Normalized defining polynomial
\( x^{22} - 140 x^{20} + 8475 x^{18} - 291035 x^{16} + 6250874 x^{14} - 87255611 x^{12} + 795482411 x^{10} - 4618766140 x^{8} + 15964116759 x^{6} + \cdots - 3015217921 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[22, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(170364212909807025886845565709641708504367497216\) \(\medspace = 2^{22}\cdot 43^{2}\cdot 1277^{2}\cdot 1297^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(140.24\) | 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: | not computed | ||
Ramified primes: | \(2\), \(43\), \(1277\), \(1297\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5}a^{14}+\frac{2}{5}a^{10}-\frac{2}{5}a^{8}+\frac{1}{5}a^{4}-\frac{1}{5}a^{2}-\frac{2}{5}$, $\frac{1}{5}a^{15}+\frac{2}{5}a^{11}-\frac{2}{5}a^{9}+\frac{1}{5}a^{5}-\frac{1}{5}a^{3}-\frac{2}{5}a$, $\frac{1}{5}a^{16}+\frac{2}{5}a^{12}-\frac{2}{5}a^{10}+\frac{1}{5}a^{6}-\frac{1}{5}a^{4}-\frac{2}{5}a^{2}$, $\frac{1}{5}a^{17}+\frac{2}{5}a^{13}-\frac{2}{5}a^{11}+\frac{1}{5}a^{7}-\frac{1}{5}a^{5}-\frac{2}{5}a^{3}$, $\frac{1}{145}a^{18}-\frac{3}{145}a^{16}-\frac{14}{145}a^{14}-\frac{63}{145}a^{12}+\frac{14}{145}a^{10}-\frac{37}{145}a^{8}+\frac{71}{145}a^{6}+\frac{11}{29}a^{4}+\frac{42}{145}a^{2}-\frac{63}{145}$, $\frac{1}{145}a^{19}-\frac{3}{145}a^{17}-\frac{14}{145}a^{15}-\frac{63}{145}a^{13}+\frac{14}{145}a^{11}-\frac{37}{145}a^{9}+\frac{71}{145}a^{7}+\frac{11}{29}a^{5}+\frac{42}{145}a^{3}-\frac{63}{145}a$, $\frac{1}{20\!\cdots\!05}a^{20}-\frac{60\!\cdots\!22}{20\!\cdots\!05}a^{18}+\frac{59\!\cdots\!86}{40\!\cdots\!61}a^{16}+\frac{10\!\cdots\!38}{20\!\cdots\!05}a^{14}-\frac{11\!\cdots\!68}{40\!\cdots\!61}a^{12}-\frac{23\!\cdots\!42}{20\!\cdots\!05}a^{10}+\frac{64\!\cdots\!19}{20\!\cdots\!05}a^{8}-\frac{99\!\cdots\!97}{20\!\cdots\!05}a^{6}+\frac{57\!\cdots\!40}{40\!\cdots\!61}a^{4}-\frac{40\!\cdots\!34}{40\!\cdots\!61}a^{2}+\frac{19\!\cdots\!72}{36\!\cdots\!55}$, $\frac{1}{20\!\cdots\!05}a^{21}-\frac{60\!\cdots\!22}{20\!\cdots\!05}a^{19}+\frac{59\!\cdots\!86}{40\!\cdots\!61}a^{17}+\frac{10\!\cdots\!38}{20\!\cdots\!05}a^{15}-\frac{11\!\cdots\!68}{40\!\cdots\!61}a^{13}-\frac{23\!\cdots\!42}{20\!\cdots\!05}a^{11}+\frac{64\!\cdots\!19}{20\!\cdots\!05}a^{9}-\frac{99\!\cdots\!97}{20\!\cdots\!05}a^{7}+\frac{57\!\cdots\!40}{40\!\cdots\!61}a^{5}-\frac{40\!\cdots\!34}{40\!\cdots\!61}a^{3}+\frac{19\!\cdots\!72}{36\!\cdots\!55}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $21$ | 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: | $\frac{74\!\cdots\!52}{20\!\cdots\!05}a^{20}-\frac{19\!\cdots\!39}{40\!\cdots\!61}a^{18}+\frac{55\!\cdots\!01}{20\!\cdots\!05}a^{16}-\frac{17\!\cdots\!31}{20\!\cdots\!05}a^{14}+\frac{33\!\cdots\!69}{20\!\cdots\!05}a^{12}-\frac{40\!\cdots\!06}{20\!\cdots\!05}a^{10}+\frac{29\!\cdots\!32}{20\!\cdots\!05}a^{8}-\frac{12\!\cdots\!82}{20\!\cdots\!05}a^{6}+\frac{25\!\cdots\!16}{20\!\cdots\!05}a^{4}-\frac{16\!\cdots\!57}{20\!\cdots\!05}a^{2}+\frac{54\!\cdots\!44}{36\!\cdots\!55}$, $\frac{50\!\cdots\!82}{20\!\cdots\!05}a^{20}-\frac{67\!\cdots\!24}{20\!\cdots\!05}a^{18}+\frac{37\!\cdots\!28}{20\!\cdots\!05}a^{16}-\frac{11\!\cdots\!91}{20\!\cdots\!05}a^{14}+\frac{22\!\cdots\!36}{20\!\cdots\!05}a^{12}-\frac{26\!\cdots\!54}{20\!\cdots\!05}a^{10}+\frac{19\!\cdots\!42}{20\!\cdots\!05}a^{8}-\frac{82\!\cdots\!86}{20\!\cdots\!05}a^{6}+\frac{34\!\cdots\!82}{40\!\cdots\!61}a^{4}-\frac{10\!\cdots\!19}{20\!\cdots\!05}a^{2}+\frac{35\!\cdots\!88}{36\!\cdots\!55}$, $\frac{10\!\cdots\!28}{20\!\cdots\!05}a^{20}-\frac{13\!\cdots\!49}{20\!\cdots\!05}a^{18}+\frac{75\!\cdots\!42}{20\!\cdots\!05}a^{16}-\frac{23\!\cdots\!32}{20\!\cdots\!05}a^{14}+\frac{89\!\cdots\!74}{40\!\cdots\!61}a^{12}-\frac{10\!\cdots\!45}{40\!\cdots\!61}a^{10}+\frac{38\!\cdots\!34}{20\!\cdots\!05}a^{8}-\frac{16\!\cdots\!16}{20\!\cdots\!05}a^{6}+\frac{33\!\cdots\!64}{20\!\cdots\!05}a^{4}-\frac{21\!\cdots\!44}{20\!\cdots\!05}a^{2}+\frac{70\!\cdots\!41}{36\!\cdots\!55}$, $\frac{39\!\cdots\!96}{20\!\cdots\!05}a^{20}-\frac{52\!\cdots\!13}{20\!\cdots\!05}a^{18}+\frac{29\!\cdots\!59}{20\!\cdots\!05}a^{16}-\frac{18\!\cdots\!37}{40\!\cdots\!61}a^{14}+\frac{34\!\cdots\!54}{40\!\cdots\!61}a^{12}-\frac{20\!\cdots\!42}{20\!\cdots\!05}a^{10}+\frac{29\!\cdots\!16}{40\!\cdots\!61}a^{8}-\frac{61\!\cdots\!77}{20\!\cdots\!05}a^{6}+\frac{12\!\cdots\!12}{20\!\cdots\!05}a^{4}-\frac{78\!\cdots\!92}{20\!\cdots\!05}a^{2}+\frac{26\!\cdots\!09}{36\!\cdots\!55}$, $\frac{57\!\cdots\!44}{40\!\cdots\!61}a^{20}-\frac{38\!\cdots\!28}{20\!\cdots\!05}a^{18}+\frac{21\!\cdots\!71}{20\!\cdots\!05}a^{16}-\frac{67\!\cdots\!51}{20\!\cdots\!05}a^{14}+\frac{12\!\cdots\!28}{20\!\cdots\!05}a^{12}-\frac{15\!\cdots\!42}{20\!\cdots\!05}a^{10}+\frac{11\!\cdots\!17}{20\!\cdots\!05}a^{8}-\frac{46\!\cdots\!16}{20\!\cdots\!05}a^{6}+\frac{19\!