Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 3*x^24 + x^23 - 3*x^22 + 19*x^21 - 17*x^20 - 8*x^19 - 27*x^18 + 13*x^17 + 72*x^16 + 7*x^15 + 58*x^14 - 37*x^13 - 186*x^12 + 151*x^11 - 314*x^10 + 404*x^9 - 330*x^8 + 297*x^7 - 201*x^6 + 123*x^5 - 65*x^4 + 32*x^3 - 18*x^2 + 7*x - 1)
gp: K = bnfinit(y^25 - 3*y^24 + y^23 - 3*y^22 + 19*y^21 - 17*y^20 - 8*y^19 - 27*y^18 + 13*y^17 + 72*y^16 + 7*y^15 + 58*y^14 - 37*y^13 - 186*y^12 + 151*y^11 - 314*y^10 + 404*y^9 - 330*y^8 + 297*y^7 - 201*y^6 + 123*y^5 - 65*y^4 + 32*y^3 - 18*y^2 + 7*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^25 - 3*x^24 + x^23 - 3*x^22 + 19*x^21 - 17*x^20 - 8*x^19 - 27*x^18 + 13*x^17 + 72*x^16 + 7*x^15 + 58*x^14 - 37*x^13 - 186*x^12 + 151*x^11 - 314*x^10 + 404*x^9 - 330*x^8 + 297*x^7 - 201*x^6 + 123*x^5 - 65*x^4 + 32*x^3 - 18*x^2 + 7*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^25 - 3*x^24 + x^23 - 3*x^22 + 19*x^21 - 17*x^20 - 8*x^19 - 27*x^18 + 13*x^17 + 72*x^16 + 7*x^15 + 58*x^14 - 37*x^13 - 186*x^12 + 151*x^11 - 314*x^10 + 404*x^9 - 330*x^8 + 297*x^7 - 201*x^6 + 123*x^5 - 65*x^4 + 32*x^3 - 18*x^2 + 7*x - 1)
\( x^{25} - 3 x^{24} + x^{23} - 3 x^{22} + 19 x^{21} - 17 x^{20} - 8 x^{19} - 27 x^{18} + 13 x^{17} + \cdots - 1 \)
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: |
\(39668558936237989671952184481937449\)
\(\medspace = 3^{10}\cdot 31^{20}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(24.21\) | 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: | $3^{1/2}31^{4/5}\approx 27.01779999660944$ | ||
| Ramified primes: |
\(3\), \(31\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5}a^{18}+\frac{2}{5}a^{17}-\frac{2}{5}a^{14}-\frac{2}{5}a^{13}+\frac{2}{5}a^{12}+\frac{2}{5}a^{10}+\frac{1}{5}a^{8}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{4}-\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{19}+\frac{1}{5}a^{17}-\frac{2}{5}a^{15}+\frac{2}{5}a^{14}+\frac{1}{5}a^{13}+\frac{1}{5}a^{12}+\frac{2}{5}a^{11}+\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{2}{5}a^{8}-\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{2}{5}a^{4}-\frac{2}{5}a^{2}+\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{20}-\frac{2}{5}a^{17}-\frac{2}{5}a^{16}+\frac{2}{5}a^{15}-\frac{2}{5}a^{14}-\frac{2}{5}a^{13}+\frac{1}{5}a^{11}-\frac{1}{5}a^{10}-\frac{2}{5}a^{9}-\frac{2}{5}a^{8}-\frac{1}{5}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{21}+\frac{2}{5}a^{17}+\frac{2}{5}a^{16}-\frac{2}{5}a^{15}-\frac{1}{5}a^{14}+\frac{1}{5}a^{13}-\frac{1}{5}a^{11}+\frac{2}{5}a^{10}-\frac{2}{5}a^{9}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{2}{5}a^{6}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{1}{5}a^{2}-\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{22}-\frac{2}{5}a^{17}-\frac{2}{5}a^{16}-\frac{1}{5}a^{15}-\frac{1}{5}a^{13}+\frac{2}{5}a^{11}-\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}+\frac{2}{5}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{3}-\frac{2}{5}a^{2}-\frac{1}{5}$, $\frac{1}{5}a^{23}+\frac{2}{5}a^{17}-\frac{1}{5}a^{16}+\frac{1}{5}a^{13}+\frac{1}{5}a^{12}-\frac{1}{5}a^{11}-\frac{1}{5}a^{9}+\frac{2}{5}a^{7}-\frac{2}{5}a^{5}-\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{1}{5}$, $\frac{1}{17\!