Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 24*x^26 + 286*x^24 - 2327*x^22 + 12825*x^20 - 40336*x^18 + 39135*x^16 + 152588*x^14 - 531606*x^12 + 381077*x^10 + 2209555*x^8 - 10321444*x^6 + 18143914*x^4 - 8922944*x^2 + 1423325)
gp: K = bnfinit(y^28 - 24*y^26 + 286*y^24 - 2327*y^22 + 12825*y^20 - 40336*y^18 + 39135*y^16 + 152588*y^14 - 531606*y^12 + 381077*y^10 + 2209555*y^8 - 10321444*y^6 + 18143914*y^4 - 8922944*y^2 + 1423325, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 24*x^26 + 286*x^24 - 2327*x^22 + 12825*x^20 - 40336*x^18 + 39135*x^16 + 152588*x^14 - 531606*x^12 + 381077*x^10 + 2209555*x^8 - 10321444*x^6 + 18143914*x^4 - 8922944*x^2 + 1423325);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 24*x^26 + 286*x^24 - 2327*x^22 + 12825*x^20 - 40336*x^18 + 39135*x^16 + 152588*x^14 - 531606*x^12 + 381077*x^10 + 2209555*x^8 - 10321444*x^6 + 18143914*x^4 - 8922944*x^2 + 1423325)
\( x^{28} - 24 x^{26} + 286 x^{24} - 2327 x^{22} + 12825 x^{20} - 40336 x^{18} + 39135 x^{16} + \cdots + 1423325 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $28$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(537788304782666838980279974392697944146481853517\)
\(\medspace = 19^{14}\cdot 197^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(50.66\) | 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: | $19^{1/2}197^{1/2}\approx 61.180062111769715$ | ||
| Ramified primes: |
\(19\), \(197\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{197}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{10}a^{19}-\frac{1}{10}a^{17}-\frac{1}{5}a^{15}+\frac{3}{10}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}+\frac{2}{5}a^{7}+\frac{1}{10}a^{5}-\frac{1}{2}a^{4}-\frac{3}{10}a^{3}-\frac{1}{2}a^{2}-\frac{3}{10}a-\frac{1}{2}$, $\frac{1}{190}a^{20}+\frac{2}{95}a^{18}+\frac{1}{5}a^{16}-\frac{27}{190}a^{14}-\frac{5}{38}a^{12}-\frac{1}{2}a^{11}-\frac{3}{38}a^{10}-\frac{23}{95}a^{8}+\frac{33}{95}a^{6}-\frac{1}{2}a^{5}-\frac{13}{190}a^{4}-\frac{1}{2}a^{3}-\frac{63}{190}a^{2}-\frac{13}{38}$, $\frac{1}{190}a^{21}+\frac{2}{95}a^{19}+\frac{1}{5}a^{17}-\frac{27}{190}a^{15}-\frac{5}{38}a^{13}-\frac{1}{2}a^{12}-\frac{3}{38}a^{11}-\frac{23}{95}a^{9}+\frac{33}{95}a^{7}-\frac{1}{2}a^{6}-\frac{13}{190}a^{5}-\frac{1}{2}a^{4}-\frac{63}{190}a^{3}-\frac{13}{38}a$, $\frac{1}{190}a^{22}+\frac{11}{95}a^{18}+\frac{11}{190}a^{16}-\frac{6}{95}a^{14}-\frac{1}{2}a^{13}+\frac{17}{38}a^{12}-\frac{1}{2}a^{11}-\frac{81}{190}a^{10}-\frac{7}{38}a^{8}-\frac{1}{2}a^{7}-\frac{87}{190}a^{6}+\frac{42}{95}a^{4}-\frac{3}{190}a^{2}-\frac{5}{38}$, $\frac{1}{190}a^{23}+\frac{3}{190}a^{19}+\frac{3}{19}a^{17}+\frac{13}{95}a^{15}+\frac{14}{95}a^{13}-\frac{1}{2}a^{12}-\frac{81}{190}a^{11}-\frac{7}{38}a^{9}+\frac{27}{190}a^{7}-\frac{3}{19}a^{5}+\frac{27}{95}a^{3}-\frac{1}{2}a^{2}+\frac{16}{95}a$, $\frac{1}{334970}a^{24}-\frac{92}{167485}a^{22}-\frac{436}{167485}a^{20}+\frac{79787}{334970}a^{18}-\frac{37574}{167485}a^{16}-\frac{27872}{167485}a^{14}-\frac{1}{2}a^{13}+\frac{51242}{167485}a^{12}+\frac{76412}{167485}a^{10}+\frac{64541}{167485}a^{8}-\frac{143707}{334970}a^{6}-\frac{2536}{167485}a^{4}+\frac{39369}{334970}a^{2}-\frac{1}{2}a-\frac{6553}{66994}$, $\frac{1}{5694490}a^{25}+\frac{12157}{5694490}a^{23}-\frac{3962}{2847245}a^{21}+\frac{10804}{2847245}a^{19}-\frac{91081}{569449}a^{17}-\frac{917851}{5694490}a^{15}+\frac{42335}{569449}a^{13}+\frac{598863}{5694490}a^{11}-\frac{1}{2}a^{10}+\frac{261997}{2847245}a^{9}-\frac{1}{2}a^{8}+\frac{397986}{2847245}a^{7}-\frac{1}{2}a^{6}+\frac{98926}{569449}a^{5}-\frac{1}{2}a^{4}+\frac{1686011}{5694490}a^{3}-\frac{410413}{2847245}a-\frac{1}{2}$, $\frac{1}{25\!\cdots\!50}a^{26}-\frac{34\!\cdots\!33}{25\!\cdots\!50}a^{24}-\frac{41\!\cdots\!17}{25\!\cdots\!50}a^{22}+\frac{26\!\cdots\!88}{12\!\cdots\!25}a^{20}+\frac{25\!\cdots\!83}{12\!\cdots\!25}a^{18}-\frac{44\!\cdots\!38}{25\!\cdots\!25}a^{16}+\frac{55\!\cdots\!78}{25\!\cdots\!25}a^{14}-\frac{1}{2}a^{13}+\frac{55\!\cdots\!84}{12\!\cdots\!25}a^{12}+\frac{81\!\cdots\!57}{25\!\cdots\!50}a^{10}-\frac{1}{2}a^{9}-\frac{31\!\cdots\!19}{13\!\cdots\!50}a^{8}+\frac{10\!\cdots\!29}{25\!\cdots\!50}a^{6}-\frac{1}{2}a^{5}+\frac{20\!\cdots\!09}{50\!\cdots\!50}a^{4}-\frac{99\!\cdots\!91}{25\!\cdots\!50}a^{2}-\frac{1}{2}a-\frac{12\!\cdots\!69}{59\!\cdots\!30}$, $\frac{1}{25\!\cdots\!50}a^{27}+\frac{15\!\cdots\!67}{25\!\cdots\!50}a^{25}-\frac{46\!\cdots\!21}{12\!\cdots\!25}a^{23}-\frac{74\!\cdots\!37}{12\!\cdots\!25}a^{21}+\frac{57\!\cdots\!41}{25\!\cdots\!50}a^{19}+\frac{86\!\cdots\!79}{50\!\cdots\!50}a^{17}-\frac{16\!\cdots\!71}{26\!\cdots\!50}a^{15}+\frac{56\!\cdots\!93}{25\!\cdots\!50}a^{13}+\frac{87\!\cdots\!76}{30\!\cdots\!25}a^{11}-\frac{1}{2}a^{10}+\frac{18\!\cdots\!89}{25\!\cdots\!50}a^{9}-\frac{1}{2}a^{8}+\frac{12\!\cdots\!29}{25\!\cdots\!50}a^{7}+\frac{16\!\cdots\!52}{25\!\cdots\!25}a^{5}-\frac{55\!\cdots\!08}{12\!\cdots\!25}a^{3}-\frac{22\!\cdots\!49}{50\!\cdots\!05}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $5$ |
Class group and class number
$C_{29}$, which has order $29$ (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: | $13$ | 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: |
$\frac{76\!\cdots\!91}{23\!\cdots\!75}a^{27}+\frac{25\!\cdots\!71}{20\!\cdots\!42}a^{26}-\frac{14\!\cdots\!73}{23\!\cdots\!75}a^{25}-\frac{13\!\cdots\!59}{50\!\cdots\!05}a^{24}+\frac{12\!\cdots\!38}{23\!\cdots\!75}a^{23}+\frac{62\!\cdots\!85}{20\!\cdots\!42}a^{22}-\frac{16\!\cdots\!13}{46\!\cdots\!50}a^{21}-\frac{24\!\cdots\!95}{10\!\cdots\!21}a^{20}+\frac{55\!\cdots\!97}{46\!\cdots\!50}a^{19}+\frac{12\!\cdots\!51}{10\!\cdots\!10}a^{18}+\frac{10\!\cdots\!11}{97\!\cdots\!38}a^{17}-\frac{31\!\cdots\!71}{10\!\cdots\!10}a^{16}-\frac{80\!\cdots\!04}{46\!\cdots\!55}a^{15}+\frac{74\!\cdots\!83}{10\!\cdots\!10}a^{14}+\frac{13\!\cdots\!01}{46\!\cdots\!50}a^{13}+\frac{17\!\cdots\!13}{10\!\cdots\!10}a^{12}+\frac{90\!\cdots\!77}{23\!\cdots\!75}a^{11}-\frac{39\!\cdots\!59}{10\!\cdots\!21}a^{10}-\frac{23\!\cdots\!14}{12\!\cdots\!25}a^{9}+\frac{35\!\cdots\!16}{10\!\cdots\!21}a^{8}+\frac{22\!\cdots\!43}{46\!\cdots\!50}a^{7}+\frac{25\!\cdots\!89}{10\!\cdots\!10}a^{6}-\frac{95\!\cdots\!79}{92\!\cdots\!10}a^{5}-\frac{44\!\cdots\!07}{50\!\cdots\!05}a^{4}-\frac{42\!\cdots\!18}{13\!\cdots\!75}a^{3}+\frac{51\!\cdots\!91}{50\!\cdots\!05}a^{2}+\frac{22\!\cdots\!33}{92\!\cdots\!10}a-\frac{15\!\cdots\!33}{59\!\cdots\!13}$, $\frac{64\!\cdots\!73}{23\!\cdots\!50}a^{27}-\frac{51\!\cdots\!48}{25\!\cdots\!25}a^{26}-\frac{76\!\cdots\!17}{11\!\cdots\!75}a^{25}+\frac{24\!\cdots\!53}{50\!\cdots\!50}a^{24}+\frac{89\!\cdots\!67}{11\!\cdots\!75}a^{23}-\frac{28\!\cdots\!73}{50\!\cdots\!50}a^{22}-\frac{72\!\cdots\!51}{11\!\cdots\!75}a^{21}+\frac{22\!\cdots\!49}{50\!\cdots\!50}a^{20}+\frac{77\!\cdots\!93}{23\!\cdots\!50}a^{19}-\frac{60\!\cdots\!18}{25\!\cdots\!25}a^{18}-\frac{23\!\cdots\!59}{23\!\cdots\!75}a^{17}+\frac{17\!\cdots\!21}{24\!\cdots\!10}a^{16}+\frac{34\!\cdots\!93}{47\!\cdots\!50}a^{15}-\frac{24\!\cdots\!17}{50\!\cdots\!05}a^{14}+\frac{52\!\cdots\!82}{11\!\cdots\!75}a^{13}-\frac{42\!\cdots\!51}{13\!\cdots\!75}a^{12}-\frac{30\!\cdots\!89}{23\!\cdots\!50}a^{11}+\frac{46\!\cdots\!33}{50\!\cdots\!50}a^{10}+\frac{13\!\cdots\!47}{23\!\cdots\!50}a^{9}-\frac{95\!\cdots\!57}{25\!\cdots\!25}a^{8}+\frac{77\!\cdots\!43}{12\!\cdots\!50}a^{7}-\frac{23\!\cdots\!59}{50\!\cdots\!50}a^{6}-\frac{61\!\cdots\!49}{23\!\cdots\!75}a^{5}+\frac{93\!\cdots\!13}{50\!\cdots\!05}a^{4}+\frac{95\!\cdots\!57}{23\!\cdots\!50}a^{3}-\frac{14\!\cdots\!49}{50\!\cdots\!50}a^{2}-\frac{95\!\cdots\!07}{94\!\cdots\!03}a+\frac{89\!\cdots\!83}{11\!\cdots\!26}$, $\frac{24\!\cdots\!03}{87\!\cdots\!50}a^{27}-\frac{27\!\cdots\!58}{12\!\cdots\!25}a^{26}-\frac{29\!\cdots\!62}{43\!\cdots\!25}a^{25}+\frac{66\!\cdots\!37}{13\!\cdots\!50}a^{24}+\frac{33\!\cdots\!12}{43\!\cdots\!25}a^{23}-\frac{72\!\cdots\!89}{12\!\cdots\!25}a^{22}-\frac{26\!\cdots\!11}{43\!\cdots\!25}a^{21}+\frac{57\!\cdots\!17}{12\!\cdots\!25}a^{20}+\frac{28\!\cdots\!23}{87\!\cdots\!50}a^{19}-\frac{14\!\cdots\!67}{58\!\cdots\!50}a^{18}-\frac{84\!\cdots\!19}{87\!\cdots\!25}a^{17}+\frac{36\!\cdots\!81}{50\!\cdots\!50}a^{16}+\frac{53\!\cdots\!99}{87\!\cdots\!25}a^{15}-\frac{23\!\cdots\!61}{50\!\cdots\!50}a^{14}+\frac{19\!\cdots\!77}{43\!\cdots\!25}a^{13}-\frac{41\!\cdots\!19}{12\!\cdots\!25}a^{12}-\frac{55\!\cdots\!77}{43\!\cdots\!25}a^{11}+\frac{23\!\cdots\!13}{25\!\cdots\!50}a^{10}+\frac{19\!\cdots\!96}{43\!\cdots\!25}a^{9}-\frac{89\!\cdots\!49}{25\!\cdots\!50}a^{8}+\frac{56\!\cdots\!37}{87\!\cdots\!50}a^{7}-\frac{12\!\cdots\!89}{25\!\cdots\!50}a^{6}-\frac{22\!\cdots\!69}{87\!\cdots\!25}a^{5}+\frac{95\!\cdots\!91}{50\!\cdots\!50}a^{4}+\frac{16\!\cdots\!76}{43\!\cdots\!25}a^{3}-\frac{36\!\cdots\!72}{12\!\cdots\!25}a^{2}-\frac{11\!\cdots\!68}{17\!\cdots\!45}a+\frac{17\!\cdots\!02}{29\!\cdots\!65}$, $\frac{20\!\cdots\!06}{46\!\cdots\!55}a^{27}-\frac{47\!\cdots\!11}{50\!\cdots\!50}a^{26}-\frac{22\!\cdots\!17}{22\!\cdots\!10}a^{25}+\frac{59\!\cdots\!57}{26\!\cdots\!50}a^{24}+\frac{97\!\cdots\!23}{92\!\cdots\!10}a^{23}-\frac{13\!\cdots\!33}{50\!\cdots\!50}a^{22}-\frac{35\!\cdots\!44}{46\!\cdots\!55}a^{21}+\frac{53\!\cdots\!12}{25\!\cdots\!25}a^{20}+\frac{32\!\cdots\!99}{92\!\cdots\!10}a^{19}-\frac{57\!\cdots\!41}{50\!\cdots\!50}a^{18}-\frac{56\!\cdots\!91}{92\!