\cdots\!00}{40\!\cdots\!61}a^{4}-\frac{59\!\cdots\!77}{20\!\cdots\!05}a^{2}+\frac{40\!\cdots\!90}{72\!\cdots\!51}$, $\frac{43\!\cdots\!64}{20\!\cdots\!05}a^{20}-\frac{57\!\cdots\!24}{20\!\cdots\!05}a^{18}+\frac{32\!\cdots\!58}{20\!\cdots\!05}a^{16}-\frac{20\!\cdots\!05}{40\!\cdots\!61}a^{14}+\frac{19\!\cdots\!73}{20\!\cdots\!05}a^{12}-\frac{22\!\cdots\!09}{20\!\cdots\!05}a^{10}+\frac{33\!\cdots\!75}{40\!\cdots\!61}a^{8}-\frac{69\!\cdots\!74}{20\!\cdots\!05}a^{6}+\frac{14\!\cdots\!54}{20\!\cdots\!05}a^{4}-\frac{90\!\cdots\!16}{20\!\cdots\!05}a^{2}+\frac{30\!\cdots\!28}{36\!\cdots\!55}$, $\frac{11\!\cdots\!89}{20\!\cdots\!05}a^{20}-\frac{15\!\cdots\!38}{20\!\cdots\!05}a^{18}+\frac{85\!\cdots\!32}{20\!\cdots\!05}a^{16}-\frac{26\!\cdots\!03}{20\!\cdots\!05}a^{14}+\frac{50\!\cdots\!64}{20\!\cdots\!05}a^{12}-\frac{60\!\cdots\!97}{20\!\cdots\!05}a^{10}+\frac{43\!\cdots\!46}{20\!\cdots\!05}a^{8}-\frac{18\!\cdots\!96}{20\!\cdots\!05}a^{6}+\frac{37\!\cdots\!48}{20\!\cdots\!05}a^{4}-\frac{23\!\cdots\!34}{20\!\cdots\!05}a^{2}+\frac{79\!\cdots\!88}{36\!\cdots\!55}$, $\frac{16\!\cdots\!96}{20\!\cdots\!05}a^{20}-\frac{21\!\cdots\!92}{20\!\cdots\!05}a^{18}+\frac{12\!\cdots\!88}{20\!\cdots\!05}a^{16}-\frac{75\!\cdots\!27}{40\!\cdots\!61}a^{14}+\frac{72\!\cdots\!76}{20\!\cdots\!05}a^{12}-\frac{85\!\cdots\!89}{20\!\cdots\!05}a^{10}+\frac{12\!\cdots\!60}{40\!\cdots\!61}a^{8}-\frac{26\!\cdots\!59}{20\!\cdots\!05}a^{6}+\frac{53\!\cdots\!34}{20\!\cdots\!05}a^{4}-\frac{33\!\cdots\!13}{20\!\cdots\!05}a^{2}+\frac{11\!\cdots\!38}{36\!\cdots\!55}$, $\frac{43\!\cdots\!64}{20\!\cdots\!05}a^{20}-\frac{57\!\cdots\!24}{20\!\cdots\!05}a^{18}+\frac{32\!\cdots\!58}{20\!\cdots\!05}a^{16}-\frac{20\!\cdots\!05}{40\!\cdots\!61}a^{14}+\frac{19\!\cdots\!73}{20\!\cdots\!05}a^{12}-\frac{22\!\cdots\!09}{20\!\cdots\!05}a^{10}+\frac{33\!\cdots\!75}{40\!\cdots\!61}a^{8}-\frac{69\!\cdots\!74}{20\!\cdots\!05}a^{6}+\frac{14\!\cdots\!54}{20\!\cdots\!05}a^{4}-\frac{90\!\cdots\!16}{20\!\cdots\!05}a^{2}+\frac{30\!\cdots\!73}{36\!\cdots\!55}$, $\frac{46\!\cdots\!32}{46\!\cdots\!35}a^{20}-\frac{61\!\cdots\!89}{46\!\cdots\!35}a^{18}+\frac{34\!\cdots\!71}{46\!\cdots\!35}a^{16}-\frac{10\!\cdots\!41}{46\!\cdots\!35}a^{14}+\frac{20\!\cdots\!62}{46\!\cdots\!35}a^{12}-\frac{49\!\cdots\!57}{93\!\cdots\!27}a^{10}+\frac{17\!\cdots\!57}{46\!\cdots\!35}a^{8}-\frac{75\!\cdots\!73}{46\!\cdots\!35}a^{6}+\frac{15\!\cdots\!02}{46\!\cdots\!35}a^{4}-\frac{19\!\cdots\!49}{93\!\cdots\!27}a^{2}+\frac{14\!\cdots\!69}{36\!\cdots\!55}$, $\frac{63\!\cdots\!22}{20\!\cdots\!05}a^{21}-\frac{20\!\cdots\!32}{20\!\cdots\!05}a^{20}-\frac{82\!\cdots\!36}{20\!\cdots\!05}a^{19}+\frac{54\!\cdots\!96}{40\!\cdots\!61}a^{18}+\frac{45\!\cdots\!34}{20\!\cdots\!05}a^{17}-\frac{15\!\cdots\!43}{20\!\cdots\!05}a^{16}-\frac{13\!\cdots\!36}{20\!\cdots\!05}a^{15}+\frac{48\!\cdots\!48}{20\!\cdots\!05}a^{14}+\frac{25\!\cdots\!07}{20\!\cdots\!05}a^{13}-\frac{91\!\cdots\!28}{20\!\cdots\!05}a^{12}-\frac{29\!\cdots\!83}{20\!\cdots\!05}a^{11}+\frac{10\!\cdots\!54}{20\!\cdots\!05}a^{10}+\frac{20\!\cdots\!82}{20\!\cdots\!05}a^{9}-\frac{79\!\cdots\!26}{20\!\cdots\!05}a^{8}-\frac{78\!\cdots\!38}{20\!\cdots\!05}a^{7}+\frac{66\!\cdots\!34}{40\!\cdots\!61}a^{6}+\frac{14\!\cdots\!97}{20\!\cdots\!05}a^{5}-\frac{68\!\cdots\!82}{20\!\cdots\!05}a^{4}-\frac{13\!\cdots\!45}{40\!\cdots\!61}a^{3}+\frac{42\!\cdots\!24}{20\!\cdots\!05}a^{2}+\frac{35\!\cdots\!42}{72\!\cdots\!51}a-\frac{14\!\cdots\!93}{36\!\cdots\!55}$, $\frac{98\!\cdots\!52}{20\!\cdots\!05}a^{21}+\frac{29\!\cdots\!95}{40\!\cdots\!61}a^{20}-\frac{13\!\cdots\!31}{20\!\cdots\!05}a^{19}-\frac{19\!\cdots\!69}{20\!\cdots\!05}a^{18}+\frac{72\!\cdots\!41}{20\!\cdots\!05}a^{17}+\frac{21\!\cdots\!05}{40\!\cdots\!61}a^{16}-\frac{22\!\cdots\!52}{20\!\cdots\!05}a^{15}-\frac{33\!\cdots\!91}{20\!\cdots\!05}a^{14}+\frac{42\!\cdots\!91}{20\!\cdots\!05}a^{13}+\frac{63\!\cdots\!43}{20\!\cdots\!05}a^{12}-\frac{50\!\cdots\!44}{20\!\cdots\!05}a^{11}-\frac{73\!\cdots\!71}{20\!\cdots\!05}a^{10}+\frac{36\!\cdots\!24}{20\!\cdots\!05}a^{9}+\frac{52\!\cdots\!87}{20\!\cdots\!05}a^{8}-\frac{15\!\cdots\!46}{20\!\cdots\!05}a^{7}-\frac{21\!\cdots\!26}{20\!\cdots\!05}a^{6}+\frac{30\!\cdots\!14}{20\!\cdots\!05}a^{5}+\frac{88\!\cdots\!39}{40\!\cdots\!61}a^{4}-\frac{19\!\cdots\!93}{20\!\cdots\!05}a^{3}-\frac{27\!\cdots\!02}{20\!\cdots\!05}a^{2}+\frac{65\!\cdots\!82}{36\!\cdots\!55}a+\frac{92\!\cdots\!91}{36\!\cdots\!55}$, $\frac{30\!\cdots\!93}{20\!\cdots\!05}a^{21}+\frac{87\!\cdots\!79}{20\!\cdots\!05}a^{20}-\frac{40\!\cdots\!67}{20\!\cdots\!05}a^{19}-\frac{11\!\cdots\!98}{20\!\cdots\!05}a^{18}+\frac{23\!\cdots\!23}{20\!\cdots\!05}a^{17}+\frac{13\!\cdots\!21}{40\!\cdots\!61}a^{16}-\frac{72\!\cdots\!52}{20\!\cdots\!05}a^{15}-\frac{20\!\cdots\!73}{20\!\cdots\!05}a^{14}+\frac{13\!\cdots\!13}{20\!\cdots\!05}a^{13}+\frac{78\!\cdots\!47}{40\!\cdots\!61}a^{12}-\frac{32\!\cdots\!90}{40\!\cdots\!61}a^{11}-\frac{46\!\cdots\!03}{20\!\cdots\!05}a^{10}+\frac{12\!\cdots\!89}{20\!\cdots\!05}a^{9}+\frac{34\!\cdots\!56}{20\!\cdots\!05}a^{8}-\frac{50\!\cdots\!42}{20\!\cdots\!05}a^{7}-\frac{14\!\cdots\!18}{20\!\cdots\!05}a^{6}+\frac{21\!\cdots\!20}{40\!\cdots\!61}a^{5}+\frac{59\!\cdots\!25}{40\!\cdots\!61}a^{4}-\frac{66\!\cdots\!22}{20\!\cdots\!05}a^{3}-\frac{37\!\cdots\!28}{40\!\cdots\!61}a^{2}+\frac{22\!\cdots\!84}{36\!\cdots\!55}a+\frac{63\!\cdots\!98}{36\!\cdots\!55}$, $\frac{58\!\cdots\!64}{20\!\cdots\!05}a^{21}+\frac{16\!\cdots\!02}{20\!\cdots\!05}a^{20}-\frac{77\!\cdots\!21}{20\!\cdots\!05}a^{19}-\frac{21\!\cdots\!59}{20\!\cdots\!05}a^{18}+\frac{43\!\cdots\!86}{20\!\cdots\!05}a^{17}+\frac{12\!\cdots\!42}{20\!\cdots\!05}a^{16}-\frac{13\!\cdots\!38}{20\!\cdots\!05}a^{15}-\frac{77\!\cdots\!53}{40\!\cdots\!61}a^{14}+\frac{26\!\cdots\!18}{20\!\cdots\!05}a^{13}+\frac{73\!\cdots\!84}{20\!\cdots\!05}a^{12}-\frac{31\!\cdots\!98}{20\!\cdots\!05}a^{11}-\frac{17\!\cdots\!09}{40\!\cdots\!61}a^{10}+\frac{22\!\cdots\!36}{20\!\cdots\!05}a^{9}+\frac{12\!\cdots\!38}{40\!\cdots\!61}a^{8}-\frac{96\!\cdots\!44}{20\!\cdots\!05}a^{7}-\frac{27\!\cdots\!82}{20\!\cdots\!05}a^{6}+\frac{19\!\cdots\!46}{20\!\cdots\!05}a^{5}+\frac{56\!\cdots\!57}{20\!\cdots\!05}a^{4}-\frac{12\!\cdots\!33}{20\!\cdots\!05}a^{3}-\frac{35\!\cdots\!78}{20\!\cdots\!05}a^{2}+\frac{42\!\cdots\!46}{36\!\cdots\!55}a+\frac{11\!\cdots\!96}{36\!\cdots\!55}$, $\frac{17\!\cdots\!09}{40\!\cdots\!61}a^{21}+\frac{26\!\cdots\!76}{20\!\cdots\!05}a^{20}-\frac{22\!\cdots\!64}{40\!\cdots\!61}a^{19}-\frac{34\!\cdots\!72}{20\!\cdots\!05}a^{18}+\frac{63\!\cdots\!64}{20\!\cdots\!05}a^{17}+\frac{19\!\cdots\!53}{20\!\cdots\!05}a^{16}-\frac{19\!\cdots\!04}{20\!\cdots\!05}a^{15}-\frac{61\!\cdots\!51}{20\!\cdots\!05}a^{14}+\frac{37\!\cdots\!03}{20\!\cdots\!05}a^{13}+\frac{11\!\cdots\!96}{20\!\cdots\!05}a^{12}-\frac{45\!\cdots\!06}{20\!\cdots\!05}a^{11}-\frac{13\!\cdots\!41}{20\!\cdots\!05}a^{10}+\frac{33\!\cdots\!88}{20\!\cdots\!05}a^{9}+\frac{10\!\cdots\!32}{20\!\cdots\!05}a^{8}-\frac{13\!\cdots\!71}{20\!\cdots\!05}a^{7}-\frac{42\!\cdots\!64}{20\!\cdots\!05}a^{6}+\frac{28\!\cdots\!37}{20\!\cdots\!05}a^{5}+\frac{88\!\cdots\!38}{20\!\cdots\!05}a^{4}-\frac{17\!\cdots\!99}{20\!\cdots\!05}a^{3}-\frac{55\!\cdots\!07}{20\!\cdots\!05}a^{2}+\frac{58\!\cdots\!88}{36\!\cdots\!55}a+\frac{37\!\cdots\!85}{72\!\cdots\!51}$, $\frac{11\!\cdots\!86}{20\!\cdots\!05}a^{21}+\frac{51\!\cdots\!96}{20\!\cdots\!05}a^{20}-\frac{14\!\cdots\!28}{20\!\cdots\!05}a^{19}-\frac{68\!\cdots\!02}{20\!\cdots\!05}a^{18}+\frac{16\!\cdots\!12}{40\!\cdots\!61}a^{17}+\frac{38\!\cdots\!07}{20\!\cdots\!05}a^{16}-\frac{52\!\cdots\!38}{40\!\cdots\!61}a^{15}-\frac{12\!\cdots\!06}{20\!\cdots\!05}a^{14}+\frac{49\!\cdots\!92}{20\!\cdots\!05}a^{13}+\frac{23\!\cdots\!69}{20\!\cdots\!05}a^{12}-\frac{59\!\cdots\!29}{20\!\cdots\!05}a^{11}-\frac{27\!\cdots\!54}{20\!\cdots\!05}a^{10}+\frac{88\!\cdots\!33}{40\!\cdots\!61}a^{9}+\frac{20\!\cdots\!02}{20\!\cdots\!05}a^{8}-\frac{18\!\cdots\!06}{20\!\cdots\!05}a^{7}-\frac{17\!\cdots\!30}{40\!\cdots\!61}a^{6}+\frac{39\!\cdots\!41}{20\!\cdots\!05}a^{5}+\frac{17\!\cdots\!04}{20\!\cdots\!05}a^{4}-\frac{24\!\cdots\!59}{20\!\cdots\!05}a^{3}-\frac{22\!\cdots\!43}{40\!\cdots\!61}a^{2}+\frac{83\!\cdots\!84}{36\!\cdots\!55}a+\frac{76\!\cdots\!10}{72\!\cdots\!51}$, $\frac{41\!\cdots\!51}{31\!\cdots\!93}a^{21}+\frac{71\!\cdots\!82}{20\!\cdots\!05}a^{20}-\frac{27\!\cdots\!32}{15\!\cdots\!65}a^{19}-\frac{94\!\cdots\!72}{20\!\cdots\!05}a^{18}+\frac{30\!\cdots\!21}{31\!\cdots\!93}a^{17}+\frac{10\!\cdots\!73}{40\!\cdots\!61}a^{16}-\frac{48\!\cdots\!97}{15\!\cdots\!65}a^{15}-\frac{33\!\cdots\!87}{40\!\cdots\!61}a^{14}+\frac{92\!\cdots\!49}{15\!\cdots\!65}a^{13}+\frac{32\!\cdots\!41}{20\!\cdots\!05}a^{12}-\frac{11\!\cdots\!21}{15\!\cdots\!65}a^{11}-\frac{38\!\cdots\!79}{20\!\cdots\!05}a^{10}+\frac{81\!\cdots\!09}{15\!\cdots\!65}a^{9}+\frac{56\!\cdots\!12}{40\!\cdots\!61}a^{8}-\frac{34\!\cdots\!23}{15\!\cdots\!65}a^{7}-\frac{11\!\cdots\!61}{20\!\cdots\!05}a^{6}+\frac{71\!\cdots\!21}{15\!\cdots\!65}a^{5}+\frac{24\!\cdots\!11}{20\!\cdots\!05}a^{4}-\frac{44\!\cdots\!67}{15\!\cdots\!65}a^{3}-\frac{30\!\cdots\!34}{40\!\cdots\!61}a^{2}+\frac{19\!\cdots\!67}{36\!\cdots\!55}a+\frac{51\!\cdots\!24}{36\!\cdots\!55}$, $\frac{29\!\cdots\!79}{20\!\cdots\!05}a^{21}-\frac{11\!\cdots\!09}{20\!\cdots\!05}a^{20}-\frac{77\!\cdots\!81}{40\!\cdots\!61}a^{19}+\frac{15\!\cdots\!24}{20\!\cdots\!05}a^{18}+\frac{21\!\cdots\!59}{20\!\cdots\!05}a^{17}-\frac{85\!\cdots\!07}{20\!\cdots\!05}a^{16}-\frac{13\!\cdots\!05}{40\!\cdots\!61}a^{15}+\frac{27\!\cdots\!74}{20\!\cdots\!05}a^{14}+\frac{12\!\cdots\!52}{20\!