Normalized defining polynomial
\( x^{20} - 4 x^{19} + 50 x^{18} - 160 x^{17} + 1183 x^{16} - 3204 x^{15} + 16682 x^{14} - 38000 x^{13} + \cdots + 95829823 \)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(97756416090547441671665711234313879552\)
\(\medspace = 2^{55}\cdot 3^{10}\cdot 11^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(79.34\) | 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: | $2^{11/4}3^{1/2}11^{4/5}\approx 79.34275267180627$ | ||
| Ramified primes: |
\(2\), \(3\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $20$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(528=2^{4}\cdot 3\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{528}(1,·)$, $\chi_{528}(389,·)$, $\chi_{528}(289,·)$, $\chi_{528}(265,·)$, $\chi_{528}(269,·)$, $\chi_{528}(25,·)$, $\chi_{528}(221,·)$, $\chi_{528}(5,·)$, $\chi_{528}(97,·)$, $\chi_{528}(485,·)$, $\chi_{528}(433,·)$, $\chi_{528}(169,·)$, $\chi_{528}(317,·)$, $\chi_{528}(49,·)$, $\chi_{528}(245,·)$, $\chi_{528}(361,·)$, $\chi_{528}(313,·)$, $\chi_{528}(509,·)$, $\chi_{528}(125,·)$, $\chi_{528}(53,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{69}a^{15}+\frac{5}{69}a^{14}-\frac{3}{23}a^{13}+\frac{7}{69}a^{12}+\frac{5}{69}a^{10}-\frac{11}{23}a^{9}+\frac{25}{69}a^{8}+\frac{32}{69}a^{7}-\frac{8}{23}a^{6}+\frac{28}{69}a^{5}+\frac{34}{69}a^{4}-\frac{5}{23}a^{3}-\frac{4}{23}a^{2}-\frac{9}{23}a+\frac{2}{23}$, $\frac{1}{69}a^{16}-\frac{11}{69}a^{14}+\frac{2}{23}a^{13}+\frac{11}{69}a^{12}+\frac{5}{69}a^{11}+\frac{11}{69}a^{10}+\frac{2}{23}a^{9}-\frac{8}{23}a^{8}-\frac{13}{69}a^{6}+\frac{32}{69}a^{5}-\frac{1}{69}a^{4}-\frac{29}{69}a^{3}+\frac{11}{23}a^{2}+\frac{26}{69}a-\frac{7}{69}$, $\frac{1}{69}a^{17}-\frac{8}{69}a^{14}+\frac{4}{69}a^{13}-\frac{10}{69}a^{12}+\frac{11}{69}a^{11}-\frac{8}{69}a^{10}+\frac{4}{69}a^{9}-\frac{1}{69}a^{8}+\frac{17}{69}a^{7}-\frac{25}{69}a^{6}-\frac{5}{23}a^{5}+\frac{1}{3}a^{4}-\frac{17}{69}a^{3}-\frac{14}{69}a^{2}-\frac{28}{69}a-\frac{26}{69}$, $\frac{1}{31\!\cdots\!81}a^{18}+\frac{76\!\cdots\!58}{10\!\cdots\!27}a^{17}+\frac{21\!\cdots\!04}{31\!\cdots\!81}a^{16}-\frac{40\!\cdots\!76}{31\!\cdots\!81}a^{15}-\frac{58\!\cdots\!55}{46\!\cdots\!49}a^{14}-\frac{26\!\cdots\!67}{31\!\cdots\!81}a^{13}-\frac{22\!\cdots\!39}{31\!\cdots\!81}a^{12}+\frac{23\!\cdots\!34}{31\!\cdots\!81}a^{11}+\frac{46\!\cdots\!84}{31\!\cdots\!81}a^{10}+\frac{99\!\cdots\!14}{10\!\cdots\!27}a^{9}-\frac{27\!\cdots\!23}{10\!\cdots\!27}a^{8}+\frac{94\!\cdots\!30}{31\!\cdots\!81}a^{7}+\frac{15\!\cdots\!22}{10\!\cdots\!27}a^{6}-\frac{13\!\cdots\!51}{31\!\cdots\!81}a^{5}-\frac{91\!\cdots\!35}{31\!\cdots\!81}a^{4}+\frac{57\!\cdots\!28}{31\!\cdots\!81}a^{3}-\frac{80\!\cdots\!02}{31\!\cdots\!81}a^{2}-\frac{16\!\cdots\!19}{10\!\cdots\!27}a-\frac{10\!\cdots\!78}{31\!\cdots\!81}$, $\frac{1}{10\!\cdots\!07}a^{19}-\frac{12\!\cdots\!96}{10\!\cdots\!07}a^{18}+\frac{39\!\cdots\!55}{10\!\cdots\!07}a^{17}-\frac{13\!\cdots\!18}{36\!\cdots\!69}a^{16}+\frac{15\!\cdots\!25}{36\!\cdots\!69}a^{15}-\frac{17\!\cdots\!60}{10\!\cdots\!07}a^{14}+\frac{13\!\cdots\!38}{10\!\cdots\!07}a^{13}-\frac{13\!\cdots\!21}{10\!\cdots\!07}a^{12}+\frac{14\!\cdots\!68}{10\!\cdots\!07}a^{11}+\frac{25\!\cdots\!52}{10\!\cdots\!07}a^{10}+\frac{15\!\cdots\!84}{10\!\cdots\!07}a^{9}+\frac{53\!\cdots\!72}{10\!\cdots\!07}a^{8}+\frac{42\!\cdots\!05}{36\!\cdots\!69}a^{7}-\frac{22\!\cdots\!15}{10\!\cdots\!07}a^{6}+\frac{12\!\cdots\!92}{36\!\cdots\!69}a^{5}+\frac{41\!\cdots\!49}{10\!\cdots\!07}a^{4}+\frac{20\!\cdots\!20}{10\!\cdots\!07}a^{3}+\frac{29\!\cdots\!43}{36\!\cdots\!69}a^{2}+\frac{13\!\cdots\!69}{36\!\cdots\!69}a+\frac{50\!\cdots\!08}{10\!\cdots\!07}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{131042}$, which has order $131042$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: |
$\frac{62\!\cdots\!56}{36\!\cdots\!69}a^{19}-\frac{42\!\cdots\!52}{36\!\cdots\!69}a^{18}+\frac{25\!\cdots\!04}{36\!\cdots\!69}a^{17}-\frac{66\!\cdots\!69}{36\!\cdots\!69}a^{16}+\frac{52\!\cdots\!76}{36\!\cdots\!69}a^{15}+\frac{48\!\cdots\!80}{36\!\cdots\!69}a^{14}+\frac{63\!\cdots\!84}{36\!\cdots\!69}a^{13}+\frac{28\!\cdots\!60}{36\!\cdots\!69}a^{12}+\frac{49\!\cdots\!48}{36\!\cdots\!69}a^{11}+\frac{36\!\cdots\!08}{36\!\cdots\!69}a^{10}+\frac{25\!\cdots\!60}{36\!\cdots\!69}a^{9}+\frac{21\!\cdots\!55}{36\!\cdots\!69}a^{8}+\frac{12\!\cdots\!40}{36\!\cdots\!69}a^{7}+\frac{80\!\cdots\!64}{36\!\cdots\!69}a^{6}+\frac{59\!\cdots\!48}{36\!\cdots\!69}a^{5}+\frac{59\!\cdots\!32}{36\!\cdots\!69}a^{4}+\frac{13\!\cdots\!36}{36\!\cdots\!69}a^{3}+\frac{86\!\cdots\!96}{36\!\cdots\!69}a^{2}+\frac{16\!\cdots\!04}{36\!\cdots\!69}a-\frac{16\!\cdots\!61}{36\!\cdots\!69}$, $\frac{49\!\cdots\!32}{36\!\cdots\!69}a^{19}+\frac{69\!\cdots\!98}{36\!\cdots\!69}a^{18}+\frac{16\!\cdots\!04}{36\!\cdots\!69}a^{17}+\frac{43\!\cdots\!68}{36\!\cdots\!69}a^{16}+\frac{27\!\cdots\!68}{36\!\cdots\!69}a^{15}+\frac{11\!\cdots\!26}{36\!\cdots\!69}a^{14}+\frac{24\!\cdots\!72}{36\!\cdots\!69}a^{13}+\frac{16\!\cdots\!16}{36\!\cdots\!69}a^{12}+\frac{98\!\cdots\!24}{36\!\cdots\!69}a^{11}+\frac{14\!\cdots\!74}{36\!\cdots\!69}a^{10}-\frac{15\!\cdots\!88}{36\!\cdots\!69}a^{9}+\frac{85\!\cdots\!76}{36\!\cdots\!69}a^{8}-\frac{39\!\cdots\!64}{36\!\cdots\!69}a^{7}+\frac{39\!\cdots\!12}{36\!\cdots\!69}a^{6}+\frac{34\!\cdots\!60}{36\!\cdots\!69}a^{5}+\frac{20\!\cdots\!55}{36\!\cdots\!69}a^{4}-\frac{40\!\cdots\!36}{36\!\cdots\!69}a^{3}+\frac{35\!\cdots\!26}{36\!\cdots\!69}a^{2}-\frac{10\!\cdots\!04}{36\!\cdots\!69}a+\frac{75\!\cdots\!12}{36\!\cdots\!69}$, $\frac{27\!\cdots\!96}{36\!\cdots\!69}a^{19}-\frac{99\!\cdots\!16}{36\!\cdots\!69}a^{18}+\frac{13\!\cdots\!76}{36\!\cdots\!69}a^{17}-\frac{41\!\cdots\!12}{36\!\cdots\!69}a^{16}+\frac{33\!\cdots\!20}{36\!\cdots\!69}a^{15}-\frac{84\!\cdots\!48}{36\!\cdots\!69}a^{14}+\frac{48\!\cdots\!44}{36\!\cdots\!69}a^{13}-\frac{10\!\cdots\!69}{36\!\cdots\!69}a^{12}+\frac{44\!\cdots\!36}{36\!\cdots\!69}a^{11}-\frac{77\!\cdots\!74}{36\!\cdots\!69}a^{10}+\frac{27\!\cdots\!56}{36\!\cdots\!69}a^{9}-\frac{39\!\cdots\!56}{36\!\cdots\!69}a^{8}+\frac{13\!\cdots\!04}{36\!\cdots\!69}a^{7}-\frac{16\!\cdots\!82}{36\!\cdots\!69}a^{6}+\frac{59\!\cdots\!32}{36\!\cdots\!69}a^{5}-\frac{59\!\cdots\!53}{36\!\cdots\!69}a^{4}+\frac{17\!\cdots\!60}{36\!\cdots\!69}a^{3}-\frac{11\!\cdots\!94}{36\!\cdots\!69}a^{2}+\frac{17\!\cdots\!28}{36\!\cdots\!69}a-\frac{13\!\cdots\!24}{36\!\cdots\!69}$, $\frac{27\!\cdots\!96}{36\!\cdots\!69}a^{19}-\frac{99\!\cdots\!16}{36\!\cdots\!69}a^{18}+\frac{13\!\cdots\!76}{36\!\cdots\!69}a^{17}-\frac{41\!\cdots\!12}{36\!\cdots\!69}a^{16}+\frac{33\!\cdots\!20}{36\!\cdots\!69}a^{15}-\frac{84\!\cdots\!48}{36\!\cdots\!69}a^{14}+\frac{48\!\cdots\!44}{36\!\cdots\!69}a^{13}-\frac{10\!\cdots\!69}{36\!\cdots\!69}a^{12}+\frac{44\!\cdots\!36}{36\!\cdots\!69}a^{11}-\frac{77\!\cdots\!74}{36\!\cdots\!69}a^{10}+\frac{27\!\cdots\!56}{36\!\cdots\!69}a^{9}-\frac{39\!\cdots\!56}{36\!\cdots\!69}a^{8}+\frac{13\!\cdots\!04}{36\!\cdots\!69}a^{7}-\frac{16\!\cdots\!82}{36\!\cdots\!69}a^{6}+\frac{59\!\cdots\!32}{36\!\cdots\!69}a^{5}-\frac{59\!\cdots\!53}{36\!\cdots\!69}a^{4}+\frac{17\!\cdots\!60}{36\!\cdots\!69}a^{3}-\frac{11\!\cdots\!94}{36\!\cdots\!69}a^{2}+\frac{17\!\cdots\!28}{36\!\cdots\!69}a+\frac{22\!\cdots\!45}{36\!\cdots\!69}$, $\frac{29015885365676}{73\!\cdots\!63}a^{19}+\frac{344395780525762}{73\!\cdots\!63}a^{18}-\frac{381019137243520}{73\!\cdots\!63}a^{17}+\frac{13\!\cdots\!11}{73\!\cdots\!63}a^{16}-\frac{76\!\cdots\!20}{24\!\cdots\!21}a^{15}+\frac{25\!\cdots\!14}{73\!\cdots\!63}a^{14}-\frac{47\!\cdots\!30}{73\!\cdots\!63}a^{13}+\frac{28\!\cdots\!17}{73\!\cdots\!63}a^{12}-\frac{46\!\cdots\!76}{73\!\cdots\!63}a^{11}+\frac{18\!\cdots\!56}{73\!\cdots\!63}a^{10}-\frac{25\!\cdots\!56}{73\!\cdots\!63}a^{9}+\frac{29\!\cdots\!05}{24\!\cdots\!21}a^{8}-\frac{95\!\cdots\!68}{73\!\cdots\!63}a^{7}+\frac{14\!\cdots\!50}{24\!\cdots\!21}a^{6}-\frac{15\!\cdots\!02}{24\!\cdots\!21}a^{5}+\frac{12\!\cdots\!66}{73\!\cdots\!63}a^{4}-\frac{81\!\cdots\!16}{73\!\cdots\!63}a^{3}+\frac{27\!\cdots\!21}{73\!\cdots\!63}a^{2}-\frac{38\!\cdots\!48}{24\!\cdots\!21}a+\frac{33\!\cdots\!20}{73\!\cdots\!63}$, $\frac{14\!\cdots\!32}{10\!\cdots\!07}a^{19}+\frac{15\!\cdots\!76}{10\!\cdots\!07}a^{18}+\frac{50\!\cdots\!92}{10\!\cdots\!07}a^{17}+\frac{10\!\cdots\!25}{10\!\cdots\!07}a^{16}+\frac{28\!\cdots\!48}{36\!\cdots\!69}a^{15}+\frac{29\!\cdots\!32}{10\!\cdots\!07}a^{14}+\frac{80\!\cdots\!86}{10\!\cdots\!07}a^{13}+\frac{45\!\cdots\!35}{10\!\cdots\!07}a^{12}+\frac{36\!\cdots\!36}{10\!\cdots\!07}a^{11}+\frac{42\!\cdots\!38}{10\!\cdots\!07}a^{10}+\frac{33\!\cdots\!20}{10\!\cdots\!07}a^{9}+\frac{81\!\cdots\!31}{36\!\cdots\!69}a^{8}+\frac{23\!\cdots\!60}{10\!\cdots\!07}a^{7}+\frac{37\!\cdots\!62}{36\!\cdots\!69}a^{6}+\frac{57\!\cdots\!38}{36\!\cdots\!69}a^{5}+\frac{58\!\cdots\!91}{10\!\cdots\!07}a^{4}-\frac{11\!\cdots\!84}{10\!\cdots\!07}a^{3}+\frac{10\!\cdots\!09}{10\!\cdots\!07}a^{2}-\frac{46\!\cdots\!32}{36\!\cdots\!69}a+\frac{69\!\cdots\!49}{10\!\cdots\!07}$, $\frac{97\!\cdots\!12}{10\!\cdots\!07}a^{19}-\frac{14\!\cdots\!43}{36\!\cdots\!69}a^{18}+\frac{52\!\cdots\!28}{10\!\cdots\!07}a^{17}-\frac{19\!\cdots\!32}{10\!\cdots\!07}a^{16}+\frac{13\!\cdots\!12}{10\!\cdots\!07}a^{15}-\frac{42\!\cdots\!78}{10\!\cdots\!07}a^{14}+\frac{65\!\cdots\!08}{36\!\cdots\!69}a^{13}-\frac{53\!\cdots\!80}{10\!\cdots\!07}a^{12}+\frac{18\!\cdots\!68}{10\!\cdots\!07}a^{11}-\frac{41\!\cdots\!81}{10\!\cdots\!07}a^{10}+\frac{11\!\cdots\!70}{10\!\cdots\!07}a^{9}-\frac{19\!\cdots\!59}{10\!\cdots\!07}a^{8}+\frac{14\!\cdots\!40}{36\!\cdots\!69}a^{7}-\frac{62\!\cdots\!08}{10\!\cdots\!07}a^{6}+\frac{15\!\cdots\!34}{10\!\cdots\!07}a^{5}-\frac{72\!\cdots\!13}{36\!\cdots\!69}a^{4}+\frac{42\!\cdots\!32}{10\!\cdots\!07}a^{3}-\frac{34\!\cdots\!29}{10\!\cdots\!07}a^{2}+\frac{37\!\cdots\!22}{10\!\cdots\!07}a+\frac{59\!\cdots\!22}{10\!\cdots\!07}$, $\frac{20\!\cdots\!16}{10\!\cdots\!07}a^{19}-\frac{80\!\cdots\!62}{10\!\cdots\!07}a^{18}+\frac{10\!\cdots\!56}{10\!\cdots\!07}a^{17}-\frac{34\!\cdots\!65}{10\!\cdots\!07}a^{16}+\frac{87\!\cdots\!72}{36\!\cdots\!69}a^{15}-\frac{71\!\cdots\!14}{10\!\cdots\!07}a^{14}+\frac{38\!\cdots\!50}{10\!\cdots\!07}a^{13}-\frac{89\!\cdots\!67}{10\!\cdots\!07}a^{12}+\frac{35\!\cdots\!56}{10\!\cdots\!07}a^{11}-\frac{69\!\cdots\!00}{10\!\cdots\!07}a^{10}+\frac{21\!\cdots\!72}{10\!\cdots\!07}a^{9}-\frac{11\!\cdots\!44}{36\!\cdots\!69}a^{8}+\frac{94\!\cdots\!00}{10\!\cdots\!07}a^{7}-\frac{47\!\cdots\!10}{36\!\cdots\!69}a^{6}+\frac{12\!\cdots\!66}{36\!\cdots\!69}a^{5}-\frac{55\!\cdots\!50}{10\!\cdots\!07}a^{4}+\frac{11\!\cdots\!16}{10\!\cdots\!07}a^{3}-\frac{10\!\cdots\!51}{10\!\cdots\!07}a^{2}+\frac{36\!\cdots\!60}{36\!\cdots\!69}a+\frac{23\!\cdots\!41}{10\!\cdots\!07}$, $\frac{82\!\cdots\!76}{36\!\cdots\!69}a^{19}-\frac{13\!\cdots\!64}{10\!\cdots\!07}a^{18}+\frac{47\!\cdots\!44}{36\!\cdots\!69}a^{17}-\frac{18\!\cdots\!39}{36\!\cdots\!69}a^{16}+\frac{36\!\cdots\!68}{10\!\cdots\!07}a^{15}-\frac{11\!\cdots\!28}{10\!\cdots\!07}a^{14}+\frac{18\!\cdots\!16}{36\!\cdots\!69}a^{13}-\frac{14\!\cdots\!42}{10\!\cdots\!07}a^{12}+\frac{52\!\cdots\!64}{10\!\cdots\!07}a^{11}-\frac{38\!\cdots\!79}{36\!\cdots\!69}a^{10}+\frac{10\!\cdots\!52}{36\!\cdots\!69}a^{9}-\frac{19\!\cdots\!91}{36\!\cdots\!69}a^{8}+\frac{46\!\cdots\!32}{36\!\cdots\!69}a^{7}-\frac{22\!\cdots\!47}{10\!\cdots\!07}a^{6}+\frac{19\!\cdots\!36}{36\!\cdots\!69}a^{5}-\frac{28\!\cdots\!20}{36\!\cdots\!69}a^{4}+\frac{16\!\cdots\!28}{10\!\cdots\!07}a^{3}-\frac{52\!\cdots\!85}{36\!\cdots\!69}a^{2}+\frac{16\!\cdots\!34}{10\!\cdots\!07}a-\frac{10\!\cdots\!62}{10\!\cdots\!07}$
| 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: | \( 530208.250733 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 530208.250733 \cdot 131042}{2\cdot\sqrt{97756416090547441671665711234313879552}}\cr\approx \mathstrut & 0.336940368674 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.0.18432.2, \(\Q(\zeta_{11})^+\), 10.10.7024111812608.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | $20$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | $20$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/23.1.0.1}{1} }^{20}$ | $20$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | $20$ | $20$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $20$ | $4$ | $5$ | $55$ | |||
|
\(3\)
| 3.20.10.2 | $x^{20} + 162 x^{12} - 486 x^{10} + 1458 x^{8} - 19683 x^{2} + 118098$ | $2$ | $10$ | $10$ | 20T1 | $[\ ]_{2}^{10}$ |
|
\(11\)
| 11.20.16.1 | $x^{20} + 40 x^{18} + 50 x^{17} + 650 x^{16} + 1644 x^{15} + 6440 x^{14} + 18720 x^{13} + 39010 x^{12} + 137400 x^{11} + 283734 x^{10} + 677980 x^{9} + 845540 x^{8} + 1499320 x^{7} + 2045360 x^{6} + 2492700 x^{5} + 5064740 x^{4} + 1884960 x^{3} + 9207500 x^{2} - 2109440 x + 7196441$ | $5$ | $4$ | $16$ | 20T1 | $[\ ]_{5}^{4}$ |