\cdots\!10}a^{17}+\frac{39\!\cdots\!51}{11\!\cdots\!35}a^{16}-\frac{34\!\cdots\!21}{18\!\cdots\!22}a^{15}-\frac{89\!\cdots\!73}{50\!\cdots\!05}a^{14}+\frac{86\!\cdots\!94}{92\!\cdots\!11}a^{13}-\frac{44\!\cdots\!84}{25\!\cdots\!25}a^{12}-\frac{27\!\cdots\!29}{46\!\cdots\!55}a^{11}+\frac{11\!\cdots\!34}{25\!\cdots\!25}a^{10}-\frac{29\!\cdots\!47}{92\!\cdots\!10}a^{9}-\frac{20\!\cdots\!32}{25\!\cdots\!25}a^{8}+\frac{27\!\cdots\!87}{24\!\cdots\!45}a^{7}-\frac{12\!\cdots\!79}{50\!\cdots\!50}a^{6}-\frac{10\!\cdots\!22}{46\!\cdots\!55}a^{5}+\frac{92\!\cdots\!31}{10\!\cdots\!10}a^{4}-\frac{88\!\cdots\!46}{63\!\cdots\!05}a^{3}-\frac{31\!\cdots\!37}{25\!\cdots\!25}a^{2}+\frac{95\!\cdots\!97}{92\!\cdots\!10}a-\frac{74\!\cdots\!49}{31\!\cdots\!27}$, $\frac{31\!\cdots\!51}{12\!\cdots\!25}a^{27}+\frac{11\!\cdots\!68}{12\!\cdots\!25}a^{26}-\frac{74\!\cdots\!58}{12\!\cdots\!25}a^{25}-\frac{55\!\cdots\!13}{25\!\cdots\!50}a^{24}+\frac{17\!\cdots\!91}{25\!\cdots\!50}a^{23}+\frac{66\!\cdots\!63}{25\!\cdots\!50}a^{22}-\frac{37\!\cdots\!71}{66\!\cdots\!75}a^{21}-\frac{26\!\cdots\!82}{12\!\cdots\!25}a^{20}+\frac{38\!\cdots\!16}{12\!\cdots\!25}a^{19}+\frac{14\!\cdots\!13}{12\!\cdots\!25}a^{18}-\frac{24\!\cdots\!23}{26\!\cdots\!50}a^{17}-\frac{18\!\cdots\!41}{50\!\cdots\!50}a^{16}+\frac{18\!\cdots\!76}{25\!\cdots\!25}a^{15}+\frac{17\!\cdots\!11}{50\!\cdots\!50}a^{14}+\frac{10\!\cdots\!61}{25\!\cdots\!50}a^{13}+\frac{18\!\cdots\!49}{12\!\cdots\!25}a^{12}-\frac{15\!\cdots\!68}{12\!\cdots\!25}a^{11}-\frac{63\!\cdots\!24}{12\!\cdots\!25}a^{10}+\frac{16\!\cdots\!03}{25\!\cdots\!50}a^{9}+\frac{41\!\cdots\!27}{12\!\cdots\!25}a^{8}+\frac{71\!\cdots\!04}{12\!\cdots\!25}a^{7}+\frac{53\!\cdots\!19}{25\!\cdots\!50}a^{6}-\frac{61\!\cdots\!61}{25\!\cdots\!25}a^{5}-\frac{24\!\cdots\!98}{25\!\cdots\!25}a^{4}+\frac{98\!\cdots\!43}{25\!\cdots\!50}a^{3}+\frac{21\!\cdots\!12}{12\!\cdots\!25}a^{2}-\frac{23\!\cdots\!27}{20\!\cdots\!42}a-\frac{17\!\cdots\!81}{31\!\cdots\!70}$, $\frac{33\!\cdots\!73}{25\!\cdots\!50}a^{27}+\frac{17\!\cdots\!03}{12\!\cdots\!25}a^{26}-\frac{45\!\cdots\!77}{14\!\cdots\!50}a^{25}-\frac{64\!\cdots\!23}{25\!\cdots\!50}a^{24}+\frac{88\!\cdots\!09}{25\!\cdots\!50}a^{23}+\frac{33\!\cdots\!67}{13\!\cdots\!50}a^{22}-\frac{34\!\cdots\!26}{12\!\cdots\!25}a^{21}-\frac{22\!\cdots\!47}{12\!\cdots\!25}a^{20}+\frac{19\!\cdots\!47}{13\!\cdots\!50}a^{19}+\frac{19\!\cdots\!71}{25\!\cdots\!50}a^{18}-\frac{10\!\cdots\!34}{25\!\cdots\!25}a^{17}-\frac{61\!\cdots\!11}{50\!\cdots\!50}a^{16}+\frac{68\!\cdots\!34}{25\!\cdots\!25}a^{15}-\frac{21\!\cdots\!89}{50\!\cdots\!50}a^{14}+\frac{23\!\cdots\!82}{12\!\cdots\!25}a^{13}+\frac{11\!\cdots\!83}{25\!\cdots\!50}a^{12}-\frac{11\!\cdots\!89}{25\!\cdots\!50}a^{11}-\frac{27\!\cdots\!54}{12\!\cdots\!25}a^{10}-\frac{11\!\cdots\!53}{25\!\cdots\!50}a^{9}+\frac{12\!\cdots\!59}{25\!\cdots\!50}a^{8}+\frac{41\!\cdots\!01}{14\!\cdots\!50}a^{7}+\frac{15\!\cdots\!49}{25\!\cdots\!50}a^{6}-\frac{59\!\cdots\!23}{50\!\cdots\!50}a^{5}-\frac{91\!\cdots\!58}{25\!\cdots\!25}a^{4}+\frac{52\!\cdots\!07}{25\!\cdots\!50}a^{3}+\frac{89\!\cdots\!47}{30\!\cdots\!25}a^{2}-\frac{64\!\cdots\!53}{10\!\cdots\!10}a-\frac{13\!\cdots\!32}{29\!\cdots\!65}$, $\frac{10\!\cdots\!29}{50\!\cdots\!50}a^{27}+\frac{90\!\cdots\!47}{25\!\cdots\!50}a^{26}-\frac{24\!\cdots\!77}{50\!\cdots\!50}a^{25}-\frac{21\!\cdots\!01}{25\!\cdots\!50}a^{24}+\frac{14\!\cdots\!21}{25\!\cdots\!25}a^{23}+\frac{12\!\cdots\!63}{12\!\cdots\!25}a^{22}-\frac{24\!\cdots\!21}{50\!\cdots\!50}a^{21}-\frac{19\!\cdots\!53}{25\!\cdots\!50}a^{20}+\frac{66\!\cdots\!27}{25\!\cdots\!25}a^{19}+\frac{53\!\cdots\!51}{12\!\cdots\!25}a^{18}-\frac{41\!\cdots\!18}{50\!\cdots\!05}a^{17}-\frac{62\!\cdots\!47}{50\!\cdots\!50}a^{16}+\frac{38\!\cdots\!01}{50\!\cdots\!05}a^{15}+\frac{39\!\cdots\!87}{50\!\cdots\!50}a^{14}+\frac{16\!\cdots\!57}{50\!\cdots\!50}a^{13}+\frac{14\!\cdots\!71}{25\!\cdots\!50}a^{12}-\frac{27\!\cdots\!66}{25\!\cdots\!25}a^{11}-\frac{40\!\cdots\!21}{25\!\cdots\!50}a^{10}+\frac{18\!\cdots\!03}{25\!\cdots\!25}a^{9}+\frac{15\!\cdots\!83}{25\!\cdots\!50}a^{8}+\frac{11\!\cdots\!13}{25\!\cdots\!25}a^{7}+\frac{10\!\cdots\!77}{13\!\cdots\!50}a^{6}-\frac{10\!\cdots\!01}{50\!\cdots\!05}a^{5}-\frac{83\!\cdots\!76}{25\!\cdots\!25}a^{4}+\frac{96\!\cdots\!59}{26\!\cdots\!50}a^{3}+\frac{61\!\cdots\!24}{12\!\cdots\!25}a^{2}-\frac{15\!\cdots\!03}{10\!\cdots\!10}a-\frac{73\!\cdots\!39}{72\!\cdots\!65}$, $\frac{13\!\cdots\!14}{12\!\cdots\!25}a^{27}-\frac{22\!\cdots\!31}{13\!\cdots\!50}a^{26}-\frac{19\!\cdots\!61}{74\!\cdots\!25}a^{25}+\frac{52\!\cdots\!06}{12\!\cdots\!25}a^{24}+\frac{38\!\cdots\!12}{12\!\cdots\!25}a^{23}-\frac{65\!\cdots\!31}{12\!\cdots\!25}a^{22}-\frac{63\!\cdots\!97}{25\!\cdots\!50}a^{21}+\frac{55\!\cdots\!68}{12\!\cdots\!25}a^{20}+\frac{17\!\cdots\!74}{12\!\cdots\!25}a^{19}-\frac{33\!\cdots\!71}{13\!\cdots\!50}a^{18}-\frac{21\!\cdots\!53}{50\!\cdots\!50}a^{17}+\frac{43\!\cdots\!19}{50\!\cdots\!50}a^{16}+\frac{96\!\cdots\!84}{25\!\cdots\!25}a^{15}-\frac{58\!\cdots\!19}{50\!\cdots\!50}a^{14}+\frac{43\!\cdots\!79}{25\!\cdots\!50}a^{13}-\frac{62\!\cdots\!77}{25\!\cdots\!50}a^{12}-\frac{71\!\cdots\!52}{12\!\cdots\!25}a^{11}+\frac{14\!\cdots\!26}{12\!\cdots\!25}a^{10}+\frac{45\!\cdots\!46}{12\!\cdots\!25}a^{9}-\frac{31\!\cdots\!21}{25\!\cdots\!50}a^{8}+\frac{18\!\cdots\!93}{74\!\cdots\!25}a^{7}-\frac{93\!\cdots\!31}{25\!\cdots\!50}a^{6}-\frac{55\!\cdots\!73}{50\!\cdots\!50}a^{5}+\frac{28\!\cdots\!88}{13\!\cdots\!75}a^{4}+\frac{23\!\cdots\!01}{12\!\cdots\!25}a^{3}-\frac{11\!\cdots\!01}{25\!\cdots\!50}a^{2}-\frac{77\!\cdots\!65}{10\!\cdots\!21}a+\frac{18\!\cdots\!11}{59\!\cdots\!30}$, $\frac{10\!\cdots\!29}{50\!\cdots\!50}a^{27}-\frac{90\!\cdots\!47}{25\!\cdots\!50}a^{26}-\frac{24\!\cdots\!77}{50\!\cdots\!50}a^{25}+\frac{21\!\cdots\!01}{25\!\cdots\!50}a^{24}+\frac{14\!\cdots\!21}{25\!\cdots\!25}a^{23}-\frac{12\!\cdots\!63}{12\!\cdots\!25}a^{22}-\frac{24\!\cdots\!21}{50\!\cdots\!50}a^{21}+\frac{19\!\cdots\!53}{25\!\cdots\!50}a^{20}+\frac{66\!\cdots\!27}{25\!\cdots\!25}a^{19}-\frac{53\!\cdots\!51}{12\!\cdots\!25}a^{18}-\frac{41\!\cdots\!18}{50\!\cdots\!05}a^{17}+\frac{62\!\cdots\!47}{50\!\cdots\!50}a^{16}+\frac{38\!\cdots\!01}{50\!