\cdots\!05}a^{13}-\frac{52\!\cdots\!86}{20\!\cdots\!05}a^{12}-\frac{14\!\cdots\!52}{20\!\cdots\!05}a^{11}+\frac{12\!\cdots\!81}{40\!\cdots\!61}a^{10}+\frac{21\!\cdots\!03}{40\!\cdots\!61}a^{9}-\frac{46\!\cdots\!18}{20\!\cdots\!05}a^{8}-\frac{45\!\cdots\!52}{20\!\cdots\!05}a^{7}+\frac{39\!\cdots\!54}{40\!\cdots\!61}a^{6}+\frac{92\!\cdots\!67}{20\!\cdots\!05}a^{5}-\frac{41\!\cdots\!16}{20\!\cdots\!05}a^{4}-\frac{11\!\cdots\!09}{40\!\cdots\!61}a^{3}+\frac{51\!\cdots\!98}{40\!\cdots\!61}a^{2}+\frac{19\!\cdots\!94}{36\!\cdots\!55}a-\frac{87\!\cdots\!31}{36\!\cdots\!55}$, $\frac{76\!\cdots\!57}{20\!\cdots\!05}a^{21}+\frac{41\!\cdots\!04}{40\!\cdots\!61}a^{20}-\frac{10\!\cdots\!42}{20\!\cdots\!05}a^{19}-\frac{55\!\cdots\!22}{40\!\cdots\!61}a^{18}+\frac{57\!\cdots\!37}{20\!\cdots\!05}a^{17}+\frac{15\!\cdots\!42}{20\!\cdots\!05}a^{16}-\frac{17\!\cdots\!62}{20\!\cdots\!05}a^{15}-\frac{97\!\cdots\!30}{40\!\cdots\!61}a^{14}+\frac{68\!\cdots\!68}{40\!\cdots\!61}a^{13}+\frac{93\!\cdots\!29}{20\!\cdots\!05}a^{12}-\frac{40\!\cdots\!87}{20\!\cdots\!05}a^{11}-\frac{11\!\cdots\!74}{20\!\cdots\!05}a^{10}+\frac{29\!\cdots\!24}{20\!\cdots\!05}a^{9}+\frac{16\!\cdots\!84}{40\!\cdots\!61}a^{8}-\frac{12\!\cdots\!09}{20\!\cdots\!05}a^{7}-\frac{34\!\cdots\!88}{20\!\cdots\!05}a^{6}+\frac{25\!\cdots\!57}{20\!\cdots\!05}a^{5}+\frac{70\!\cdots\!68}{20\!\cdots\!05}a^{4}-\frac{16\!\cdots\!97}{20\!\cdots\!05}a^{3}-\frac{44\!\cdots\!39}{20\!\cdots\!05}a^{2}+\frac{54\!\cdots\!53}{36\!\cdots\!55}a+\frac{29\!\cdots\!56}{72\!\cdots\!51}$, $\frac{35\!\cdots\!68}{20\!\cdots\!05}a^{21}+\frac{20\!\cdots\!86}{40\!\cdots\!61}a^{20}-\frac{47\!\cdots\!31}{20\!\cdots\!05}a^{19}-\frac{13\!\cdots\!77}{20\!\cdots\!05}a^{18}+\frac{26\!\cdots\!21}{20\!\cdots\!05}a^{17}+\frac{77\!\cdots\!09}{20\!\cdots\!05}a^{16}-\frac{82\!\cdots\!92}{20\!\cdots\!05}a^{15}-\frac{48\!\cdots\!65}{40\!\cdots\!61}a^{14}+\frac{15\!\cdots\!97}{20\!\cdots\!05}a^{13}+\frac{46\!\cdots\!27}{20\!\cdots\!05}a^{12}-\frac{37\!\cdots\!10}{40\!\cdots\!61}a^{11}-\frac{11\!\cdots\!06}{40\!\cdots\!61}a^{10}+\frac{13\!\cdots\!04}{20\!\cdots\!05}a^{9}+\frac{81\!\cdots\!13}{40\!\cdots\!61}a^{8}-\frac{11\!\cdots\!37}{40\!\cdots\!61}a^{7}-\frac{17\!\cdots\!84}{20\!\cdots\!05}a^{6}+\frac{11\!\cdots\!23}{20\!\cdots\!05}a^{5}+\frac{35\!\cdots\!04}{20\!\cdots\!05}a^{4}-\frac{69\!\cdots\!61}{20\!\cdots\!05}a^{3}-\frac{22\!\cdots\!27}{20\!