5.2825761.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| 1.164.10t1.a.d | $1$ | $ 2^{2} \cdot 41 $ | 10.0.8176563434619904.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.164.10t1.a.a | $1$ | $ 2^{2} \cdot 41 $ | 10.0.8176563434619904.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| * | 1.41.5t1.a.d | $1$ | $ 41 $ | 5.5.2825761.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| 1.164.10t1.a.c | $1$ | $ 2^{2} \cdot 41 $ | 10.0.8176563434619904.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.164.10t1.a.b | $1$ | $ 2^{2} \cdot 41 $ | 10.0.8176563434619904.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| * | 2.6724.10t6.a.d | $2$ | $ 2^{2} \cdot 41^{2}$ | 25.5.188919613181312032574569023867244773376.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.6724.10t6.a.c | $2$ | $ 2^{2} \cdot 41^{2}$ | 25.5.188919613181312032574569023867244773376.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.164.10t6.a.b | $2$ | $ 2^{2} \cdot 41 $ | 25.5.188919613181312032574569023867244773376.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.164.10t6.a.a | $2$ | $ 2^{2} \cdot 41 $ | 25.5.188919613181312032574569023867244773376.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.6724.5t2.a.b | $2$ | $ 2^{2} \cdot 41^{2}$ | 5.1.45212176.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.6724.10t6.a.b | $2$ | $ 2^{2} \cdot 41^{2}$ | 25.5.188919613181312032574569023867244773376.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.164.10t6.a.d | $2$ | $ 2^{2} \cdot 41 $ | 25.5.188919613181312032574569023867244773376.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.6724.5t2.a.a | $2$ | $ 2^{2} \cdot 41^{2}$ | 5.1.45212176.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.164.10t6.a.c | $2$ | $ 2^{2} \cdot 41 $ | 25.5.188919613181312032574569023867244773376.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.6724.10t6.a.a | $2$ | $ 2^{2} \cdot 41^{2}$ | 25.5.188919613181312032574569023867244773376.1 | $C_5\times D_5$ (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 *.