\cdots\!05}a^{15}-\frac{39\!\cdots\!87}{50\!\cdots\!50}a^{14}+\frac{16\!\cdots\!57}{50\!\cdots\!50}a^{13}-\frac{14\!\cdots\!71}{25\!\cdots\!50}a^{12}-\frac{27\!\cdots\!66}{25\!\cdots\!25}a^{11}+\frac{40\!\cdots\!21}{25\!\cdots\!50}a^{10}+\frac{18\!\cdots\!03}{25\!\cdots\!25}a^{9}-\frac{15\!\cdots\!83}{25\!\cdots\!50}a^{8}+\frac{11\!\cdots\!13}{25\!\cdots\!25}a^{7}-\frac{10\!\cdots\!77}{13\!\cdots\!50}a^{6}-\frac{10\!\cdots\!01}{50\!\cdots\!05}a^{5}+\frac{83\!\cdots\!76}{25\!\cdots\!25}a^{4}+\frac{96\!\cdots\!59}{26\!\cdots\!50}a^{3}-\frac{61\!\cdots\!24}{12\!\cdots\!25}a^{2}-\frac{15\!\cdots\!03}{10\!\cdots\!10}a+\frac{73\!\cdots\!39}{72\!\cdots\!65}$, $\frac{28\!\cdots\!38}{12\!\cdots\!25}a^{27}+\frac{27\!\cdots\!47}{25\!\cdots\!50}a^{26}-\frac{66\!\cdots\!29}{12\!\cdots\!25}a^{25}-\frac{31\!\cdots\!38}{12\!\cdots\!25}a^{24}+\frac{76\!\cdots\!29}{12\!\cdots\!25}a^{23}+\frac{35\!\cdots\!13}{12\!\cdots\!25}a^{22}-\frac{12\!\cdots\!49}{25\!\cdots\!50}a^{21}-\frac{27\!\cdots\!89}{12\!\cdots\!25}a^{20}+\frac{64\!\cdots\!91}{25\!\cdots\!50}a^{19}+\frac{28\!\cdots\!27}{25\!\cdots\!50}a^{18}-\frac{18\!\cdots\!18}{25\!\cdots\!25}a^{17}-\frac{36\!\cdots\!09}{11\!\cdots\!50}a^{16}+\frac{11\!\cdots\!63}{29\!\cdots\!50}a^{15}+\frac{51\!\cdots\!07}{50\!\cdots\!50}a^{14}+\frac{27\!\cdots\!77}{74\!\cdots\!25}a^{13}+\frac{20\!\cdots\!98}{12\!\cdots\!25}a^{12}-\frac{29\!\cdots\!99}{30\!\cdots\!25}a^{11}-\frac{95\!\cdots\!71}{25\!\cdots\!50}a^{10}+\frac{56\!\cdots\!89}{25\!\cdots\!50}a^{9}+\frac{14\!\cdots\!33}{25\!\cdots\!50}a^{8}+\frac{33\!\cdots\!58}{66\!\cdots\!75}a^{7}+\frac{28\!\cdots\!69}{12\!\cdots\!25}a^{6}-\frac{98\!\cdots\!81}{50\!\cdots\!50}a^{5}-\frac{42\!\cdots\!87}{50\!\cdots\!50}a^{4}+\frac{68\!\cdots\!59}{25\!\cdots\!50}a^{3}+\frac{13\!\cdots\!99}{12\!\cdots\!25}a^{2}-\frac{32\!\cdots\!90}{10\!\cdots\!21}a+\frac{11\!\cdots\!26}{29\!\cdots\!65}$, $\frac{16\!\cdots\!93}{20\!\cdots\!42}a^{27}-\frac{41\!\cdots\!99}{25\!\cdots\!50}a^{26}-\frac{24\!\cdots\!28}{12\!\cdots\!05}a^{25}+\frac{14\!\cdots\!46}{12\!\cdots\!25}a^{24}+\frac{23\!\cdots\!21}{10\!\cdots\!10}a^{23}-\frac{48\!\cdots\!17}{25\!\cdots\!50}a^{22}-\frac{18\!\cdots\!79}{10\!\cdots\!10}a^{21}+\frac{22\!\cdots\!38}{12\!\cdots\!25}a^{20}+\frac{50\!\cdots\!32}{50\!\cdots\!05}a^{19}-\frac{15\!\cdots\!17}{12\!\cdots\!25}a^{18}-\frac{31\!\cdots\!79}{10\!\cdots\!10}a^{17}+\frac{25\!\cdots\!99}{50\!\cdots\!50}a^{16}+\frac{52\!\cdots\!31}{20\!\cdots\!42}a^{15}-\frac{19\!\cdots\!77}{25\!\cdots\!25}a^{14}+\frac{12\!\cdots\!49}{10\!\cdots\!10}a^{13}-\frac{15\!\cdots\!66}{12\!\cdots\!25}a^{12}-\frac{40\!\cdots\!11}{10\!\cdots\!10}a^{11}+\frac{13\!\cdots\!57}{25\!\cdots\!50}a^{10}+\frac{13\!\cdots\!62}{50\!\cdots\!05}a^{9}-\frac{22\!\cdots\!11}{25\!\cdots\!50}a^{8}+\frac{18\!\cdots\!81}{10\!\cdots\!10}a^{7}-\frac{45\!\cdots\!23}{12\!\cdots\!25}a^{6}-\frac{40\!\cdots\!13}{50\!\cdots\!05}a^{5}+\frac{62\!\cdots\!59}{50\!\cdots\!50}a^{4}+\frac{13\!\cdots\!42}{10\!\cdots\!21}a^{3}-\frac{36\!\cdots\!83}{12\!\cdots\!25}a^{2}-\frac{54\!\cdots\!23}{10\!\cdots\!10}a+\frac{34\!\cdots\!27}{15\!\cdots\!35}$, $\frac{16\!\cdots\!93}{20\!\cdots\!42}a^{27}+\frac{41\!\cdots\!99}{25\!\cdots\!50}a^{26}-\frac{24\!\cdots\!28}{12\!\cdots\!05}a^{25}-\frac{14\!\cdots\!46}{12\!\cdots\!25}a^{24}+\frac{23\!\cdots\!21}{10\!\cdots\!10}a^{23}+\frac{48\!\cdots\!17}{25\!\cdots\!50}a^{22}-\frac{18\!\cdots\!79}{10\!\cdots\!10}a^{21}-\frac{22\!\cdots\!38}{12\!\cdots\!25}a^{20}+\frac{50\!\cdots\!32}{50\!\cdots\!05}a^{19}+\frac{15\!\cdots\!17}{12\!\cdots\!25}a^{18}-\frac{31\!\cdots\!79}{10\!\cdots\!10}a^{17}-\frac{25\!\cdots\!99}{50\!\cdots\!50}a^{16}+\frac{52\!\cdots\!31}{20\!\cdots\!42}a^{15}+\frac{19\!\cdots\!77}{25\!\cdots\!25}a^{14}+\frac{12\!\cdots\!49}{10\!\cdots\!10}a^{13}+\frac{15\!\cdots\!66}{12\!\cdots\!25}a^{12}-\frac{40\!\cdots\!11}{10\!\cdots\!10}a^{11}-\frac{13\!\cdots\!57}{25\!\cdots\!50}a^{10}+\frac{13\!\cdots\!62}{50\!\cdots\!05}a^{9}+\frac{22\!\cdots\!11}{25\!\cdots\!50}a^{8}+\frac{18\!\cdots\!81}{10\!\cdots\!10}a^{7}+\frac{45\!\cdots\!23}{12\!\cdots\!25}a^{6}-\frac{40\!\cdots\!13}{50\!\cdots\!05}a^{5}-\frac{62\!\cdots\!59}{50\!\cdots\!50}a^{4}+\frac{13\!\cdots\!42}{10\!\cdots\!21}a^{3}+\frac{36\!\cdots\!83}{12\!\cdots\!25}a^{2}-\frac{54\!\cdots\!23}{10\!\cdots\!10}a-\frac{34\!\cdots\!27}{15\!\cdots\!35}$, $\frac{31\!\cdots\!83}{29\!\cdots\!50}a^{27}+\frac{14\!\cdots\!21}{12\!\cdots\!25}a^{26}-\frac{15\!\cdots\!38}{58\!\cdots\!75}a^{25}-\frac{69\!\cdots\!43}{12\!\cdots\!25}a^{24}+\frac{78\!\cdots\!19}{25\!\cdots\!25}a^{23}+\frac{10\!\cdots\!43}{12\!\cdots\!25}a^{22}-\frac{34\!\cdots\!48}{13\!\cdots\!75}a^{21}-\frac{20\!\cdots\!83}{25\!\cdots\!50}a^{20}+\frac{72\!\cdots\!41}{50\!\cdots\!50}a^{19}+\frac{13\!\cdots\!47}{25\!\cdots\!50}a^{18}-\frac{23\!\cdots\!09}{50\!\cdots\!05}a^{17}-\frac{10\!\cdots\!77}{50\!\cdots\!50}a^{16}+\frac{23\!\cdots\!04}{50\!\cdots\!05}a^{15}+\frac{67\!\cdots\!36}{25\!\cdots\!25}a^{14}+\frac{89\!\cdots\!63}{50\!\cdots\!50}a^{13}+\frac{20\!\cdots\!81}{25\!\cdots\!50}a^{12}-\frac{32\!\cdots\!73}{50\!\cdots\!50}a^{11}-\frac{72\!\cdots\!81}{25\!\cdots\!50}a^{10}+\frac{12\!\cdots\!82}{25\!\cdots\!25}a^{9}+\frac{28\!\cdots\!13}{25\!\cdots\!50}a^{8}+\frac{13\!\cdots\!29}{50\!\cdots\!50}a^{7}+\frac{21\!\cdots\!43}{25\!\cdots\!50}a^{6}-\frac{12\!\cdots\!69}{10\!\cdots\!10}a^{5}-\frac{11\!\cdots\!31}{25\!\cdots\!25}a^{4}+\frac{10\!\cdots\!89}{50\!\cdots\!50}a^{3}+\frac{12\!\cdots\!89}{12\!\cdots\!25}a^{2}-\frac{30\!\cdots\!96}{50\!\cdots\!05}a-\frac{16\!\cdots\!73}{59\!\cdots\!30}$
| 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: | \( 337496841957.8353 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 337496841957.8353 \cdot 29}{2\cdot\sqrt{537788304782666838980279974392697944146481853517}}\cr\approx \mathstrut & 0.997355979843452 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^28 - 24*x^26 + 286*x^24 - 2327*x^22 + 12825*x^20 - 40336*x^18 + 39135*x^16 + 152588*x^14 - 531606*x^12 + 381077*x^10 + 2209555*x^8 - 10321444*x^6 + 18143914*x^4 - 8922944*x^2 + 1423325)
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^28 - 24*x^26 + 286*x^24 - 2327*x^22 + 12825*x^20 - 40336*x^18 + 39135*x^16 + 152588*x^14 - 531606*x^12 + 381077*x^10 + 2209555*x^8 - 10321444*x^6 + 18143914*x^4 - 8922944*x^2 + 1423325, 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^28 - 24*x^26 + 286*x^24 - 2327*x^22 + 12825*x^20 - 40336*x^18 + 39135*x^16 + 152588*x^14 - 531606*x^12 + 381077*x^10 + 2209555*x^8 - 10321444*x^6 + 18143914*x^4 - 8922944*x^2 + 1423325);
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^28 - 24*x^26 + 286*x^24 - 2327*x^22 + 12825*x^20 - 40336*x^18 + 39135*x^16 + 152588*x^14 - 531606*x^12 + 381077*x^10 + 2209555*x^8 - 10321444*x^6 + 18143914*x^4 - 8922944*x^2 + 1423325);
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
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 56 |
| The 17 conjugacy class representatives for $D_{28}$ |
| Character table for $D_{28}$ |
Intermediate fields
| \(\Q(\sqrt{-19}) \), 4.0.71117.1, 7.1.52439613407.1, 14.0.52248348031236668805331.1 |
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 28 sibling: | 28.2.5576015581167650909427113418703236578781943428571.1 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $28$ | $28$ | ${\href{/padicField/5.2.0.1}{2} }^{13}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.14.0.1}{14} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{13}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $28$ | ${\href{/padicField/17.2.0.1}{2} }^{13}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/23.14.0.1}{14} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{14}$ | $28$ | ${\href{/padicField/37.2.0.1}{2} }^{14}$ | ${\href{/padicField/41.2.0.1}{2} }^{14}$ | ${\href{/padicField/43.2.0.1}{2} }^{14}$ | ${\href{/padicField/47.7.0.1}{7} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{14}$ | ${\href{/padicField/59.2.0.1}{2} }^{14}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(19\)
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(197\)
| $\Q_{197}$ | $x + 195$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{197}$ | $x + 195$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 197.2.1.1 | $x^{2} + 197$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 197.2.1.1 | $x^{2} + 197$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 197.2.1.1 | $x^{2} + 197$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 197.2.1.1 | $x^{2} + 197$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 197.2.1.1 | $x^{2} + 197$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 197.2.1.1 | $x^{2} + 197$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 197.2.1.1 | $x^{2} + 197$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 197.2.1.1 | $x^{2} + 197$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 197.2.1.1 | $x^{2} + 197$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 197.2.1.1 | $x^{2} + 197$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 197.2.1.1 | $x^{2} + 197$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 197.2.1.1 | $x^{2} + 197$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 197.2.1.1 | $x^{2} + 197$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
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 *.