\cdots\!05}a^{2}+\frac{21\!\cdots\!98}{36\!\cdots\!55}a+\frac{76\!\cdots\!67}{36\!\cdots\!55}$, $\frac{17\!\cdots\!02}{20\!\cdots\!05}a^{21}-\frac{10\!\cdots\!77}{40\!\cdots\!61}a^{20}-\frac{23\!\cdots\!48}{20\!\cdots\!05}a^{19}+\frac{69\!\cdots\!43}{20\!\cdots\!05}a^{18}+\frac{13\!\cdots\!91}{20\!\cdots\!05}a^{17}-\frac{38\!\cdots\!42}{20\!\cdots\!05}a^{16}-\frac{81\!\cdots\!20}{40\!\cdots\!61}a^{15}+\frac{12\!\cdots\!72}{20\!\cdots\!05}a^{14}+\frac{15\!\cdots\!67}{40\!\cdots\!61}a^{13}-\frac{45\!\cdots\!98}{40\!\cdots\!61}a^{12}-\frac{90\!\cdots\!32}{20\!\cdots\!05}a^{11}+\frac{26\!\cdots\!06}{20\!\cdots\!05}a^{10}+\frac{13\!\cdots\!99}{40\!\cdots\!61}a^{9}-\frac{19\!\cdots\!29}{20\!\cdots\!05}a^{8}-\frac{27\!\cdots\!89}{20\!\cdots\!05}a^{7}+\frac{16\!\cdots\!15}{40\!\cdots\!61}a^{6}+\frac{55\!\cdots\!24}{20\!\cdots\!05}a^{5}-\frac{16\!\cdots\!83}{20\!\cdots\!05}a^{4}-\frac{34\!\cdots\!19}{20\!\cdots\!05}a^{3}+\frac{10\!\cdots\!28}{20\!\cdots\!05}a^{2}+\frac{23\!\cdots\!50}{72\!\cdots\!51}a-\frac{34\!\cdots\!12}{36\!\cdots\!55}$ (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: | \( 49537806486500000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{22}\cdot(2\pi)^{0}\cdot 49537806486500000 \cdot 1}{2\cdot\sqrt{170364212909807025886845565709641708504367497216}}\cr\approx \mathstrut & 0.251696685006847 \end{aligned}\] (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: | 22.22.220962384144019712575238698725405295930164643889152.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{9}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }^{5}$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.11.0.1}{11} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{10}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/47.11.0.1}{11} }^{2}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}$ |
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 $22$ | $2$ | $11$ | $22$ | |||
\(43\) | 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
43.4.2.1 | $x^{4} + 84 x^{3} + 1856 x^{2} + 3864 x + 77452$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
43.4.0.1 | $x^{4} + 5 x^{2} + 42 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(1277\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
\(1297\) | $\Q_{1297}$ | $x$ | $1$ | $1$ | $0$ | 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 $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ |