Normalized defining polynomial
\( x^{22} - 35 x^{20} - 72 x^{19} + 349 x^{18} + 1479 x^{17} + 303 x^{16} - 6784 x^{15} - 5684 x^{14} + \cdots + 509 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[10, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(22661033510180079603495293971842498241\) \(\medspace = 1297^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(49.88\) | 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: | $1297^{3/4}\approx 216.12498794754728$ | ||
Ramified primes: | \(1297\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5}a^{16}-\frac{2}{5}a^{14}+\frac{2}{5}a^{12}-\frac{1}{5}a^{11}-\frac{2}{5}a^{10}-\frac{1}{5}a^{9}+\frac{2}{5}a^{8}-\frac{2}{5}a^{7}+\frac{2}{5}a^{6}+\frac{1}{5}a^{5}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{17}-\frac{2}{5}a^{15}+\frac{2}{5}a^{13}-\frac{1}{5}a^{12}-\frac{2}{5}a^{11}-\frac{1}{5}a^{10}+\frac{2}{5}a^{9}-\frac{2}{5}a^{8}+\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a$, $\frac{1}{5}a^{18}-\frac{2}{5}a^{14}-\frac{1}{5}a^{13}+\frac{2}{5}a^{12}+\frac{2}{5}a^{11}-\frac{2}{5}a^{10}+\frac{1}{5}a^{9}+\frac{1}{5}a^{8}+\frac{2}{5}a^{7}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}-\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{19}-\frac{2}{5}a^{15}-\frac{1}{5}a^{14}+\frac{2}{5}a^{13}+\frac{2}{5}a^{12}-\frac{2}{5}a^{11}+\frac{1}{5}a^{10}+\frac{1}{5}a^{9}+\frac{2}{5}a^{8}-\frac{1}{5}a^{7}-\frac{2}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{7975}a^{20}-\frac{664}{7975}a^{19}+\frac{4}{1595}a^{18}+\frac{22}{725}a^{17}-\frac{499}{7975}a^{16}+\frac{3893}{7975}a^{15}+\frac{259}{1595}a^{14}+\frac{1483}{7975}a^{13}+\frac{2974}{7975}a^{12}+\frac{2137}{7975}a^{11}-\frac{3966}{7975}a^{10}-\frac{2661}{7975}a^{9}-\frac{3302}{7975}a^{8}+\frac{112}{319}a^{7}-\frac{3582}{7975}a^{6}+\frac{2986}{7975}a^{5}-\frac{372}{1595}a^{4}-\frac{118}{7975}a^{3}+\frac{3508}{7975}a^{2}-\frac{71}{725}a+\frac{214}{7975}$, $\frac{1}{23\!\cdots\!75}a^{21}+\frac{55\!\cdots\!28}{93\!\cdots\!39}a^{20}-\frac{13\!\cdots\!21}{23\!\cdots\!75}a^{19}-\frac{21\!\cdots\!43}{23\!\cdots\!75}a^{18}+\frac{12\!\cdots\!24}{23\!\cdots\!75}a^{17}+\frac{52\!\cdots\!07}{23\!\cdots\!75}a^{16}+\frac{62\!\cdots\!76}{13\!\cdots\!75}a^{15}-\frac{28\!\cdots\!37}{23\!\cdots\!75}a^{14}-\frac{41\!\cdots\!94}{23\!\cdots\!75}a^{13}-\frac{46\!\cdots\!57}{23\!\cdots\!75}a^{12}-\frac{10\!\cdots\!58}{23\!\cdots\!75}a^{11}+\frac{14\!\cdots\!53}{46\!\cdots\!95}a^{10}-\frac{28\!\cdots\!46}{23\!\cdots\!75}a^{9}-\frac{80\!\cdots\!78}{23\!\cdots\!75}a^{8}-\frac{64\!\cdots\!72}{23\!\cdots\!75}a^{7}+\frac{29\!\cdots\!78}{23\!\cdots\!75}a^{6}-\frac{99\!\cdots\!91}{23\!\cdots\!75}a^{5}-\frac{43\!\cdots\!73}{23\!\cdots\!75}a^{4}-\frac{23\!\cdots\!54}{23\!\cdots\!75}a^{3}-\frac{74\!\cdots\!44}{23\!\cdots\!75}a^{2}+\frac{28\!\cdots\!99}{46\!\cdots\!95}a-\frac{32\!\cdots\!36}{80\!\cdots\!75}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $15$ | 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: | $\frac{25\!\cdots\!82}{23\!\cdots\!75}a^{21}-\frac{53\!\cdots\!09}{23\!\cdots\!75}a^{20}-\frac{91\!\cdots\!76}{23\!\cdots\!75}a^{19}-\frac{16\!\cdots\!76}{23\!\cdots\!75}a^{18}+\frac{38\!\cdots\!68}{93\!\cdots\!39}a^{17}+\frac{14\!\cdots\!97}{93\!\cdots\!39}a^{16}-\frac{60\!\cdots\!84}{13\!\cdots\!75}a^{15}-\frac{17\!\cdots\!39}{23\!\cdots\!75}a^{14}-\frac{42\!\cdots\!17}{93\!\cdots\!39}a^{13}+\frac{18\!\cdots\!49}{46\!\cdots\!95}a^{12}+\frac{20\!\cdots\!51}{23\!\cdots\!75}a^{11}+\frac{11\!\cdots\!54}{23\!\cdots\!75}a^{10}-\frac{15\!\cdots\!38}{23\!\cdots\!75}a^{9}-\frac{24\!\cdots\!33}{23\!\cdots\!75}a^{8}-\frac{12\!\cdots\!39}{23\!\cdots\!75}a^{7}-\frac{36\!\cdots\!81}{23\!\cdots\!75}a^{6}+\frac{10\!\cdots\!04}{23\!\cdots\!75}a^{5}+\frac{13\!\cdots\!69}{23\!\cdots\!75}a^{4}-\frac{49\!\cdots\!66}{23\!\cdots\!75}a^{3}-\frac{24\!\cdots\!71}{93\!\cdots\!39}a^{2}+\frac{55\!\cdots\!49}{23\!\cdots\!75}a+\frac{47\!\cdots\!01}{23\!\cdots\!75}$, $\frac{12\!\cdots\!51}{46\!\cdots\!95}a^{21}-\frac{35\!\cdots\!83}{46\!\cdots\!95}a^{20}-\frac{43\!\cdots\!31}{46\!\cdots\!95}a^{19}-\frac{76\!\cdots\!42}{46\!\cdots\!95}a^{18}+\frac{90\!\cdots\!03}{93\!\cdots\!39}a^{17}+\frac{16\!\cdots\!39}{46\!\cdots\!95}a^{16}-\frac{49\!\cdots\!46}{27\!\cdots\!35}a^{15}-\frac{82\!\cdots\!57}{46\!\cdots\!95}a^{14}-\frac{45\!\cdots\!74}{46\!\cdots\!95}a^{13}+\frac{41\!\cdots\!82}{46\!\cdots\!95}a^{12}+\frac{95\!\cdots\!63}{46\!\cdots\!95}a^{11}+\frac{54\!\cdots\!27}{46\!\cdots\!95}a^{10}-\frac{12\!\cdots\!01}{93\!\cdots\!39}a^{9}-\frac{10\!\cdots\!19}{46\!\cdots\!95}a^{8}-\frac{62\!\cdots\!54}{46\!\cdots\!95}a^{7}-\frac{27\!\cdots\!16}{46\!\cdots\!95}a^{6}+\frac{68\!\cdots\!37}{85\!\cdots\!49}a^{5}+\frac{53\!\cdots\!64}{46\!\cdots\!95}a^{4}-\frac{20\!\cdots\!09}{46\!\cdots\!95}a^{3}-\frac{22\!\cdots\!99}{46\!\cdots\!95}a^{2}+\frac{86\!\cdots\!69}{93\!\cdots\!39}a+\frac{20\!\cdots\!90}{93\!\cdots\!39}$, $\frac{40\!\cdots\!44}{23\!\cdots\!75}a^{21}-\frac{31\!\cdots\!25}{93\!\cdots\!39}a^{20}-\frac{14\!\cdots\!89}{23\!\cdots\!75}a^{19}-\frac{26\!\cdots\!57}{23\!\cdots\!75}a^{18}+\frac{14\!\cdots\!96}{23\!\cdots\!75}a^{17}+\frac{56\!\cdots\!13}{23\!\cdots\!75}a^{16}+\frac{14\!\cdots\!19}{13\!\cdots\!75}a^{15}-\frac{27\!\cdots\!93}{23\!\cdots\!75}a^{14}-\frac{17\!\cdots\!71}{23\!\cdots\!75}a^{13}+\frac{13\!\cdots\!02}{23\!\cdots\!75}a^{12}+\frac{32\!\cdots\!88}{23\!\cdots\!75}a^{11}+\frac{81\!\cdots\!36}{93\!\cdots\!39}a^{10}-\frac{20\!\cdots\!79}{23\!\cdots\!75}a^{9}-\frac{37\!\cdots\!47}{23\!\cdots\!75}a^{8}-\frac{22\!\cdots\!38}{23\!\cdots\!75}a^{7}-\frac{89\!\cdots\!98}{23\!\cdots\!75}a^{6}+\frac{12\!\cdots\!01}{23\!\cdots\!75}a^{5}+\frac{18\!\cdots\!08}{23\!\cdots\!75}a^{4}-\frac{54\!\cdots\!61}{23\!\cdots\!75}a^{3}-\frac{88\!\cdots\!66}{23\!\cdots\!75}a^{2}+\frac{24\!\cdots\!82}{46\!\cdots\!95}a+\frac{45\!\cdots\!64}{23\!\cdots\!75}$, $\frac{66\!\cdots\!29}{23\!\cdots\!75}a^{21}-\frac{19\!\cdots\!19}{23\!\cdots\!75}a^{20}-\frac{23\!\cdots\!83}{23\!\cdots\!75}a^{19}-\frac{40\!\cdots\!47}{23\!\cdots\!75}a^{18}+\frac{24\!\cdots\!93}{23\!\cdots\!75}a^{17}+\frac{90\!\cdots\!34}{23\!\cdots\!75}a^{16}-\frac{33\!\cdots\!52}{13\!\cdots\!75}a^{15}-\frac{44\!\cdots\!48}{23\!\cdots\!75}a^{14}-\frac{24\!\cdots\!73}{23\!\cdots\!75}a^{13}+\frac{22\!\cdots\!51}{23\!\cdots\!75}a^{12}+\frac{10\!\cdots\!48}{46\!\cdots\!95}a^{11}+\frac{28\!\cdots\!19}{23\!\cdots\!75}a^{10}-\frac{75\!\cdots\!74}{46\!\cdots\!95}a^{9}-\frac{58\!\cdots\!39}{23\!\cdots\!75}a^{8}-\frac{31\!\cdots\!23}{23\!\cdots\!75}a^{7}-\frac{22\!\cdots\!02}{46\!\cdots\!95}a^{6}+\frac{20\!\cdots\!92}{21\!\cdots\!25}a^{5}+\frac{30\!\cdots\!88}{23\!\cdots\!75}a^{4}-\frac{11\!\cdots\!34}{23\!\cdots\!75}a^{3}-\frac{14\!\cdots\!83}{23\!\cdots\!75}a^{2}+\frac{86\!\cdots\!51}{80\!\cdots\!75}a+\frac{13\!\cdots\!13}{23\!\cdots\!75}$, $\frac{45\!\cdots\!74}{23\!\cdots\!75}a^{21}-\frac{20\!\cdots\!04}{23\!\cdots\!75}a^{20}-\frac{15\!\cdots\!28}{23\!\cdots\!75}a^{19}-\frac{26\!\cdots\!37}{23\!\cdots\!75}a^{18}+\frac{17\!\cdots\!18}{23\!\cdots\!75}a^{17}+\frac{63\!\cdots\!39}{23\!\cdots\!75}a^{16}-\frac{86\!\cdots\!22}{13\!\cdots\!75}a^{15}-\frac{34\!\cdots\!63}{23\!\cdots\!75}a^{14}-\frac{18\!\cdots\!03}{23\!\cdots\!75}a^{13}+\frac{17\!\cdots\!41}{23\!\cdots\!75}a^{12}+\frac{14\!\cdots\!57}{93\!\cdots\!39}a^{11}+\frac{89\!\cdots\!84}{23\!\cdots\!75}a^{10}-\frac{11\!\cdots\!69}{46\!\cdots\!95}a^{9}-\frac{64\!\cdots\!44}{23\!\cdots\!75}a^{8}-\frac{85\!\cdots\!53}{23\!\cdots\!75}a^{7}+\frac{56\!\cdots\!14}{46\!\cdots\!95}a^{6}+\frac{37\!\cdots\!67}{21\!\cdots\!25}a^{5}+\frac{35\!\cdots\!68}{23\!\cdots\!75}a^{4}-\frac{14\!\cdots\!89}{23\!\cdots\!75}a^{3}-\frac{27\!\cdots\!88}{23\!\cdots\!75}a^{2}+\frac{32\!\cdots\!54}{23\!\cdots\!75}a+\frac{63\!\cdots\!88}{23\!\cdots\!75}$, $\frac{39\!\cdots\!19}{23\!\cdots\!75}a^{21}-\frac{47\!\cdots\!24}{80\!\cdots\!75}a^{20}-\frac{27\!\cdots\!92}{46\!\cdots\!95}a^{19}-\frac{23\!\cdots\!67}{23\!\cdots\!75}a^{18}+\frac{14\!\cdots\!09}{23\!\cdots\!75}a^{17}+\frac{53\!\cdots\!27}{23\!\cdots\!75}a^{16}-\frac{87\!\cdots\!79}{27\!\cdots\!35}a^{15}-\frac{24\!\cdots\!83}{21\!\cdots\!25}a^{14}-\frac{12\!\cdots\!14}{23\!\cdots\!75}a^{13}+\frac{13\!\cdots\!13}{23\!\cdots\!75}a^{12}+\frac{29\!\cdots\!11}{23\!\cdots\!75}a^{11}+\frac{14\!\cdots\!61}{23\!\cdots\!75}a^{10}-\frac{23\!\cdots\!48}{23\!\cdots\!75}a^{9}-\frac{60\!\cdots\!99}{42\!\cdots\!45}a^{8}-\frac{15\!\cdots\!53}{21\!\cdots\!25}a^{7}-\frac{55\!\cdots\!66}{23\!\cdots\!75}a^{6}+\frac{29\!\cdots\!36}{46\!\cdots\!95}a^{5}+\frac{15\!\cdots\!63}{21\!\cdots\!25}a^{4}-\frac{81\!\cdots\!28}{23\!\cdots\!75}a^{3}-\frac{76\!\cdots\!84}{23\!\cdots\!75}a^{2}+\frac{13\!\cdots\!51}{23\!\cdots\!75}a+\frac{13\!\cdots\!77}{46\!\cdots\!95}$, $\frac{13\!\cdots\!37}{23\!\cdots\!75}a^{21}-\frac{83\!\cdots\!87}{23\!\cdots\!75}a^{20}-\frac{47\!\cdots\!84}{23\!\cdots\!75}a^{19}-\frac{69\!\cdots\!91}{23\!\cdots\!75}a^{18}+\frac{52\!\cdots\!09}{23\!\cdots\!75}a^{17}+\frac{15\!\cdots\!82}{21\!\cdots\!25}a^{16}-\frac{34\!\cdots\!01}{12\!\cdots\!25}a^{15}-\frac{88\!\cdots\!74}{23\!\cdots\!75}a^{14}-\frac{21\!\cdots\!89}{23\!\cdots\!75}a^{13}+\frac{46\!\cdots\!68}{23\!\cdots\!75}a^{12}+\frac{18\!\cdots\!52}{46\!\cdots\!95}a^{11}+\frac{32\!\cdots\!07}{23\!\cdots\!75}a^{10}-\frac{15\!\cdots\!82}{46\!\cdots\!95}a^{9}-\frac{89\!\cdots\!12}{23\!\cdots\!75}a^{8}-\frac{44\!