\cdots\!75}a^{24}+\frac{63\!\cdots\!48}{17\!\cdots\!75}a^{23}-\frac{15\!\cdots\!61}{17\!\cdots\!75}a^{22}+\frac{41\!\cdots\!16}{17\!\cdots\!75}a^{21}+\frac{58\!\cdots\!99}{70\!\cdots\!71}a^{20}+\frac{96\!\cdots\!43}{17\!\cdots\!75}a^{19}-\frac{29\!\cdots\!43}{35\!\cdots\!55}a^{18}-\frac{25\!\cdots\!82}{17\!\cdots\!75}a^{17}+\frac{61\!\cdots\!81}{17\!\cdots\!75}a^{16}+\frac{77\!\cdots\!63}{17\!\cdots\!75}a^{15}+\frac{16\!\cdots\!61}{35\!\cdots\!55}a^{14}-\frac{35\!\cdots\!67}{17\!\cdots\!75}a^{13}-\frac{64\!\cdots\!69}{17\!\cdots\!75}a^{12}-\frac{19\!\cdots\!24}{35\!\cdots\!55}a^{11}+\frac{75\!\cdots\!96}{17\!\cdots\!75}a^{10}-\frac{74\!\cdots\!33}{17\!\cdots\!75}a^{9}+\frac{52\!\cdots\!61}{17\!\cdots\!75}a^{8}+\frac{81\!\cdots\!56}{17\!\cdots\!75}a^{7}-\frac{64\!\cdots\!47}{17\!\cdots\!75}a^{6}-\frac{44\!\cdots\!18}{17\!\cdots\!75}a^{5}-\frac{38\!\cdots\!84}{35\!\cdots\!55}a^{4}+\frac{41\!\cdots\!37}{35\!\cdots\!55}a^{3}+\frac{18\!\cdots\!27}{17\!\cdots\!75}a^{2}+\frac{51\!\cdots\!59}{17\!\cdots\!75}a-\frac{47\!\cdots\!99}{17\!\cdots\!75}$
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: |
$\frac{38\!\cdots\!28}{17\!\cdots\!75}a^{24}-\frac{10\!\cdots\!46}{17\!\cdots\!75}a^{23}+\frac{19\!\cdots\!62}{17\!\cdots\!75}a^{22}-\frac{10\!\cdots\!12}{17\!\cdots\!75}a^{21}+\frac{28\!\cdots\!16}{70\!\cdots\!71}a^{20}-\frac{52\!\cdots\!06}{17\!\cdots\!75}a^{19}-\frac{82\!\cdots\!84}{35\!\cdots\!55}a^{18}-\frac{11\!\cdots\!46}{17\!\cdots\!75}a^{17}+\frac{33\!\cdots\!23}{17\!\cdots\!75}a^{16}+\frac{28\!\cdots\!84}{17\!\cdots\!75}a^{15}+\frac{15\!\cdots\!46}{35\!\cdots\!55}a^{14}+\frac{22\!\cdots\!44}{17\!\cdots\!75}a^{13}-\frac{11\!\cdots\!32}{17\!\cdots\!75}a^{12}-\frac{14\!\cdots\!18}{35\!\cdots\!55}a^{11}+\frac{44\!\cdots\!38}{17\!\cdots\!75}a^{10}-\frac{11\!\cdots\!04}{17\!\cdots\!75}a^{9}+\frac{13\!\cdots\!48}{17\!\cdots\!75}a^{8}-\frac{10\!\cdots\!02}{17\!\cdots\!75}a^{7}+\frac{95\!\cdots\!54}{17\!\cdots\!75}a^{6}-\frac{59\!\cdots\!84}{17\!\cdots\!75}a^{5}+\frac{70\!\cdots\!36}{35\!\cdots\!55}a^{4}-\frac{36\!\cdots\!38}{35\!\cdots\!55}a^{3}+\frac{81\!\cdots\!26}{17\!\cdots\!75}a^{2}-\frac{53\!\cdots\!48}{17\!\cdots\!75}a+\frac{15\!\cdots\!68}{17\!\cdots\!75}$, $\frac{22\!\cdots\!91}{17\!\cdots\!75}a^{24}-\frac{60\!\cdots\!92}{17\!\cdots\!75}a^{23}+\frac{63\!\cdots\!24}{17\!\cdots\!75}a^{22}-\frac{66\!\cdots\!04}{17\!\cdots\!75}a^{21}+\frac{81\!\cdots\!03}{35\!\cdots\!55}a^{20}-\frac{23\!\cdots\!42}{17\!\cdots\!75}a^{19}-\frac{11\!\cdots\!52}{70\!\cdots\!71}a^{18}-\frac{70\!\cdots\!07}{17\!\cdots\!75}a^{17}+\frac{48\!\cdots\!31}{17\!\cdots\!75}a^{16}+\frac{16\!\cdots\!43}{17\!\cdots\!75}a^{15}+\frac{14\!\cdots\!09}{35\!\cdots\!55}a^{14}+\frac{15\!\cdots\!68}{17\!\cdots\!75}a^{13}-\frac{28\!\cdots\!39}{17\!\cdots\!75}a^{12}-\frac{87\!\cdots\!79}{35\!\cdots\!55}a^{11}+\frac{19\!\cdots\!26}{17\!\cdots\!75}a^{10}-\frac{63\!\cdots\!08}{17\!\cdots\!75}a^{9}+\frac{68\!\cdots\!61}{17\!\cdots\!75}a^{8}-\frac{49\!\cdots\!44}{17\!\cdots\!75}a^{7}+\frac{47\!\cdots\!48}{17\!\cdots\!75}a^{6}-\frac{27\!\cdots\!88}{17\!\cdots\!75}a^{5}+\frac{34\!\cdots\!19}{35\!\cdots\!55}a^{4}-\frac{15\!\cdots\!74}{35\!\cdots\!55}a^{3}+\frac{45\!\cdots\!12}{17\!\cdots\!75}a^{2}-\frac{26\!\cdots\!86}{17\!\cdots\!75}a+\frac{74\!\cdots\!01}{17\!\cdots\!75}$, $\frac{48\!\cdots\!27}{17\!\cdots\!75}a^{24}-\frac{16\!\cdots\!94}{17\!\cdots\!75}a^{23}+\frac{86\!\cdots\!03}{17\!\cdots\!75}a^{22}-\frac{10\!\cdots\!88}{17\!\cdots\!75}a^{21}+\frac{19\!\cdots\!77}{35\!\cdots\!55}a^{20}-\frac{11\!\cdots\!44}{17\!\cdots\!75}a^{19}-\frac{87\!\cdots\!48}{35\!\cdots\!55}a^{18}-\frac{95\!\cdots\!29}{17\!\cdots\!75}a^{17}+\frac{15\!\cdots\!42}{17\!\cdots\!75}a^{16}+\frac{38\!\cdots\!96}{17\!\cdots\!75}a^{15}-\frac{22\!\cdots\!41}{35\!\cdots\!55}a^{14}+\frac{89\!\cdots\!01}{17\!\cdots\!75}a^{13}-\frac{39\!\cdots\!53}{17\!\cdots\!75}a^{12}-\frac{19\!\cdots\!16}{35\!\cdots\!55}a^{11}+\frac{11\!\cdots\!67}{17\!\cdots\!75}a^{10}-\frac{13\!\cdots\!36}{17\!\cdots\!75}a^{9}+\frac{24\!\cdots\!57}{17\!\cdots\!75}a^{8}-\frac{18\!\cdots\!68}{17\!\cdots\!75}a^{7}+\frac{15\!\cdots\!71}{17\!\cdots\!75}a^{6}-\frac{12\!\cdots\!26}{17\!\cdots\!75}a^{5}+\frac{14\!\cdots\!98}{35\!\cdots\!55}a^{4}-\frac{90\!\cdots\!62}{35\!\cdots\!55}a^{3}+\frac{22\!\cdots\!39}{17\!\cdots\!75}a^{2}-\frac{12\!\cdots\!72}{17\!\cdots\!75}a+\frac{45\!\cdots\!62}{17\!\cdots\!75}$, $a$, $\frac{97\!\cdots\!19}{17\!\cdots\!75}a^{24}-\frac{20\!\cdots\!83}{17\!\cdots\!75}a^{23}-\frac{11\!\cdots\!54}{17\!\cdots\!75}a^{22}-\frac{32\!\cdots\!01}{17\!\cdots\!75}a^{21}+\frac{32\!\cdots\!03}{35\!\cdots\!55}a^{20}-\frac{14\!\cdots\!68}{17\!\cdots\!75}a^{19}-\frac{29\!\cdots\!76}{35\!\cdots\!55}a^{18}-\frac{38\!\cdots\!48}{17\!\cdots\!75}a^{17}-\frac{16\!\cdots\!41}{17\!\cdots\!75}a^{16}+\frac{68\!\cdots\!02}{17\!\cdots\!75}a^{15}+\frac{14\!\cdots\!63}{35\!\cdots\!55}a^{14}+\frac{95\!\cdots\!97}{17\!\cdots\!75}a^{13}+\frac{27\!\cdots\!19}{17\!\cdots\!75}a^{12}-\frac{74\!\cdots\!55}{70\!\cdots\!71}a^{11}-\frac{17\!\cdots\!06}{17\!\cdots\!75}a^{10}-\frac{26\!\cdots\!32}{17\!\cdots\!75}a^{9}+\frac{16\!\cdots\!94}{17\!\cdots\!75}a^{8}-\frac{95\!\cdots\!21}{17\!\cdots\!75}a^{7}+\frac{13\!\cdots\!22}{17\!\cdots\!75}a^{6}-\frac{32\!\cdots\!57}{17\!\cdots\!75}a^{5}+\frac{72\!\cdots\!99}{35\!\cdots\!55}a^{4}-\frac{10\!\cdots\!56}{35\!\cdots\!55}a^{3}+\frac{60\!\cdots\!73}{17\!\cdots\!75}a^{2}-\frac{42\!\cdots\!34}{17\!\cdots\!75}a-\frac{74\!\cdots\!31}{17\!\cdots\!75}$, $\frac{20\!\cdots\!47}{17\!\cdots\!75}a^{24}-\frac{54\!\cdots\!09}{17\!\cdots\!75}a^{23}+\frac{40\!\cdots\!98}{17\!\cdots\!75}a^{22}-\frac{58\!\cdots\!38}{17\!\cdots\!75}a^{21}+\frac{74\!\cdots\!82}{35\!\cdots\!55}a^{20}-\frac{21\!\cdots\!04}{17\!\cdots\!75}a^{19}-\frac{10\!\cdots\!11}{70\!\cdots\!71}a^{18}-\frac{63\!\cdots\!29}{17\!\cdots\!75}a^{17}+\frac{42\!\cdots\!67}{17\!\cdots\!75}a^{16}+\frac{15\!\cdots\!21}{17\!\cdots\!75}a^{15}+\frac{14\!\cdots\!04}{35\!\cdots\!55}a^{14}+\frac{13\!\cdots\!26}{17\!\cdots\!75}a^{13}-\frac{37\!\cdots\!73}{17\!\cdots\!75}a^{12}-\frac{83\!\cdots\!19}{35\!\cdots\!55}a^{11}+\frac{16\!\cdots\!17}{17\!\cdots\!75}a^{10}-\frac{56\!\cdots\!86}{17\!\cdots\!75}a^{9}+\frac{63\!\cdots\!22}{17\!\cdots\!75}a^{8}-\frac{40\!\cdots\!78}{17\!\cdots\!75}a^{7}+\frac{42\!\cdots\!06}{17\!\cdots\!75}a^{6}-\frac{22\!\cdots\!01}{17\!\cdots\!75}a^{5}+\frac{47\!\cdots\!44}{70\!\cdots\!71}a^{4}-\frac{12\!\cdots\!47}{35\!\cdots\!55}a^{3}+\frac{24\!\cdots\!44}{17\!\cdots\!75}a^{2}-\frac{17\!\cdots\!22}{17\!\cdots\!75}a+\frac{34\!\cdots\!12}{17\!\cdots\!75}$, $\frac{14\!\cdots\!02}{17\!\cdots\!75}a^{24}-\frac{38\!\cdots\!24}{17\!\cdots\!75}a^{23}+\frac{23\!\cdots\!63}{17\!\cdots\!75}a^{22}-\frac{46\!\cdots\!68}{17\!\cdots\!75}a^{21}+\frac{10\!\cdots\!54}{70\!\cdots\!71}a^{20}-\frac{15\!\cdots\!54}{17\!\cdots\!75}a^{19}-\frac{29\!\cdots\!58}{35\!\cdots\!55}a^{18}-\frac{46\!\cdots\!84}{17\!\cdots\!75}a^{17}-\frac{54\!\cdots\!63}{17\!\cdots\!75}a^{16}+\frac{10\!\cdots\!71}{17\!\cdots\!75}a^{15}+\frac{19\!\cdots\!36}{70\!\cdots\!71}a^{14}+\frac{11\!\cdots\!91}{17\!\cdots\!75}a^{13}-\frac{14\!\cdots\!68}{17\!\cdots\!75}a^{12}-\frac{55\!\cdots\!41}{35\!\cdots\!55}a^{11}+\frac{12\!\cdots\!72}{17\!\cdots\!75}a^{10}-\frac{44\!\cdots\!76}{17\!\cdots\!75}a^{9}+\frac{45\!\cdots\!57}{17\!\cdots\!75}a^{8}-\frac{34\!\cdots\!33}{17\!\cdots\!75}a^{7}+\frac{33\!\cdots\!01}{17\!\cdots\!75}a^{6}-\frac{18\!\cdots\!71}{17\!\cdots\!75}a^{5}+\frac{44\!\cdots\!92}{70\!\cdots\!71}a^{4}-\frac{96\!\cdots\!11}{35\!\cdots\!55}a^{3}+\frac{27\!\cdots\!39}{17\!\cdots\!75}a^{2}-\frac{14\!\cdots\!42}{17\!\cdots\!75}a+\frac{26\!\cdots\!82}{17\!\cdots\!75}$, $\frac{14\!\cdots\!76}{17\!\cdots\!75}a^{24}-\frac{36\!\cdots\!87}{17\!\cdots\!75}a^{23}-\frac{34\!\cdots\!06}{17\!\cdots\!75}a^{22}-\frac{45\!\cdots\!74}{17\!\cdots\!75}a^{21}+\frac{51\!\cdots\!56}{35\!\cdots\!55}a^{20}-\frac{12\!\cdots\!92}{17\!\cdots\!75}a^{19}-\frac{35\!\cdots\!33}{35\!\cdots\!55}a^{18}-\frac{48\!\cdots\!37}{17\!\cdots\!75}a^{17}-\frac{49\!\cdots\!84}{17\!\cdots\!75}a^{16}+\frac{10\!\cdots\!68}{17\!\cdots\!75}a^{15}+\frac{12\!\cdots\!13}{35\!\cdots\!55}a^{14}+\frac{11\!\cdots\!58}{17\!\cdots\!75}a^{13}+\frac{46\!\cdots\!11}{17\!\cdots\!75}a^{12}-\frac{10\!\cdots\!20}{70\!\cdots\!71}a^{11}+\frac{84\!\cdots\!96}{17\!\cdots\!75}a^{10}-\frac{41\!\cdots\!33}{17\!\cdots\!75}a^{9}+\frac{38\!\cdots\!51}{17\!\cdots\!75}a^{8}-\frac{29\!\cdots\!59}{17\!\cdots\!75}a^{7}+\frac{29\!\cdots\!58}{17\!\cdots\!75}a^{6}-\frac{15\!\cdots\!88}{17\!\cdots\!75}a^{5}+\frac{21\!\cdots\!46}{35\!\cdots\!55}a^{4}-\frac{95\!\cdots\!26}{35\!\cdots\!55}a^{3}+\frac{23\!\cdots\!82}{17\!\cdots\!75}a^{2}-\frac{12\!\cdots\!76}{17\!\cdots\!75}a+\frac{37\!\cdots\!41}{17\!\cdots\!75}$, $\frac{73\!\cdots\!54}{70\!\cdots\!71}a^{24}-\frac{82\!\cdots\!77}{35\!\cdots\!55}a^{23}-\frac{26\!\cdots\!19}{35\!\cdots\!55}a^{22}-\frac{13\!\cdots\!61}{35\!\cdots\!55}a^{21}+\frac{60\!\cdots\!47}{35\!\cdots\!55}a^{20}-\frac{15\!\cdots\!02}{35\!\cdots\!55}a^{19}-\frac{41\!\cdots\!12}{35\!\cdots\!55}a^{18}-\frac{13\!\cdots\!68}{35\!\cdots\!55}a^{17}-\frac{52\!\cdots\!71}{35\!\cdots\!55}a^{16}+\frac{23\!\cdots\!04}{35\!\cdots\!55}a^{15}+\frac{21\!\cdots\!52}{35\!\cdots\!55}a^{14}+\frac{37\!\cdots\!04}{35\!\cdots\!55}a^{13}+\frac{11\!\cdots\!07}{35\!\cdots\!55}a^{12}-\frac{63\!\cdots\!97}{35\!\cdots\!55}a^{11}+\frac{44\!\cdots\!69}{35\!\cdots\!55}a^{10}-\frac{11\!\cdots\!11}{35\!\cdots\!55}a^{9}+\frac{72\!\cdots\!76}{35\!\cdots\!55}a^{8}-\frac{65\!\cdots\!47}{35\!\cdots\!55}a^{7}+\frac{13\!\cdots\!75}{70\!\cdots\!71}a^{6}-\frac{28\!\cdots\!83}{35\!\cdots\!55}a^{5}+\frac{54\!\cdots\!97}{70\!\cdots\!71}a^{4}-\frac{80\!\cdots\!46}{35\!\cdots\!55}a^{3}+\frac{13\!\cdots\!82}{70\!\cdots\!71}a^{2}-\frac{28\!\cdots\!06}{35\!\cdots\!55}a+\frac{42\!\cdots\!76}{35\!\cdots\!55}$, $\frac{13\!\cdots\!22}{35\!\cdots\!55}a^{24}-\frac{35\!\cdots\!84}{35\!\cdots\!55}a^{23}+\frac{16\!\cdots\!21}{35\!\cdots\!55}a^{22}-\frac{38\!\cdots\!08}{35\!\cdots\!55}a^{21}+\frac{47\!\cdots\!38}{70\!\cdots\!71}a^{20}-\frac{14\!\cdots\!02}{35\!\cdots\!55}a^{19}-\frac{15\!\cdots\!88}{35\!\cdots\!55}a^{18}-\frac{40\!\cdots\!84}{35\!\cdots\!55}a^{17}+\frac{42\!\cdots\!71}{35\!\cdots\!55}a^{16}+\frac{98\!\cdots\!94}{35\!\cdots\!55}a^{15}+\frac{82\!\cdots\!46}{70\!\cdots\!71}a^{14}+\frac{89\!\cdots\!41}{35\!\cdots\!55}a^{13}-\frac{22\!\cdots\!82}{35\!\cdots\!55}a^{12}-\frac{51\!\cdots\!61}{70\!\cdots\!71}a^{11}+\frac{23\!\cdots\!60}{70\!\cdots\!71}a^{10}-\frac{37\!\cdots\!86}{35\!\cdots\!55}a^{9}+\frac{41\!\cdots\!37}{35\!\cdots\!55}a^{8}-\frac{29\!\cdots\!51}{35\!\cdots\!55}a^{7}+\frac{29\!\cdots\!93}{35\!\cdots\!55}a^{6}-\frac{16\!\cdots\!91}{35\!\cdots\!55}a^{5}+\frac{10\!\cdots\!74}{35\!\cdots\!55}a^{4}-\frac{48\!\cdots\!27}{35\!\cdots\!55}a^{3}+\frac{22\!\cdots\!04}{35\!\cdots\!55}a^{2}-\frac{14\!\cdots\!62}{35\!\cdots\!55}a+\frac{33\!\cdots\!03}{35\!\cdots\!55}$, $\frac{23\!\cdots\!56}{17\!\cdots\!75}a^{24}-\frac{53\!\cdots\!17}{17\!\cdots\!75}a^{23}-\frac{16\!\cdots\!96}{17\!\cdots\!75}a^{22}-\frac{71\!\cdots\!24}{17\!\cdots\!75}a^{21}+\frac{78\!\cdots\!02}{35\!\cdots\!55}a^{20}-\frac{12\!\cdots\!72}{17\!\cdots\!75}a^{19}-\frac{65\!\cdots\!14}{35\!\cdots\!55}a^{18}-\frac{82\!\cdots\!17}{17\!\cdots\!75}a^{17}-\frac{17\!\cdots\!34}{17\!\cdots\!75}a^{16}+\frac{16\!\cdots\!53}{17\!\cdots\!75}a^{15}+\frac{25\!\cdots\!81}{35\!\cdots\!55}a^{14}+\frac{19\!\cdots\!13}{17\!\cdots\!75}a^{13}+\frac{21\!\cdots\!16}{17\!\cdots\!75}a^{12}-\frac{88\!\cdots\!26}{35\!\cdots\!55}a^{11}+\frac{51\!\cdots\!41}{17\!\cdots\!75}a^{10}-\frac{61\!\cdots\!58}{17\!\cdots\!75}a^{9}+\frac{51\!\cdots\!86}{17\!\cdots\!75}a^{8}-\frac{33\!\cdots\!89}{17\!\cdots\!75}a^{7}+\frac{39\!\cdots\!93}{17\!\cdots\!75}a^{6}-\frac{17\!\cdots\!68}{17\!\cdots\!75}a^{5}+\frac{54\!\cdots\!91}{70\!\cdots\!71}a^{4}-\frac{11\!\cdots\!39}{35\!\cdots\!55}a^{3}+\frac{36\!\cdots\!82}{17\!\cdots\!75}a^{2}-\frac{21\!\cdots\!21}{17\!\cdots\!75}a+\frac{30\!\cdots\!21}{17\!\cdots\!75}$, $\frac{21\!\cdots\!21}{17\!\cdots\!75}a^{24}-\frac{64\!\cdots\!72}{17\!\cdots\!75}a^{23}+\frac{18\!\cdots\!69}{17\!\cdots\!75}a^{22}-\frac{58\!\cdots\!24}{17\!\cdots\!75}a^{21}+\frac{16\!\cdots\!10}{70\!\cdots\!71}a^{20}-\frac{35\!\cdots\!92}{17\!\cdots\!75}a^{19}-\frac{44\!\cdots\!66}{35\!\cdots\!55}a^{18}-\frac{57\!\cdots\!77}{17\!\cdots\!75}a^{17}+\frac{30\!\cdots\!61}{17\!\cdots\!75}a^{16}+\frac{16\!\cdots\!58}{17\!\cdots\!75}a^{15}+\frac{39\!\cdots\!41}{35\!\cdots\!55}a^{14}+\frac{10\!\cdots\!88}{17\!\cdots\!75}a^{13}-\frac{91\!\cdots\!04}{17\!\cdots\!75}a^{12}-\frac{85\!\cdots\!74}{35\!\cdots\!55}a^{11}+\frac{31\!\cdots\!96}{17\!\cdots\!75}a^{10}-\frac{63\!\cdots\!38}{17\!\cdots\!75}a^{9}+\frac{85\!\cdots\!71}{17\!\cdots\!75}a^{8}-\frac{62\!\cdots\!99}{17\!\cdots\!75}a^{7}+\frac{57\!\cdots\!68}{17\!\cdots\!75}a^{6}-\frac{37\!\cdots\!48}{17\!\cdots\!75}a^{5}+\frac{84\!\cdots\!62}{70\!\cdots\!71}a^{4}-\frac{46\!\cdots\!03}{70\!\cdots\!71}a^{3}+\frac{48\!\cdots\!97}{17\!\cdots\!75}a^{2}-\frac{31\!\cdots\!26}{17\!\cdots\!75}a+\frac{10\!\cdots\!31}{17\!\cdots\!75}$, $\frac{39\!\cdots\!74}{17\!\cdots\!75}a^{24}-\frac{15\!\cdots\!33}{17\!\cdots\!75}a^{23}+\frac{30\!\cdots\!86}{17\!\cdots\!75}a^{22}+\frac{15\!\cdots\!94}{17\!\cdots\!75}a^{21}+\frac{21\!\cdots\!00}{70\!\cdots\!71}a^{20}-\frac{23\!\cdots\!88}{17\!\cdots\!75}a^{19}+\frac{53\!\cdots\!14}{35\!\cdots\!55}a^{18}+\frac{20\!\cdots\!47}{17\!\cdots\!75}a^{17}+\frac{53\!\cdots\!74}{17\!\cdots\!75}a^{16}+\frac{24\!\cdots\!32}{17\!\cdots\!75}a^{15}-\frac{18\!\cdots\!08}{35\!\cdots\!55}a^{14}-\frac{90\!\cdots\!13}{17\!\cdots\!75}a^{13}-\frac{13\!\cdots\!51}{17\!\cdots\!75}a^{12}-\frac{11\!\cdots\!99}{35\!\cdots\!55}a^{11}+\frac{25\!\cdots\!64}{17\!\cdots\!75}a^{10}-\frac{51\!\cdots\!22}{17\!\cdots\!75}a^{9}+\frac{39\!\cdots\!44}{17\!\cdots\!75}a^{8}-\frac{22\!\cdots\!21}{17\!\cdots\!75}a^{7}+\frac{18\!\cdots\!17}{17\!\cdots\!75}a^{6}-\frac{18\!\cdots\!77}{17\!\cdots\!75}a^{5}+\frac{11\!\cdots\!14}{35\!\cdots\!55}a^{4}-\frac{13\!\cdots\!89}{35\!\cdots\!55}a^{3}+\frac{99\!\cdots\!48}{17\!\cdots\!75}a^{2}-\frac{17\!\cdots\!79}{17\!\cdots\!75}a+\frac{48\!\cdots\!09}{17\!\cdots\!75}$, $\frac{14\!\cdots\!39}{17\!\cdots\!75}a^{24}-\frac{35\!\cdots\!23}{17\!\cdots\!75}a^{23}-\frac{37\!\cdots\!49}{17\!\cdots\!75}a^{22}-\frac{43\!\cdots\!51}{17\!\cdots\!75}a^{21}+\frac{98\!\cdots\!92}{70\!\cdots\!71}a^{20}-\frac{11\!\cdots\!58}{17\!\cdots\!75}a^{19}-\frac{70\!\cdots\!77}{70\!\cdots\!71}a^{18}-\frac{46\!\cdots\!88}{17\!\cdots\!75}a^{17}-\frac{44\!\cdots\!06}{17\!\cdots\!75}a^{16}+\frac{10\!\cdots\!97}{17\!\cdots\!75}a^{15}+\frac{12\!\cdots\!16}{35\!\cdots\!55}a^{14}+\frac{10\!\cdots\!57}{17\!\cdots\!75}a^{13}-\frac{75\!\cdots\!76}{17\!\cdots\!75}a^{12}-\frac{53\!\cdots\!44}{35\!\cdots\!55}a^{11}+\frac{78\!\cdots\!24}{17\!\cdots\!75}a^{10}-\frac{39\!\cdots\!97}{17\!\cdots\!75}a^{9}+\frac{37\!\cdots\!04}{17\!\cdots\!75}a^{8}-\frac{26\!\cdots\!56}{17\!\cdots\!75}a^{7}+\frac{28\!\cdots\!77}{17\!\cdots\!75}a^{6}-\frac{13\!\cdots\!82}{17\!\cdots\!75}a^{5}+\frac{19\!\cdots\!31}{35\!\cdots\!55}a^{4}-\frac{82\!\cdots\!38}{35\!\cdots\!55}a^{3}+\frac{22\!\cdots\!63}{17\!\cdots\!75}a^{2}-\frac{13\!\cdots\!89}{17\!\cdots\!75}a+\frac{23\!\cdots\!04}{17\!\cdots\!75}$
| 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: | \( 68852831.13170123 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{10}\cdot 68852831.13170123 \cdot 1}{2\cdot\sqrt{39668558936237989671952184481937449}}\cr\approx \mathstrut & 0.530416782562469 \end{aligned}\] (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 + x^23 - 3*x^22 + 19*x^21 - 17*x^20 - 8*x^19 - 27*x^18 + 13*x^17 + 72*x^16 + 7*x^15 + 58*x^14 - 37*x^13 - 186*x^12 + 151*x^11 - 314*x^10 + 404*x^9 - 330*x^8 + 297*x^7 - 201*x^6 + 123*x^5 - 65*x^4 + 32*x^3 - 18*x^2 + 7*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 + x^23 - 3*x^22 + 19*x^21 - 17*x^20 - 8*x^19 - 27*x^18 + 13*x^17 + 72*x^16 + 7*x^15 + 58*x^14 - 37*x^13 - 186*x^12 + 151*x^11 - 314*x^10 + 404*x^9 - 330*x^8 + 297*x^7 - 201*x^6 + 123*x^5 - 65*x^4 + 32*x^3 - 18*x^2 + 7*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 + x^23 - 3*x^22 + 19*x^21 - 17*x^20 - 8*x^19 - 27*x^18 + 13*x^17 + 72*x^16 + 7*x^15 + 58*x^14 - 37*x^13 - 186*x^12 + 151*x^11 - 314*x^10 + 404*x^9 - 330*x^8 + 297*x^7 - 201*x^6 + 123*x^5 - 65*x^4 + 32*x^3 - 18*x^2 + 7*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 + x^23 - 3*x^22 + 19*x^21 - 17*x^20 - 8*x^19 - 27*x^18 + 13*x^17 + 72*x^16 + 7*x^15 + 58*x^14 - 37*x^13 - 186*x^12 + 151*x^11 - 314*x^10 + 404*x^9 - 330*x^8 + 297*x^7 - 201*x^6 + 123*x^5 - 65*x^4 + 32*x^3 - 18*x^2 + 7*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
| 5.5.923521.1, 5.1.8311689.1 |
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: | 10.0.224415603.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}{,}\,{\href{/padicField/2.5.0.1}{5} }$ | R | ${\href{/padicField/5.2.0.1}{2} }^{10}{,}\,{\href{/padicField/5.1.0.1}{1} }^{5}$ | ${\href{/padicField/7.5.0.1}{5} }^{5}$ | ${\href{/padicField/11.10.0.1}{10} }^{2}{,}\,{\href{/padicField/11.5.0.1}{5} }$ | ${\href{/padicField/13.5.0.1}{5} }^{5}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.5.0.1}{5} }^{5}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}{,}\,{\href{/padicField/23.5.0.1}{5} }$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.5.0.1}{5} }$ | R | ${\href{/padicField/37.5.0.1}{5} }^{5}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.5.0.1}{5} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.10.0.1}{10} }^{2}{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.5.0.1 | $x^{5} + 2 x + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 3.10.5.2 | $x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
| 3.10.5.2 | $x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(31\)
| Deg $25$ | $5$ | $5$ | $20$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.93.10t1.a.c | $1$ | $ 3 \cdot 31 $ | 10.0.207252522098163.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| * | 1.31.5t1.a.d | $1$ | $ 31 $ | 5.5.923521.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.31.5t1.a.b | $1$ | $ 31 $ | 5.5.923521.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| 1.93.10t1.a.d | $1$ | $ 3 \cdot 31 $ | 10.0.207252522098163.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.93.10t1.a.b | $1$ | $ 3 \cdot 31 $ | 10.0.207252522098163.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| 1.93.10t1.a.a | $1$ | $ 3 \cdot 31 $ | 10.0.207252522098163.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ | |
| * | 1.31.5t1.a.a | $1$ | $ 31 $ | 5.5.923521.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.31.5t1.a.c | $1$ | $ 31 $ | 5.5.923521.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 2.93.10t6.a.a | $2$ | $ 3 \cdot 31 $ | 25.5.39668558936237989671952184481937449.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.2883.10t6.a.b | $2$ | $ 3 \cdot 31^{2}$ | 25.5.39668558936237989671952184481937449.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.2883.5t2.a.a | $2$ | $ 3 \cdot 31^{2}$ | 5.1.8311689.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.2883.10t6.a.a | $2$ | $ 3 \cdot 31^{2}$ | 25.5.39668558936237989671952184481937449.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.2883.10t6.a.c | $2$ | $ 3 \cdot 31^{2}$ | 25.5.39668558936237989671952184481937449.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.2883.10t6.a.d | $2$ | $ 3 \cdot 31^{2}$ | 25.5.39668558936237989671952184481937449.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.93.10t6.a.c | $2$ | $ 3 \cdot 31 $ | 25.5.39668558936237989671952184481937449.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.93.10t6.a.b | $2$ | $ 3 \cdot 31 $ | 25.5.39668558936237989671952184481937449.1 | $C_5\times D_5$ (as 25T3) | $0$ | $0$ |
| * | 2.2883.5t2.a.b | $2$ | $ 3 \cdot 31^{2}$ | 5.1.8311689.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
| * | 2.93.10t6.a.d | $2$ | $ 3 \cdot 31 $ | 25.5.39668558936237989671952184481937449.1 | $C_5\times D_5$ (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 *.