Normalized defining polynomial
\( x^{20} - x^{19} + 3 x^{18} - 11 x^{17} + 44 x^{16} + 300 x^{15} + 316 x^{14} - 1991 x^{13} + \cdots + 120577 \)
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: | \(120810895044353150938886048668570711901\) \(\medspace = 101^{19}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(80.19\) | 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: | $101^{19/20}\approx 80.18724729938236$ | ||
Ramified primes: | \(101\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{101}) \) | ||
$\card{ \Gal(K/\Q) }$: | $20$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(101\) | ||
Dirichlet character group: | $\lbrace$$\chi_{101}(1,·)$, $\chi_{101}(69,·)$, $\chi_{101}(6,·)$, $\chi_{101}(65,·)$, $\chi_{101}(10,·)$, $\chi_{101}(14,·)$, $\chi_{101}(17,·)$, $\chi_{101}(84,·)$, $\chi_{101}(87,·)$, $\chi_{101}(36,·)$, $\chi_{101}(91,·)$, $\chi_{101}(95,·)$, $\chi_{101}(32,·)$, $\chi_{101}(100,·)$, $\chi_{101}(39,·)$, $\chi_{101}(41,·)$, $\chi_{101}(44,·)$, $\chi_{101}(57,·)$, $\chi_{101}(60,·)$, $\chi_{101}(62,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{512}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{137}a^{16}+\frac{32}{137}a^{15}+\frac{55}{137}a^{14}-\frac{25}{137}a^{13}+\frac{60}{137}a^{12}-\frac{29}{137}a^{11}+\frac{29}{137}a^{10}-\frac{4}{137}a^{9}+\frac{29}{137}a^{8}+\frac{47}{137}a^{7}+\frac{33}{137}a^{6}-\frac{37}{137}a^{5}-\frac{12}{137}a^{4}+\frac{9}{137}a^{3}-\frac{43}{137}a^{2}-\frac{60}{137}a+\frac{59}{137}$, $\frac{1}{137}a^{17}-\frac{10}{137}a^{15}-\frac{4}{137}a^{14}+\frac{38}{137}a^{13}-\frac{31}{137}a^{12}-\frac{2}{137}a^{11}+\frac{27}{137}a^{10}+\frac{20}{137}a^{9}-\frac{59}{137}a^{8}+\frac{36}{137}a^{7}+\frac{3}{137}a^{6}-\frac{61}{137}a^{5}-\frac{18}{137}a^{4}-\frac{57}{137}a^{3}-\frac{54}{137}a^{2}+\frac{61}{137}a+\frac{30}{137}$, $\frac{1}{137}a^{18}+\frac{42}{137}a^{15}+\frac{40}{137}a^{14}-\frac{7}{137}a^{13}+\frac{50}{137}a^{12}+\frac{11}{137}a^{11}+\frac{36}{137}a^{10}+\frac{38}{137}a^{9}+\frac{52}{137}a^{8}+\frac{62}{137}a^{7}-\frac{5}{137}a^{6}+\frac{23}{137}a^{5}-\frac{40}{137}a^{4}+\frac{36}{137}a^{3}+\frac{42}{137}a^{2}-\frac{22}{137}a+\frac{42}{137}$, $\frac{1}{17\!\cdots\!91}a^{19}-\frac{47\!\cdots\!10}{17\!\cdots\!91}a^{18}+\frac{26\!\cdots\!18}{17\!\cdots\!91}a^{17}-\frac{23\!\cdots\!58}{17\!\cdots\!91}a^{16}-\frac{51\!\cdots\!90}{17\!\cdots\!91}a^{15}-\frac{32\!\cdots\!07}{17\!\cdots\!91}a^{14}-\frac{36\!\cdots\!84}{17\!\cdots\!91}a^{13}+\frac{16\!\cdots\!80}{17\!\cdots\!91}a^{12}-\frac{28\!\cdots\!49}{17\!\cdots\!91}a^{11}+\frac{66\!\cdots\!89}{17\!\cdots\!91}a^{10}-\frac{37\!\cdots\!65}{17\!\cdots\!91}a^{9}+\frac{33\!\cdots\!59}{17\!\cdots\!91}a^{8}-\frac{22\!\cdots\!99}{17\!\cdots\!91}a^{7}-\frac{29\!\cdots\!32}{17\!\cdots\!91}a^{6}+\frac{20\!\cdots\!23}{17\!\cdots\!91}a^{5}+\frac{44\!\cdots\!87}{17\!\cdots\!91}a^{4}+\frac{36\!\cdots\!80}{17\!\cdots\!91}a^{3}+\frac{18\!\cdots\!39}{17\!\cdots\!91}a^{2}-\frac{36\!\cdots\!68}{17\!\cdots\!91}a+\frac{18\!\cdots\!41}{17\!\cdots\!91}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{5}\times C_{25}$, which has order $125$ (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: | $\frac{33\!\cdots\!39}{17\!\cdots\!91}a^{19}-\frac{42\!\cdots\!63}{17\!\cdots\!91}a^{18}+\frac{68\!\cdots\!90}{17\!\cdots\!91}a^{17}-\frac{26\!\cdots\!77}{17\!\cdots\!91}a^{16}+\frac{12\!\cdots\!81}{17\!\cdots\!91}a^{15}+\frac{10\!\cdots\!04}{17\!\cdots\!91}a^{14}+\frac{46\!\cdots\!36}{17\!\cdots\!91}a^{13}-\frac{76\!\cdots\!09}{17\!\cdots\!91}a^{12}+\frac{28\!\cdots\!12}{17\!\cdots\!91}a^{11}+\frac{14\!\cdots\!78}{17\!\cdots\!91}a^{10}+\frac{57\!\cdots\!27}{17\!\cdots\!91}a^{9}+\frac{12\!\cdots\!92}{17\!\cdots\!91}a^{8}-\frac{19\!\cdots\!53}{17\!\cdots\!91}a^{7}+\frac{14\!\cdots\!35}{17\!\cdots\!91}a^{6}+\frac{42\!\cdots\!19}{17\!\cdots\!91}a^{5}+\frac{18\!\cdots\!32}{17\!\cdots\!91}a^{4}-\frac{14\!\cdots\!11}{17\!\cdots\!91}a^{3}+\frac{76\!\cdots\!62}{17\!\cdots\!91}a^{2}+\frac{43\!\cdots\!83}{17\!\cdots\!91}a+\frac{32\!\cdots\!76}{17\!\cdots\!91}$, $\frac{17\!\cdots\!55}{17\!\cdots\!91}a^{19}-\frac{12\!\cdots\!74}{17\!\cdots\!91}a^{18}+\frac{37\!\cdots\!58}{17\!\cdots\!91}a^{17}-\frac{18\!\cdots\!30}{17\!\cdots\!91}a^{16}+\frac{65\!\cdots\!17}{17\!\cdots\!91}a^{15}+\frac{54\!\cdots\!66}{17\!\cdots\!91}a^{14}+\frac{66\!\cdots\!98}{17\!\cdots\!91}a^{13}-\frac{35\!\cdots\!27}{17\!\cdots\!91}a^{12}-\frac{20\!\cdots\!21}{17\!\cdots\!91}a^{11}+\frac{36\!\cdots\!35}{17\!\cdots\!91}a^{10}+\frac{43\!\cdots\!51}{17\!\cdots\!91}a^{9}+\frac{27\!\cdots\!26}{17\!\cdots\!91}a^{8}-\frac{67\!\cdots\!22}{17\!\cdots\!91}a^{7}-\frac{77\!\cdots\!32}{17\!\cdots\!91}a^{6}+\frac{11\!\cdots\!71}{17\!\cdots\!91}a^{5}+\frac{27\!\cdots\!56}{17\!\cdots\!91}a^{4}+\frac{80\!\cdots\!34}{17\!\cdots\!91}a^{3}-\frac{17\!\cdots\!26}{17\!\cdots\!91}a^{2}-\frac{16\!\cdots\!19}{12\!\cdots\!43}a-\frac{17\!\cdots\!38}{17\!\cdots\!91}$, $\frac{11\!\cdots\!73}{24\!\cdots\!53}a^{19}-\frac{32\!\cdots\!76}{24\!\cdots\!53}a^{18}+\frac{51\!\cdots\!83}{24\!\cdots\!53}a^{17}-\frac{15\!\cdots\!86}{24\!\cdots\!53}a^{16}+\frac{28\!\cdots\!53}{24\!\cdots\!53}a^{15}+\frac{37\!\cdots\!43}{24\!\cdots\!53}a^{14}+\frac{49\!\cdots\!47}{24\!\cdots\!53}a^{13}-\frac{29\!\cdots\!32}{24\!\cdots\!53}a^{12}-\frac{54\!\cdots\!27}{24\!\cdots\!53}a^{11}-\frac{14\!\cdots\!88}{24\!\cdots\!53}a^{10}+\frac{32\!\cdots\!33}{24\!\cdots\!53}a^{9}+\frac{15\!\cdots\!79}{24\!\cdots\!53}a^{8}-\frac{11\!\cdots\!16}{24\!\cdots\!53}a^{7}-\frac{35\!\cdots\!56}{24\!\cdots\!53}a^{6}-\frac{34\!\cdots\!12}{24\!\cdots\!53}a^{5}-\frac{14\!\cdots\!40}{24\!\cdots\!53}a^{4}-\frac{30\!\cdots\!34}{24\!\cdots\!53}a^{3}-\frac{58\!\cdots\!13}{24\!\cdots\!53}a^{2}-\frac{57\!\cdots\!56}{24\!\cdots\!53}a-\frac{10\!\cdots\!24}{24\!\cdots\!53}$, $\frac{20\!\cdots\!09}{17\!\cdots\!91}a^{19}-\frac{43\!\cdots\!67}{17\!\cdots\!91}a^{18}+\frac{78\!\cdots\!09}{17\!\cdots\!91}a^{17}-\frac{25\!\cdots\!56}{17\!\cdots\!91}a^{16}+\frac{10\!\cdots\!44}{17\!\cdots\!91}a^{15}+\frac{53\!\cdots\!88}{17\!\cdots\!91}a^{14}-\frac{10\!\cdots\!29}{17\!\cdots\!91}a^{13}-\frac{48\!\cdots\!28}{17\!\cdots\!91}a^{12}+\frac{50\!\cdots\!59}{17\!\cdots\!91}a^{11}+\frac{57\!\cdots\!86}{17\!\cdots\!91}a^{10}+\frac{38\!\cdots\!03}{17\!\cdots\!91}a^{9}-\frac{35\!\cdots\!58}{17\!\cdots\!91}a^{8}-\frac{91\!\cdots\!11}{17\!\cdots\!91}a^{7}+\frac{11\!\cdots\!69}{17\!\cdots\!91}a^{6}+\frac{23\!\cdots\!32}{17\!\cdots\!91}a^{5}-\frac{86\!\cdots\!23}{17\!\cdots\!91}a^{4}-\frac{30\!\cdots\!16}{17\!\cdots\!91}a^{3}+\frac{20\!\cdots\!72}{17\!\cdots\!91}a^{2}+\frac{44\!\cdots\!86}{17\!\cdots\!91}a-\frac{25\!\cdots\!31}{17\!\cdots\!91}$, $\frac{97\!\cdots\!54}{17\!\cdots\!91}a^{19}-\frac{84\!\cdots\!29}{17\!\cdots\!91}a^{18}+\frac{24\!\cdots\!89}{12\!\cdots\!43}a^{17}-\frac{83\!\cdots\!25}{17\!\cdots\!91}a^{16}+\frac{42\!\cdots\!99}{17\!\cdots\!91}a^{15}+\frac{30\!\cdots\!50}{17\!\cdots\!91}a^{14}+\frac{34\!\cdots\!03}{17\!\cdots\!91}a^{13}-\frac{16\!\cdots\!80}{17\!\cdots\!91}a^{12}+\frac{67\!\cdots\!49}{17\!\cdots\!91}a^{11}+\frac{20\!\cdots\!28}{17\!\cdots\!91}a^{10}+\frac{19\!\cdots\!46}{17\!\cdots\!91}a^{9}+\frac{84\!\cdots\!74}{17\!\cdots\!91}a^{8}-\frac{18\!\cdots\!20}{17\!\cdots\!91}a^{7}+\frac{46\!\cdots\!14}{17\!\cdots\!91}a^{6}+\frac{15\!\cdots\!42}{17\!\cdots\!91}a^{5}+\frac{15\!\cdots\!03}{17\!\cdots\!91}a^{4}+\frac{68\!\cdots\!78}{17\!\cdots\!91}a^{3}+\frac{59\!\cdots\!46}{17\!\cdots\!91}a^{2}+\frac{55\!\cdots\!77}{17\!\cdots\!91}a+\frac{31\!\cdots\!33}{17\!\cdots\!91}$, $\frac{79\!\cdots\!08}{17\!\cdots\!91}a^{19}-\frac{13\!\cdots\!97}{17\!\cdots\!91}a^{18}+\frac{30\!\cdots\!04}{17\!\cdots\!91}a^{17}-\frac{10\!\cdots\!42}{17\!\cdots\!91}a^{16}+\frac{40\!\cdots\!73}{17\!\cdots\!91}a^{15}+\frac{21\!\cdots\!91}{17\!\cdots\!91}a^{14}+\frac{86\!\cdots\!80}{17\!\cdots\!91}a^{13}-\frac{16\!\cdots\!64}{17\!\cdots\!91}a^{12}+\frac{11\!\cdots\!16}{17\!\cdots\!91}a^{11}+\frac{12\!\cdots\!65}{17\!\cdots\!91}a^{10}+\frac{17\!\cdots\!28}{17\!\cdots\!91}a^{9}-\frac{60\!\cdots\!58}{17\!\cdots\!91}a^{8}-\frac{25\!\cdots\!53}{17\!\cdots\!91}a^{7}+\frac{18\!\cdots\!76}{17\!\cdots\!91}a^{6}+\frac{81\!\cdots\!63}{17\!\cdots\!91}a^{5}+\frac{44\!\cdots\!31}{17\!\cdots\!91}a^{4}-\frac{58\!\cdots\!06}{17\!\cdots\!91}a^{3}+\frac{38\!\cdots\!52}{17\!\cdots\!91}a^{2}+\frac{70\!\cdots\!69}{17\!\cdots\!91}a+\frac{10\!\cdots\!83}{17\!\cdots\!91}$, $\frac{75\!\cdots\!50}{17\!\cdots\!91}a^{19}-\frac{14\!\cdots\!61}{17\!\cdots\!91}a^{18}+\frac{26\!\cdots\!91}{17\!\cdots\!91}a^{17}-\frac{12\!\cdots\!91}{17\!\cdots\!91}a^{16}+\frac{39\!\cdots\!75}{17\!\cdots\!91}a^{15}+\frac{19\!\cdots\!09}{17\!\cdots\!91}a^{14}+\frac{53\!\cdots\!02}{17\!\cdots\!91}a^{13}-\frac{18\!\cdots\!26}{17\!\cdots\!91}a^{12}+\frac{42\!\cdots\!00}{17\!\cdots\!91}a^{11}+\frac{63\!\cdots\!70}{17\!\cdots\!91}a^{10}+\frac{20\!\cdots\!05}{17\!\cdots\!91}a^{9}-\frac{61\!\cdots\!96}{17\!\cdots\!91}a^{8}-\frac{30\!\cdots\!19}{17\!\cdots\!91}a^{7}-\frac{33\!\cdots\!74}{17\!\cdots\!91}a^{6}+\frac{23\!\cdots\!70}{17\!\cdots\!91}a^{5}+\frac{40\!\cdots\!72}{17\!\cdots\!91}a^{4}-\frac{89\!\cdots\!64}{17\!\cdots\!91}a^{3}-\frac{38\!\cdots\!27}{17\!\cdots\!91}a^{2}-\frac{29\!\cdots\!74}{17\!\cdots\!91}a+\frac{72\!\cdots\!16}{17\!\cdots\!91}$, $\frac{45\!\cdots\!75}{17\!\cdots\!91}a^{19}-\frac{65\!\cdots\!96}{17\!\cdots\!91}a^{18}+\frac{11\!\cdots\!53}{17\!\cdots\!91}a^{17}-\frac{44\!\cdots\!66}{17\!\cdots\!91}a^{16}+\frac{19\!\cdots\!97}{17\!\cdots\!91}a^{15}+\frac{13\!\cdots\!72}{17\!\cdots\!91}a^{14}+\frac{54\!\cdots\!80}{17\!\cdots\!91}a^{13}-\frac{10\!\cdots\!04}{17\!\cdots\!91}a^{12}+\frac{40\!\cdots\!46}{17\!\cdots\!91}a^{11}+\frac{16\!\cdots\!97}{17\!\cdots\!91}a^{10}+\frac{88\!\cdots\!02}{17\!\cdots\!91}a^{9}+\frac{15\!\cdots\!63}{17\!\cdots\!91}a^{8}-\frac{24\!\cdots\!32}{17\!\cdots\!91}a^{7}+\frac{12\!\cdots\!97}{17\!\cdots\!91}a^{6}+\frac{49\!\cdots\!90}{17\!\cdots\!91}a^{5}+\frac{52\!\cdots\!82}{17\!\cdots\!91}a^{4}-\frac{28\!\cdots\!58}{17\!\cdots\!91}a^{3}+\frac{42\!\cdots\!29}{12\!\cdots\!43}a^{2}+\frac{47\!\cdots\!37}{17\!\cdots\!91}a+\frac{26\!\cdots\!18}{17\!\cdots\!91}$, $\frac{11\!\cdots\!44}{17\!\cdots\!91}a^{19}+\frac{14\!\cdots\!34}{17\!\cdots\!91}a^{18}-\frac{30\!\cdots\!00}{17\!\cdots\!91}a^{17}-\frac{67\!\cdots\!27}{17\!\cdots\!91}a^{16}+\frac{24\!\cdots\!16}{17\!\cdots\!91}a^{15}+\frac{51\!\cdots\!72}{17\!\cdots\!91}a^{14}+\frac{98\!\cdots\!15}{17\!\cdots\!91}a^{13}-\frac{26\!\cdots\!92}{17\!\cdots\!91}a^{12}-\frac{67\!\cdots\!75}{17\!\cdots\!91}a^{11}+\frac{73\!\cdots\!50}{17\!\cdots\!91}a^{10}+\frac{29\!\cdots\!53}{17\!\cdots\!91}a^{9}+\frac{81\!\cdots\!45}{17\!\cdots\!91}a^{8}-\frac{10\!\cdots\!74}{17\!\cdots\!91}a^{7}-\frac{14\!\cdots\!47}{17\!\cdots\!91}a^{6}+\frac{64\!\cdots\!30}{17\!\cdots\!91}a^{5}+\frac{30\!\cdots\!58}{17\!\cdots\!91}a^{4}+\frac{31\!\cdots\!34}{17\!\cdots\!91}a^{3}-\frac{42\!\cdots\!83}{17\!\cdots\!91}a^{2}-\frac{47\!\cdots\!51}{17\!\cdots\!91}a-\frac{45\!\cdots\!45}{17\!\cdots\!91}$ (assuming GRH) | 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: | \( 224544554.426 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 224544554.426 \cdot 125}{2\cdot\sqrt{120810895044353150938886048668570711901}}\cr\approx \mathstrut & 0.122441369119 \end{aligned}\] (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
\(\Q(\sqrt{101}) \), 4.0.1030301.1, 5.5.104060401.1, 10.10.1093685272684360901.1 |
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 | $20$ | $20$ | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | $20$ | $20$ | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{10}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{5}$ | ${\href{/padicField/43.10.0.1}{10} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | $20$ | $20$ |
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 |
---|---|---|---|---|---|---|---|
\(101\) | 101.20.19.6 | $x^{20} + 101$ | $20$ | $1$ | $19$ | 20T1 | $[\ ]_{20}$ |