\cdots\!44}{23\!\cdots\!75}a^{7}-\frac{37\!\cdots\!17}{46\!\cdots\!95}a^{6}+\frac{47\!\cdots\!76}{23\!\cdots\!75}a^{5}+\frac{42\!\cdots\!24}{23\!\cdots\!75}a^{4}-\frac{31\!\cdots\!62}{23\!\cdots\!75}a^{3}-\frac{16\!\cdots\!09}{23\!\cdots\!75}a^{2}+\frac{82\!\cdots\!02}{23\!\cdots\!75}a-\frac{33\!\cdots\!91}{23\!\cdots\!75}$, $\frac{82\!\cdots\!52}{23\!\cdots\!75}a^{21}-\frac{18\!\cdots\!08}{23\!\cdots\!75}a^{20}-\frac{57\!\cdots\!12}{46\!\cdots\!95}a^{19}-\frac{52\!\cdots\!66}{23\!\cdots\!75}a^{18}+\frac{29\!\cdots\!07}{23\!\cdots\!75}a^{17}+\frac{11\!\cdots\!76}{23\!\cdots\!75}a^{16}+\frac{44\!\cdots\!76}{27\!\cdots\!35}a^{15}-\frac{54\!\cdots\!79}{23\!\cdots\!75}a^{14}-\frac{34\!\cdots\!27}{23\!\cdots\!75}a^{13}+\frac{27\!\cdots\!39}{23\!\cdots\!75}a^{12}+\frac{65\!\cdots\!73}{23\!\cdots\!75}a^{11}+\frac{41\!\cdots\!83}{23\!\cdots\!75}a^{10}-\frac{39\!\cdots\!84}{23\!\cdots\!75}a^{9}-\frac{14\!\cdots\!49}{46\!\cdots\!95}a^{8}-\frac{46\!\cdots\!04}{23\!\cdots\!75}a^{7}-\frac{21\!\cdots\!48}{23\!\cdots\!75}a^{6}+\frac{44\!\cdots\!71}{46\!\cdots\!95}a^{5}+\frac{36\!\cdots\!09}{23\!\cdots\!75}a^{4}-\frac{10\!\cdots\!84}{23\!\cdots\!75}a^{3}-\frac{16\!\cdots\!37}{23\!\cdots\!75}a^{2}+\frac{28\!\cdots\!63}{23\!\cdots\!75}a+\frac{25\!\cdots\!48}{46\!\cdots\!95}$, $\frac{48\!\cdots\!24}{23\!\cdots\!75}a^{21}-\frac{18\!\cdots\!86}{23\!\cdots\!75}a^{20}-\frac{61\!\cdots\!23}{85\!\cdots\!49}a^{19}-\frac{28\!\cdots\!97}{23\!\cdots\!75}a^{18}+\frac{17\!\cdots\!19}{23\!\cdots\!75}a^{17}+\frac{64\!\cdots\!22}{23\!\cdots\!75}a^{16}-\frac{24\!\cdots\!95}{55\!\cdots\!67}a^{15}-\frac{32\!\cdots\!73}{23\!\cdots\!75}a^{14}-\frac{15\!\cdots\!89}{23\!\cdots\!75}a^{13}+\frac{16\!\cdots\!28}{23\!\cdots\!75}a^{12}+\frac{36\!\cdots\!36}{23\!\cdots\!75}a^{11}+\frac{16\!\cdots\!16}{23\!\cdots\!75}a^{10}-\frac{30\!\cdots\!58}{23\!\cdots\!75}a^{9}-\frac{16\!\cdots\!59}{93\!\cdots\!39}a^{8}-\frac{19\!\cdots\!63}{23\!\cdots\!75}a^{7}-\frac{45\!\cdots\!21}{23\!\cdots\!75}a^{6}+\frac{75\!\cdots\!27}{93\!\cdots\!39}a^{5}+\frac{21\!\cdots\!73}{23\!\cdots\!75}a^{4}-\frac{10\!\cdots\!68}{23\!\cdots\!75}a^{3}-\frac{10\!\cdots\!84}{23\!\cdots\!75}a^{2}+\frac{24\!\cdots\!11}{23\!\cdots\!75}a+\frac{22\!\cdots\!18}{46\!\cdots\!95}$, $\frac{26\!\cdots\!51}{21\!\cdots\!25}a^{21}-\frac{13\!\cdots\!23}{73\!\cdots\!25}a^{20}-\frac{91\!\cdots\!43}{21\!\cdots\!25}a^{19}-\frac{17\!\cdots\!63}{21\!\cdots\!25}a^{18}+\frac{18\!\cdots\!81}{42\!\cdots\!45}a^{17}+\frac{75\!\cdots\!66}{42\!\cdots\!45}a^{16}+\frac{31\!\cdots\!23}{12\!\cdots\!25}a^{15}-\frac{17\!\cdots\!87}{21\!\cdots\!25}a^{14}-\frac{27\!\cdots\!11}{42\!\cdots\!45}a^{13}+\frac{17\!\cdots\!73}{42\!\cdots\!45}a^{12}+\frac{22\!\cdots\!93}{21\!\cdots\!25}a^{11}+\frac{15\!\cdots\!62}{21\!\cdots\!25}a^{10}-\frac{13\!\cdots\!74}{21\!\cdots\!25}a^{9}-\frac{27\!\cdots\!54}{21\!\cdots\!25}a^{8}-\frac{17\!\cdots\!87}{21\!\cdots\!25}a^{7}-\frac{54\!\cdots\!78}{21\!\cdots\!25}a^{6}+\frac{82\!\cdots\!02}{21\!\cdots\!25}a^{5}+\frac{13\!\cdots\!22}{21\!\cdots\!25}a^{4}-\frac{17\!\cdots\!43}{21\!\cdots\!25}a^{3}-\frac{15\!\cdots\!52}{42\!\cdots\!45}a^{2}+\frac{21\!\cdots\!17}{21\!\cdots\!25}a+\frac{89\!\cdots\!58}{21\!\cdots\!25}$, $\frac{37\!\cdots\!02}{23\!\cdots\!75}a^{21}-\frac{13\!\cdots\!62}{23\!\cdots\!75}a^{20}-\frac{12\!\cdots\!14}{23\!\cdots\!75}a^{19}-\frac{20\!\cdots\!81}{21\!\cdots\!25}a^{18}+\frac{12\!\cdots\!79}{23\!\cdots\!75}a^{17}+\frac{50\!\cdots\!82}{23\!\cdots\!75}a^{16}+\frac{30\!\cdots\!84}{13\!\cdots\!75}a^{15}-\frac{22\!\cdots\!79}{23\!\cdots\!75}a^{14}-\frac{16\!\cdots\!29}{23\!\cdots\!75}a^{13}+\frac{11\!\cdots\!38}{23\!\cdots\!75}a^{12}+\frac{58\!\cdots\!18}{46\!\cdots\!95}a^{11}+\frac{22\!\cdots\!92}{23\!\cdots\!75}a^{10}-\frac{50\!\cdots\!94}{93\!\cdots\!39}a^{9}-\frac{34\!\cdots\!37}{23\!\cdots\!75}a^{8}-\frac{29\!\cdots\!24}{23\!\cdots\!75}a^{7}-\frac{28\!\cdots\!84}{46\!\cdots\!95}a^{6}+\frac{85\!\cdots\!91}{23\!\cdots\!75}a^{5}+\frac{18\!\cdots\!39}{23\!\cdots\!75}a^{4}+\frac{20\!\cdots\!88}{23\!\cdots\!75}a^{3}-\frac{55\!\cdots\!19}{21\!\cdots\!25}a^{2}-\frac{17\!\cdots\!73}{23\!\cdots\!75}a-\frac{10\!\cdots\!71}{21\!\cdots\!25}$, $\frac{26\!\cdots\!79}{23\!\cdots\!75}a^{21}-\frac{76\!\cdots\!32}{23\!\cdots\!75}a^{20}-\frac{90\!\cdots\!21}{23\!\cdots\!75}a^{19}-\frac{16\!\cdots\!82}{23\!\cdots\!75}a^{18}+\frac{32\!\cdots\!38}{80\!\cdots\!75}a^{17}+\frac{35\!\cdots\!31}{23\!\cdots\!75}a^{16}-\frac{21\!\cdots\!91}{47\!\cdots\!75}a^{15}-\frac{17\!\cdots\!08}{23\!\cdots\!75}a^{14}-\frac{10\!\cdots\!32}{23\!\cdots\!75}a^{13}+\frac{87\!\cdots\!14}{23\!\cdots\!75}a^{12}+\frac{20\!\cdots\!79}{23\!\cdots\!75}a^{11}+\frac{11\!\cdots\!57}{23\!\cdots\!75}a^{10}-\frac{14\!\cdots\!22}{23\!\cdots\!75}a^{9}-\frac{23\!\cdots\!18}{23\!\cdots\!75}a^{8}-\frac{13\!\cdots\!38}{23\!\cdots\!75}a^{7}-\frac{40\!\cdots\!14}{21\!\cdots\!25}a^{6}+\frac{87\!\cdots\!89}{23\!\cdots\!75}a^{5}+\frac{12\!\cdots\!03}{23\!\cdots\!75}a^{4}-\frac{77\!\cdots\!29}{46\!\cdots\!95}a^{3}-\frac{59\!\cdots\!82}{23\!\cdots\!75}a^{2}+\frac{77\!\cdots\!67}{23\!\cdots\!75}a+\frac{64\!\cdots\!61}{23\!\cdots\!75}$, $\frac{68\!\cdots\!07}{23\!\cdots\!75}a^{21}-\frac{19\!\cdots\!97}{46\!\cdots\!95}a^{20}-\frac{22\!\cdots\!82}{23\!\cdots\!75}a^{19}-\frac{19\!\cdots\!41}{23\!\cdots\!75}a^{18}+\frac{25\!\cdots\!93}{23\!\cdots\!75}a^{17}+\frac{66\!\cdots\!89}{23\!\cdots\!75}a^{16}-\frac{30\!\cdots\!28}{13\!\cdots\!75}a^{15}-\frac{35\!\cdots\!29}{23\!\cdots\!75}a^{14}+\frac{35\!\cdots\!92}{23\!\cdots\!75}a^{13}+\frac{19\!\cdots\!61}{23\!\cdots\!75}a^{12}+\frac{34\!\cdots\!79}{23\!\cdots\!75}a^{11}+\frac{43\!\cdots\!14}{93\!\cdots\!39}a^{10}-\frac{30\!\cdots\!62}{23\!\cdots\!75}a^{9}-\frac{42\!\cdots\!56}{23\!\cdots\!75}a^{8}-\frac{34\!\cdots\!44}{23\!\cdots\!75}a^{7}-\frac{10\!\cdots\!74}{23\!\cdots\!75}a^{6}+\frac{24\!\cdots\!78}{21\!\cdots\!25}a^{5}+\frac{28\!\cdots\!79}{23\!\cdots\!75}a^{4}+\frac{94\!\cdots\!52}{23\!\cdots\!75}a^{3}+\frac{38\!\cdots\!62}{23\!\cdots\!75}a^{2}-\frac{47\!\cdots\!02}{46\!\cdots\!95}a-\frac{45\!\cdots\!72}{80\!\cdots\!75}$, $\frac{15\!\cdots\!77}{21\!\cdots\!25}a^{21}-\frac{16\!\cdots\!69}{46\!\cdots\!95}a^{20}-\frac{60\!\cdots\!82}{23\!\cdots\!75}a^{19}-\frac{97\!\cdots\!81}{23\!\cdots\!75}a^{18}+\frac{58\!\cdots\!53}{21\!\cdots\!25}a^{17}+\frac{22\!\cdots\!59}{23\!\cdots\!75}a^{16}-\frac{27\!\cdots\!03}{13\!\cdots\!75}a^{15}-\frac{11\!\cdots\!69}{23\!\cdots\!75}a^{14}-\frac{47\!\cdots\!28}{23\!\cdots\!75}a^{13}+\frac{59\!\cdots\!26}{23\!\cdots\!75}a^{12}+\frac{12\!\cdots\!24}{23\!\cdots\!75}a^{11}+\frac{22\!\cdots\!87}{93\!\cdots\!39}a^{10}-\frac{10\!\cdots\!37}{23\!\cdots\!75}a^{9}-\frac{13\!\cdots\!26}{23\!\cdots\!75}a^{8}-\frac{68\!\cdots\!64}{23\!\cdots\!75}a^{7}-\frac{20\!\cdots\!64}{23\!\cdots\!75}a^{6}+\frac{64\!\cdots\!48}{23\!\cdots\!75}a^{5}+\frac{71\!\cdots\!44}{23\!\cdots\!75}a^{4}-\frac{35\!\cdots\!98}{23\!\cdots\!75}a^{3}-\frac{31\!\cdots\!93}{23\!\cdots\!75}a^{2}+\frac{14\!\cdots\!73}{42\!\cdots\!45}a+\frac{22\!\cdots\!92}{23\!\cdots\!75}$, $\frac{23\!\cdots\!93}{23\!\cdots\!75}a^{21}-\frac{16\!\cdots\!24}{23\!\cdots\!75}a^{20}-\frac{80\!\cdots\!37}{23\!\cdots\!75}a^{19}-\frac{11\!\cdots\!69}{23\!\cdots\!75}a^{18}+\frac{89\!\cdots\!24}{23\!\cdots\!75}a^{17}+\frac{28\!\cdots\!97}{23\!\cdots\!75}a^{16}-\frac{77\!\cdots\!33}{13\!\cdots\!75}a^{15}-\frac{13\!\cdots\!26}{21\!\cdots\!25}a^{14}-\frac{28\!\cdots\!34}{23\!\cdots\!75}a^{13}+\frac{79\!\cdots\!93}{23\!\cdots\!75}a^{12}+\frac{14\!\cdots\!08}{23\!\cdots\!75}a^{11}+\frac{42\!\cdots\!24}{23\!\cdots\!75}a^{10}-\frac{15\!\cdots\!94}{23\!\cdots\!75}a^{9}-\frac{14\!\cdots\!36}{21\!\cdots\!25}a^{8}-\frac{22\!\cdots\!34}{73\!\cdots\!25}a^{7}-\frac{18\!\cdots\!58}{23\!\cdots\!75}a^{6}+\frac{92\!\cdots\!83}{23\!\cdots\!75}a^{5}+\frac{67\!\cdots\!66}{21\!\cdots\!25}a^{4}-\frac{23\!\cdots\!69}{93\!\cdots\!39}a^{3}-\frac{26\!\cdots\!59}{23\!\cdots\!75}a^{2}+\frac{78\!\cdots\!19}{23\!\cdots\!75}a+\frac{29\!\cdots\!67}{23\!\cdots\!75}$ (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: | \( 45272687172.7 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{6}\cdot 45272687172.7 \cdot 1}{2\cdot\sqrt{22661033510180079603495293971842498241}}\cr\approx \mathstrut & 0.299602606641 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.D_{11}$ (as 22T30):
A solvable group of order 22528 |
The 100 conjugacy class representatives for $C_2^{10}.D_{11}$ are not computed |
Character table for $C_2^{10}.D_{11}$ is not computed |
Intermediate fields
11.11.3670285774226257.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 22 siblings: | data not computed |
Degree 44 siblings: | data not computed |
Minimal sibling: | 22.14.17471883970840462300304775614373553.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.11.0.1}{11} }^{2}$ | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{9}$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.11.0.1}{11} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{5}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{5}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.11.0.1}{11} }^{2}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(1297\) | $\Q_{1297}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1297}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $4$ | $1$ | $3$ | ||||
Deg $4$ | $4$ | $1$ | $3$ |