Normalized defining polynomial
\( x^{20} + 5 x^{18} - 20 x^{17} + 910 x^{16} + 236 x^{15} + 26360 x^{14} + 26460 x^{13} + \cdots + 315569310049 \)
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: |
\(934801626748320922851562500000000000000000000\)
\(\medspace = 2^{20}\cdot 3^{10}\cdot 5^{34}\cdot 11^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(177.23\) | 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: | $2\cdot 3^{1/2}5^{17/10}11^{1/2}\approx 177.22948397152206$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $20$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(3300=2^{2}\cdot 3\cdot 5^{2}\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3300}(1,·)$, $\chi_{3300}(131,·)$, $\chi_{3300}(769,·)$, $\chi_{3300}(1739,·)$, $\chi_{3300}(2641,·)$, $\chi_{3300}(2771,·)$, $\chi_{3300}(1429,·)$, $\chi_{3300}(791,·)$, $\chi_{3300}(1079,·)$, $\chi_{3300}(2399,·)$, $\chi_{3300}(419,·)$, $\chi_{3300}(1321,·)$, $\chi_{3300}(1451,·)$, $\chi_{3300}(109,·)$, $\chi_{3300}(2749,·)$, $\chi_{3300}(3059,·)$, $\chi_{3300}(2089,·)$, $\chi_{3300}(2111,·)$, $\chi_{3300}(1981,·)$, $\chi_{3300}(661,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7}a^{7}-\frac{1}{7}a$, $\frac{1}{7}a^{8}-\frac{1}{7}a^{2}$, $\frac{1}{7}a^{9}-\frac{1}{7}a^{3}$, $\frac{1}{525}a^{10}-\frac{1}{15}a^{8}-\frac{2}{105}a^{7}+\frac{1}{5}a^{6}-\frac{1}{75}a^{5}+\frac{46}{105}a^{4}-\frac{4}{15}a^{3}+\frac{1}{15}a^{2}+\frac{2}{105}a-\frac{26}{75}$, $\frac{1}{525}a^{11}-\frac{1}{15}a^{9}-\frac{2}{105}a^{8}+\frac{2}{35}a^{7}-\frac{1}{75}a^{6}+\frac{46}{105}a^{5}-\frac{4}{15}a^{4}+\frac{1}{15}a^{3}+\frac{2}{105}a^{2}-\frac{107}{525}a$, $\frac{1}{525}a^{12}-\frac{2}{105}a^{9}+\frac{1}{105}a^{8}+\frac{6}{175}a^{7}+\frac{46}{105}a^{6}+\frac{4}{15}a^{5}+\frac{2}{5}a^{4}-\frac{11}{35}a^{3}-\frac{82}{525}a^{2}-\frac{1}{21}a-\frac{2}{15}$, $\frac{1}{525}a^{13}+\frac{1}{105}a^{9}-\frac{32}{525}a^{8}-\frac{4}{105}a^{7}+\frac{4}{15}a^{6}+\frac{4}{15}a^{5}+\frac{1}{15}a^{4}+\frac{31}{175}a^{3}+\frac{1}{21}a^{2}+\frac{12}{35}a-\frac{7}{15}$, $\frac{1}{3675}a^{14}-\frac{26}{525}a^{9}+\frac{1}{735}a^{8}-\frac{1}{105}a^{7}+\frac{7}{15}a^{6}-\frac{4}{15}a^{5}+\frac{32}{75}a^{4}-\frac{4}{21}a^{3}+\frac{136}{735}a^{2}+\frac{43}{105}a-\frac{7}{15}$, $\frac{1}{14700}a^{15}+\frac{1}{2100}a^{13}-\frac{1}{2100}a^{12}-\frac{1}{1050}a^{11}-\frac{1}{1050}a^{10}+\frac{1}{196}a^{9}+\frac{113}{2100}a^{8}-\frac{31}{700}a^{7}+\frac{243}{700}a^{6}+\frac{121}{2100}a^{5}+\frac{8}{21}a^{4}+\frac{2363}{7350}a^{3}-\frac{179}{1050}a^{2}+\frac{69}{350}a-\frac{49}{300}$, $\frac{1}{14700}a^{16}-\frac{1}{14700}a^{14}-\frac{1}{2100}a^{13}-\frac{1}{1050}a^{12}-\frac{1}{1050}a^{11}-\frac{3}{4900}a^{10}+\frac{1}{100}a^{9}+\frac{149}{14700}a^{8}-\frac{11}{2100}a^{7}-\frac{333}{700}a^{6}-\frac{8}{175}a^{5}+\frac{377}{2450}a^{4}+\frac{23}{150}a^{3}-\frac{1691}{7350}a^{2}+\frac{817}{2100}a-\frac{2}{75}$, $\frac{1}{632100}a^{17}-\frac{1}{105350}a^{16}-\frac{1}{210700}a^{15}+\frac{17}{210700}a^{14}-\frac{29}{45150}a^{13}+\frac{1}{1075}a^{12}+\frac{523}{632100}a^{11}+\frac{341}{632100}a^{10}-\frac{6529}{210700}a^{9}+\frac{13049}{210700}a^{8}+\frac{743}{30100}a^{7}-\frac{31}{129}a^{6}+\frac{30004}{158025}a^{5}+\frac{14459}{316050}a^{4}-\frac{6143}{105350}a^{3}+\frac{2503}{25284}a^{2}-\frac{11533}{45150}a-\frac{251}{1290}$, $\frac{1}{22\!\cdots\!00}a^{18}-\frac{6591794967976}{18\!\cdots\!75}a^{17}+\frac{8322586924292}{37\!\cdots\!75}a^{16}-\frac{31\!\cdots\!49}{22\!\cdots\!00}a^{15}+\frac{37\!\cdots\!97}{22\!\cdots\!00}a^{14}+\frac{179510164531729}{90\!\cdots\!80}a^{13}-\frac{67\!\cdots\!57}{22\!\cdots\!00}a^{12}+\frac{16\!\cdots\!33}{22\!\cdots\!00}a^{11}-\frac{20\!\cdots\!73}{26\!\cdots\!25}a^{10}-\frac{34\!\cdots\!06}{55\!\cdots\!25}a^{9}+\frac{10\!\cdots\!59}{36\!\cdots\!50}a^{8}-\frac{20\!\cdots\!17}{31\!\cdots\!00}a^{7}+\frac{10\!\cdots\!83}{22\!\cdots\!00}a^{6}+\frac{25\!\cdots\!79}{18\!\cdots\!75}a^{5}+\frac{42\!\cdots\!29}{26\!\cdots\!25}a^{4}+\frac{10\!\cdots\!07}{73\!\cdots\!00}a^{3}-\frac{52\!\cdots\!39}{11\!\cdots\!50}a^{2}-\frac{46\!\cdots\!09}{21\!\cdots\!20}a-\frac{21\!\cdots\!33}{45\!\cdots\!90}$, $\frac{1}{88\!\cdots\!00}a^{19}-\frac{10\!\cdots\!33}{14\!\cdots\!50}a^{18}-\frac{58\!\cdots\!79}{12\!\cdots\!00}a^{17}-\frac{20\!\cdots\!53}{22\!\cdots\!25}a^{16}+\frac{66\!\cdots\!42}{22\!\cdots\!25}a^{15}+\frac{12\!\cdots\!79}{98\!\cdots\!00}a^{14}-\frac{14\!\cdots\!57}{73\!\cdots\!75}a^{13}-\frac{60\!\cdots\!61}{88\!\cdots\!00}a^{12}+\frac{57\!\cdots\!01}{84\!\cdots\!20}a^{11}+\frac{24\!\cdots\!71}{73\!\cdots\!75}a^{10}+\frac{68\!\cdots\!92}{29\!\cdots\!07}a^{9}-\frac{29\!\cdots\!09}{42\!\cdots\!00}a^{8}-\frac{31\!\cdots\!89}{59\!\cdots\!40}a^{7}+\frac{67\!\cdots\!79}{29\!\cdots\!00}a^{6}-\frac{68\!\cdots\!73}{63\!\cdots\!50}a^{5}-\frac{27\!\cdots\!83}{88\!\cdots\!00}a^{4}-\frac{18\!\cdots\!14}{44\!\cdots\!05}a^{3}-\frac{26\!\cdots\!39}{63\!\cdots\!50}a^{2}-\frac{34\!\cdots\!99}{25\!\cdots\!00}a-\frac{13\!\cdots\!08}{64\!\cdots\!75}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $7$ |
Class group and class number
$C_{2}\times C_{7719688}$, which has order $15439376$ (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: |
$\frac{58\!\cdots\!84}{10\!\cdots\!25}a^{19}-\frac{16\!\cdots\!78}{10\!\cdots\!25}a^{18}+\frac{16\!\cdots\!36}{10\!\cdots\!25}a^{17}-\frac{15\!\cdots\!47}{10\!\cdots\!25}a^{16}+\frac{55\!\cdots\!88}{10\!\cdots\!25}a^{15}-\frac{12\!\cdots\!92}{10\!\cdots\!25}a^{14}+\frac{13\!\cdots\!52}{10\!\cdots\!25}a^{13}-\frac{47\!\cdots\!77}{21\!\cdots\!05}a^{12}+\frac{35\!\cdots\!44}{10\!\cdots\!25}a^{11}-\frac{60\!\cdots\!61}{10\!\cdots\!25}a^{10}+\frac{13\!\cdots\!38}{21\!\cdots\!05}a^{9}-\frac{15\!\cdots\!06}{10\!\cdots\!25}a^{8}+\frac{10\!\cdots\!64}{10\!\cdots\!25}a^{7}-\frac{26\!\cdots\!31}{10\!\cdots\!25}a^{6}+\frac{10\!\cdots\!74}{10\!\cdots\!25}a^{5}-\frac{31\!\cdots\!27}{10\!\cdots\!25}a^{4}+\frac{80\!\cdots\!08}{10\!\cdots\!25}a^{3}-\frac{22\!\cdots\!79}{10\!\cdots\!25}a^{2}+\frac{22\!\cdots\!24}{86\!\cdots\!49}a-\frac{34\!\cdots\!49}{43\!\cdots\!45}$, $\frac{21\!\cdots\!12}{73\!\cdots\!75}a^{19}+\frac{43\!\cdots\!36}{14\!\cdots\!35}a^{18}-\frac{49\!\cdots\!42}{15\!\cdots\!75}a^{17}-\frac{69\!\cdots\!41}{14\!\cdots\!35}a^{16}+\frac{14\!\cdots\!28}{73\!\cdots\!75}a^{15}+\frac{27\!\cdots\!22}{10\!\cdots\!25}a^{14}+\frac{51\!\cdots\!04}{73\!\cdots\!75}a^{13}+\frac{54\!\cdots\!24}{73\!\cdots\!75}a^{12}+\frac{37\!\cdots\!12}{15\!\cdots\!75}a^{11}+\frac{14\!\cdots\!96}{73\!\cdots\!75}a^{10}+\frac{74\!\cdots\!46}{14\!\cdots\!35}a^{9}+\frac{38\!\cdots\!61}{10\!\cdots\!25}a^{8}+\frac{10\!\cdots\!36}{14\!\cdots\!35}a^{7}+\frac{37\!\cdots\!94}{73\!\cdots\!75}a^{6}+\frac{27\!\cdots\!74}{43\!\cdots\!45}a^{5}+\frac{38\!\cdots\!79}{73\!\cdots\!75}a^{4}+\frac{29\!\cdots\!62}{73\!\cdots\!75}a^{3}+\frac{35\!\cdots\!77}{10\!\cdots\!25}a^{2}+\frac{16\!\cdots\!56}{15\!\cdots\!75}a+\frac{24\!\cdots\!02}{21\!\cdots\!25}$, $\frac{39\!\cdots\!28}{73\!\cdots\!75}a^{19}+\frac{50\!\cdots\!66}{29\!\cdots\!07}a^{18}-\frac{42\!\cdots\!94}{10\!\cdots\!25}a^{17}-\frac{96\!\cdots\!73}{73\!\cdots\!75}a^{16}+\frac{36\!\cdots\!48}{73\!\cdots\!75}a^{15}+\frac{19\!\cdots\!38}{10\!\cdots\!25}a^{14}+\frac{61\!\cdots\!76}{73\!\cdots\!75}a^{13}+\frac{42\!\cdots\!69}{73\!\cdots\!75}a^{12}+\frac{69\!\cdots\!16}{21\!\cdots\!05}a^{11}+\frac{11\!\cdots\!67}{73\!\cdots\!75}a^{10}+\frac{87\!\cdots\!56}{14\!\cdots\!35}a^{9}+\frac{70\!\cdots\!18}{24\!\cdots\!75}a^{8}+\frac{12\!\cdots\!96}{14\!\cdots\!35}a^{7}+\frac{30\!\cdots\!03}{73\!\cdots\!75}a^{6}+\frac{83\!\cdots\!04}{10\!\cdots\!25}a^{5}+\frac{34\!\cdots\!56}{73\!\cdots\!75}a^{4}+\frac{34\!\cdots\!18}{73\!\cdots\!75}a^{3}+\frac{35\!\cdots\!02}{10\!\cdots\!25}a^{2}+\frac{18\!\cdots\!08}{15\!\cdots\!75}a+\frac{29\!\cdots\!58}{21\!\cdots\!25}$, $\frac{18\!\cdots\!16}{73\!\cdots\!75}a^{19}-\frac{17\!\cdots\!06}{14\!\cdots\!35}a^{18}-\frac{15\!\cdots\!84}{42\!\cdots\!01}a^{17}+\frac{25\!\cdots\!32}{73\!\cdots\!75}a^{16}+\frac{43\!\cdots\!44}{14\!\cdots\!35}a^{15}-\frac{88\!\cdots\!84}{10\!\cdots\!25}a^{14}+\frac{94\!\cdots\!72}{73\!\cdots\!75}a^{13}-\frac{24\!\cdots\!31}{14\!\cdots\!35}a^{12}+\frac{88\!\cdots\!96}{10\!\cdots\!25}a^{11}-\frac{27\!\cdots\!29}{73\!\cdots\!75}a^{10}+\frac{24\!\cdots\!22}{29\!\cdots\!07}a^{9}-\frac{83\!\cdots\!87}{10\!\cdots\!25}a^{8}+\frac{52\!\cdots\!72}{29\!\cdots\!07}a^{7}-\frac{64\!\cdots\!91}{73\!\cdots\!75}a^{6}+\frac{16\!\cdots\!74}{10\!\cdots\!25}a^{5}-\frac{38\!\cdots\!23}{73\!\cdots\!75}a^{4}+\frac{55\!\cdots\!56}{73\!\cdots\!75}a^{3}-\frac{45\!\cdots\!55}{42\!\cdots\!01}a^{2}+\frac{17\!\cdots\!52}{15\!\cdots\!75}a+\frac{32\!\cdots\!31}{21\!\cdots\!25}$, $\frac{14\!\cdots\!33}{73\!\cdots\!75}a^{19}-\frac{58\!\cdots\!43}{29\!\cdots\!00}a^{18}+\frac{58\!\cdots\!53}{12\!\cdots\!00}a^{17}-\frac{25\!\cdots\!49}{88\!\cdots\!00}a^{16}+\frac{15\!\cdots\!43}{88\!\cdots\!00}a^{15}-\frac{73\!\cdots\!53}{42\!\cdots\!00}a^{14}+\frac{23\!\cdots\!13}{29\!\cdots\!00}a^{13}-\frac{54\!\cdots\!99}{14\!\cdots\!50}a^{12}+\frac{42\!\cdots\!02}{31\!\cdots\!75}a^{11}-\frac{19\!\cdots\!11}{22\!\cdots\!25}a^{10}+\frac{11\!\cdots\!71}{35\!\cdots\!84}a^{9}-\frac{23\!\cdots\!57}{12\!\cdots\!00}a^{8}+\frac{23\!\cdots\!51}{44\!\cdots\!50}a^{7}-\frac{24\!\cdots\!83}{88\!\cdots\!00}a^{6}+\frac{17\!\cdots\!51}{25\!\cdots\!60}a^{5}-\frac{64\!\cdots\!22}{22\!\cdots\!25}a^{4}+\frac{49\!\cdots\!93}{88\!\cdots\!00}a^{3}-\frac{23\!\cdots\!23}{12\!\cdots\!00}a^{2}+\frac{13\!\cdots\!32}{64\!\cdots\!75}a-\frac{12\!\cdots\!53}{25\!\cdots\!00}$, $\frac{26\!\cdots\!67}{44\!\cdots\!50}a^{19}+\frac{28\!\cdots\!51}{14\!\cdots\!50}a^{18}-\frac{29\!\cdots\!37}{63\!\cdots\!50}a^{17}-\frac{28\!\cdots\!59}{29\!\cdots\!00}a^{16}+\frac{64\!\cdots\!79}{11\!\cdots\!28}a^{15}+\frac{82\!\cdots\!53}{42\!\cdots\!00}a^{14}+\frac{81\!\cdots\!57}{88\!\cdots\!10}a^{13}+\frac{16\!\cdots\!91}{29\!\cdots\!00}a^{12}+\frac{22\!\cdots\!93}{63\!\cdots\!15}a^{11}+\frac{13\!\cdots\!11}{88\!\cdots\!00}a^{10}+\frac{18\!\cdots\!31}{29\!\cdots\!70}a^{9}+\frac{16\!\cdots\!21}{63\!\cdots\!50}a^{8}+\frac{46\!\cdots\!66}{51\!\cdots\!75}a^{7}+\frac{54\!\cdots\!67}{14\!\cdots\!50}a^{6}+\frac{99\!\cdots\!99}{12\!\cdots\!00}a^{5}+\frac{83\!\cdots\!77}{22\!\cdots\!25}a^{4}+\frac{69\!\cdots\!01}{14\!\cdots\!50}a^{3}+\frac{79\!\cdots\!77}{31\!\cdots\!75}a^{2}+\frac{75\!\cdots\!49}{60\!\cdots\!00}a+\frac{82\!\cdots\!77}{86\!\cdots\!00}$, $\frac{26\!\cdots\!22}{63\!\cdots\!25}a^{19}+\frac{29\!\cdots\!89}{50\!\cdots\!20}a^{18}-\frac{15\!\cdots\!89}{90\!\cdots\!75}a^{17}-\frac{30\!\cdots\!59}{50\!\cdots\!20}a^{16}+\frac{10\!\cdots\!87}{42\!\cdots\!50}a^{15}+\frac{15\!\cdots\!44}{30\!\cdots\!25}a^{14}+\frac{23\!\cdots\!71}{25\!\cdots\!00}a^{13}+\frac{91\!\cdots\!69}{63\!\cdots\!25}a^{12}+\frac{13\!\cdots\!91}{36\!\cdots\!00}a^{11}+\frac{31\!\cdots\!43}{84\!\cdots\!00}a^{10}+\frac{94\!\cdots\!01}{12\!\cdots\!05}a^{9}+\frac{64\!\cdots\!41}{90\!\cdots\!75}a^{8}+\frac{11\!\cdots\!73}{11\!\cdots\!40}a^{7}+\frac{25\!\cdots\!31}{25\!\cdots\!00}a^{6}+\frac{52\!\cdots\!19}{72\!\cdots\!60}a^{5}+\frac{43\!\cdots\!41}{42\!\cdots\!50}a^{4}+\frac{70\!\cdots\!13}{25\!\cdots\!00}a^{3}+\frac{12\!\cdots\!99}{18\!\cdots\!50}a^{2}+\frac{27\!\cdots\!11}{12\!\cdots\!25}a+\frac{45\!\cdots\!91}{24\!\cdots\!00}$, $\frac{19\!\cdots\!19}{29\!\cdots\!00}a^{19}-\frac{78\!\cdots\!51}{88\!\cdots\!00}a^{18}-\frac{21\!\cdots\!91}{12\!\cdots\!60}a^{17}-\frac{44\!\cdots\!77}{29\!\cdots\!00}a^{16}+\frac{45\!\cdots\!97}{73\!\cdots\!75}a^{15}-\frac{22\!\cdots\!61}{42\!\cdots\!00}a^{14}+\frac{83\!\cdots\!93}{68\!\cdots\!00}a^{13}-\frac{54\!\cdots\!39}{88\!\cdots\!00}a^{12}+\frac{53\!\cdots\!38}{15\!\cdots\!75}a^{11}-\frac{51\!\cdots\!29}{29\!\cdots\!00}a^{10}+\frac{27\!\cdots\!79}{44\!\cdots\!05}a^{9}-\frac{29\!\cdots\!33}{42\!\cdots\!00}a^{8}+\frac{13\!\cdots\!13}{14\!\cdots\!50}a^{7}-\frac{58\!\cdots\!01}{44\!\cdots\!50}a^{6}+\frac{11\!\cdots\!69}{12\!\cdots\!60}a^{5}-\frac{51\!\cdots\!27}{29\!\cdots\!00}a^{4}+\frac{11\!\cdots\!99}{17\!\cdots\!20}a^{3}-\frac{12\!\cdots\!98}{10\!\cdots\!25}a^{2}+\frac{12\!\cdots\!23}{60\!\cdots\!00}a-\frac{46\!\cdots\!13}{86\!\cdots\!00}$, $\frac{46\!\cdots\!39}{29\!\cdots\!00}a^{19}+\frac{10\!\cdots\!37}{22\!\cdots\!25}a^{18}+\frac{16\!\cdots\!09}{12\!\cdots\!03}a^{17}-\frac{25\!\cdots\!79}{29\!\cdots\!00}a^{16}+\frac{87\!\cdots\!63}{88\!\cdots\!00}a^{15}+\frac{16\!\cdots\!29}{42\!\cdots\!00}a^{14}+\frac{29\!\cdots\!71}{59\!\cdots\!40}a^{13}+\frac{10\!\cdots\!87}{88\!\cdots\!00}a^{12}+\frac{34\!\cdots\!69}{31\!\cdots\!75}a^{11}+\frac{18\!\cdots\!13}{73\!\cdots\!75}a^{10}+\frac{96\!\cdots\!23}{44\!\cdots\!05}a^{9}+\frac{42\!\cdots\!81}{12\!\cdots\!00}a^{8}+\frac{84\!\cdots\!47}{29\!\cdots\!00}a^{7}+\frac{28\!\cdots\!73}{14\!\cdots\!50}a^{6}+\frac{31\!\cdots\!43}{12\!\cdots\!30}a^{5}-\frac{26\!\cdots\!77}{29\!\cdots\!00}a^{4}+\frac{56\!\cdots\!19}{44\!\cdots\!50}a^{3}-\frac{27\!\cdots\!87}{12\!\cdots\!00}a^{2}+\frac{84\!\cdots\!61}{30\!\cdots\!50}a-\frac{15\!\cdots\!69}{12\!\cdots\!50}$
| 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: | \( 180801817.57689384 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 180801817.57689384 \cdot 15439376}{2\cdot\sqrt{934801626748320922851562500000000000000000000}}\cr\approx \mathstrut & 4.37765461656408 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times C_{10}$ (as 20T3):
| An abelian group of order 20 |
| The 20 conjugacy class representatives for $C_2\times C_{10}$ |
| Character table for $C_2\times C_{10}$ |
Intermediate fields
| \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{15}, \sqrt{-33})\), 5.5.390625.1, 10.0.6114905156250000000000.1, 10.10.189843750000000000.1, 10.0.122872161865234375.1 |
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 | R | ${\href{/padicField/7.1.0.1}{1} }^{20}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.1.0.1}{1} }^{20}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 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 | $x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 592 x^{5} + 656 x^{4} + 384 x^{3} - 112 x^{2} - 352 x - 1248$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
|
\(3\)
| 3.20.10.1 | $x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ |
|
\(5\)
| Deg $20$ | $10$ | $2$ | $34$ | |||
|
\(11\)
| 11.10.5.2 | $x^{10} + 55 x^{8} + 20 x^{7} + 1210 x^{6} - 202 x^{5} + 13410 x^{4} - 14080 x^{3} + 75585 x^{2} - 68970 x + 171252$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 11.10.5.2 | $x^{10} + 55 x^{8} + 20 x^{7} + 1210 x^{6} - 202 x^{5} + 13410 x^{4} - 14080 x^{3} + 75585 x^{2} - 68970 x + 171252$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |