Normalized defining polynomial
\( x^{22} - x^{20} - 251 x^{18} - 474 x^{16} + 16151 x^{14} + 58074 x^{12} - 84453 x^{10} - 36512 x^{8} + 41198 x^{6} + 7092 x^{4} - 4293 x^{2} - 729 \)
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: | \(4129233136056857981979443884256982828952059904\) \(\medspace = 2^{22}\cdot 74843^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(118.43\) | 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: | \(2\), \(74843\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{15}+\frac{1}{3}a^{13}-\frac{1}{3}a^{9}+\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{18}-\frac{1}{9}a^{16}+\frac{1}{9}a^{14}+\frac{1}{3}a^{12}-\frac{4}{9}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{9}a^{4}-\frac{4}{9}a^{2}$, $\frac{1}{27}a^{19}-\frac{1}{27}a^{17}-\frac{8}{27}a^{15}+\frac{4}{9}a^{13}+\frac{5}{27}a^{11}-\frac{1}{9}a^{9}+\frac{1}{9}a^{7}-\frac{8}{27}a^{5}-\frac{4}{27}a^{3}-\frac{1}{3}a$, $\frac{1}{48\!\cdots\!61}a^{20}-\frac{13\!\cdots\!25}{48\!\cdots\!61}a^{18}+\frac{93\!\cdots\!38}{48\!\cdots\!61}a^{16}+\frac{88\!\cdots\!61}{16\!\cdots\!87}a^{14}+\frac{10\!\cdots\!48}{48\!\cdots\!61}a^{12}-\frac{15\!\cdots\!31}{16\!\cdots\!87}a^{10}+\frac{41\!\cdots\!08}{16\!\cdots\!87}a^{8}-\frac{15\!\cdots\!82}{48\!\cdots\!61}a^{6}-\frac{92\!\cdots\!81}{48\!\cdots\!61}a^{4}-\frac{59\!\cdots\!28}{54\!\cdots\!29}a^{2}-\frac{24\!\cdots\!40}{60\!\cdots\!81}$, $\frac{1}{14\!\cdots\!83}a^{21}-\frac{13\!\cdots\!25}{14\!\cdots\!83}a^{19}+\frac{93\!\cdots\!38}{14\!\cdots\!83}a^{17}-\frac{15\!\cdots\!26}{48\!\cdots\!61}a^{15}+\frac{58\!\cdots\!09}{14\!\cdots\!83}a^{13}-\frac{15\!\cdots\!31}{48\!\cdots\!61}a^{11}+\frac{41\!\cdots\!08}{48\!\cdots\!61}a^{9}-\frac{63\!\cdots\!43}{14\!\cdots\!83}a^{7}+\frac{39\!\cdots\!80}{14\!\cdots\!83}a^{5}+\frac{53\!\cdots\!01}{16\!\cdots\!87}a^{3}-\frac{28\!\cdots\!07}{60\!\cdots\!81}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $15$ | 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: | $\frac{87\!\cdots\!50}{48\!\cdots\!61}a^{20}-\frac{61\!\cdots\!44}{48\!\cdots\!61}a^{18}-\frac{21\!\cdots\!36}{48\!\cdots\!61}a^{16}-\frac{15\!\cdots\!88}{16\!\cdots\!87}a^{14}+\frac{13\!\cdots\!00}{48\!\cdots\!61}a^{12}+\frac{18\!\cdots\!86}{16\!\cdots\!87}a^{10}-\frac{18\!\cdots\!81}{16\!\cdots\!87}a^{8}-\frac{39\!\cdots\!99}{48\!\cdots\!61}a^{6}+\frac{17\!\cdots\!95}{48\!\cdots\!61}a^{4}+\frac{15\!\cdots\!70}{18\!\cdots\!43}a^{2}-\frac{11\!\cdots\!23}{60\!\cdots\!81}$, $\frac{11\!\cdots\!31}{48\!\cdots\!61}a^{20}-\frac{10\!\cdots\!69}{48\!\cdots\!61}a^{18}-\frac{28\!\cdots\!47}{48\!\cdots\!61}a^{16}-\frac{18\!\cdots\!98}{16\!\cdots\!87}a^{14}+\frac{18\!\cdots\!94}{48\!\cdots\!61}a^{12}+\frac{22\!\cdots\!59}{16\!\cdots\!87}a^{10}-\frac{29\!\cdots\!54}{16\!\cdots\!87}a^{8}-\frac{34\!\cdots\!49}{48\!\cdots\!61}a^{6}+\frac{29\!\cdots\!77}{48\!\cdots\!61}a^{4}+\frac{97\!\cdots\!19}{54\!\cdots\!29}a^{2}-\frac{86\!\cdots\!24}{60\!\cdots\!81}$, $\frac{34\!\cdots\!83}{48\!\cdots\!61}a^{20}-\frac{29\!\cdots\!89}{48\!\cdots\!61}a^{18}-\frac{87\!\cdots\!78}{48\!\cdots\!61}a^{16}-\frac{59\!\cdots\!80}{16\!\cdots\!87}a^{14}+\frac{55\!\cdots\!49}{48\!\cdots\!61}a^{12}+\frac{69\!\cdots\!15}{16\!\cdots\!87}a^{10}-\frac{82\!\cdots\!54}{16\!\cdots\!87}a^{8}-\frac{10\!\cdots\!12}{48\!\cdots\!61}a^{6}+\frac{82\!\cdots\!40}{48\!\cdots\!61}a^{4}-\frac{29\!\cdots\!11}{54\!\cdots\!29}a^{2}-\frac{34\!\cdots\!24}{60\!\cdots\!81}$, $\frac{33\!\cdots\!20}{48\!\cdots\!61}a^{20}-\frac{43\!\cdots\!42}{48\!\cdots\!61}a^{18}-\frac{83\!\cdots\!95}{48\!\cdots\!61}a^{16}-\frac{76\!\cdots\!94}{16\!\cdots\!87}a^{14}+\frac{51\!\cdots\!11}{48\!\cdots\!61}a^{12}+\frac{79\!\cdots\!46}{16\!\cdots\!87}a^{10}-\frac{34\!\cdots\!07}{16\!\cdots\!87}a^{8}-\frac{31\!\cdots\!61}{48\!\cdots\!61}a^{6}+\frac{24\!\cdots\!65}{48\!\cdots\!61}a^{4}+\frac{87\!\cdots\!67}{60\!\cdots\!81}a^{2}+\frac{14\!\cdots\!66}{60\!\cdots\!81}$, $\frac{17\!\cdots\!15}{14\!\cdots\!83}a^{21}-\frac{21\!\cdots\!05}{14\!\cdots\!83}a^{19}-\frac{44\!\cdots\!79}{14\!\cdots\!83}a^{17}-\frac{24\!\cdots\!09}{48\!\cdots\!61}a^{15}+\frac{28\!\cdots\!82}{14\!\cdots\!83}a^{13}+\frac{32\!\cdots\!59}{48\!\cdots\!61}a^{11}-\frac{56\!\cdots\!97}{48\!\cdots\!61}a^{9}-\frac{19\!\cdots\!98}{14\!\cdots\!83}a^{7}+\frac{71\!\cdots\!93}{14\!\cdots\!83}a^{5}-\frac{91\!\cdots\!53}{16\!\cdots\!87}a^{3}-\frac{51\!\cdots\!33}{18\!\cdots\!43}a$, $\frac{16\!\cdots\!76}{48\!\cdots\!61}a^{21}-\frac{11\!\cdots\!03}{48\!\cdots\!61}a^{19}-\frac{41\!\cdots\!10}{48\!\cdots\!61}a^{17}-\frac{10\!\cdots\!77}{54\!\cdots\!29}a^{15}+\frac{26\!\cdots\!97}{48\!\cdots\!61}a^{13}+\frac{13\!\cdots\!33}{60\!\cdots\!81}a^{11}-\frac{34\!\cdots\!64}{16\!\cdots\!87}a^{9}-\frac{87\!\cdots\!56}{48\!\cdots\!61}a^{7}+\frac{29\!\cdots\!13}{48\!\cdots\!61}a^{5}+\frac{54\!\cdots\!73}{16\!\cdots\!87}a^{3}+\frac{62\!\cdots\!73}{18\!\cdots\!43}a$, $\frac{44\!\cdots\!41}{16\!\cdots\!87}a^{20}-\frac{52\!\cdots\!55}{16\!\cdots\!87}a^{18}-\frac{11\!\cdots\!66}{16\!\cdots\!87}a^{16}-\frac{21\!\cdots\!77}{18\!\cdots\!43}a^{14}+\frac{72\!\cdots\!49}{16\!\cdots\!87}a^{12}+\frac{27\!\cdots\!76}{18\!\cdots\!43}a^{10}-\frac{13\!\cdots\!81}{54\!\cdots\!29}a^{8}-\frac{91\!\cdots\!32}{16\!\cdots\!87}a^{6}+\frac{19\!\cdots\!15}{16\!\cdots\!87}a^{4}-\frac{92\!\cdots\!80}{54\!\cdots\!29}a^{2}-\frac{67\!\cdots\!17}{60\!\cdots\!81}$, $\frac{15\!\cdots\!22}{14\!\cdots\!83}a^{21}-\frac{20\!\cdots\!58}{14\!\cdots\!83}a^{19}-\frac{39\!\cdots\!45}{14\!\cdots\!83}a^{17}-\frac{20\!\cdots\!24}{48\!\cdots\!61}a^{15}+\frac{25\!\cdots\!05}{14\!\cdots\!83}a^{13}+\frac{27\!\cdots\!66}{48\!\cdots\!61}a^{11}-\frac{54\!\cdots\!57}{48\!\cdots\!61}a^{9}-\frac{34\!\cdots\!25}{14\!\cdots\!83}a^{7}+\frac{82\!\cdots\!89}{14\!\cdots\!83}a^{5}-\frac{24\!\cdots\!10}{54\!\cdots\!29}a^{3}-\frac{15\!\cdots\!70}{60\!\cdots\!81}a$, $\frac{28\!\cdots\!78}{48\!\cdots\!61}a^{20}-\frac{33\!\cdots\!98}{48\!\cdots\!61}a^{18}-\frac{71\!\cdots\!62}{48\!\cdots\!61}a^{16}-\frac{41\!\cdots\!32}{16\!\cdots\!87}a^{14}+\frac{46\!\cdots\!83}{48\!\cdots\!61}a^{12}+\frac{52\!\cdots\!73}{16\!\cdots\!87}a^{10}-\frac{89\!\cdots\!36}{16\!\cdots\!87}a^{8}-\frac{59\!\cdots\!53}{48\!\cdots\!61}a^{6}+\frac{12\!\cdots\!24}{48\!\cdots\!61}a^{4}-\frac{54\!\cdots\!42}{18\!\cdots\!43}a^{2}-\frac{14\!\cdots\!96}{60\!\cdots\!81}$, $\frac{35\!\cdots\!21}{48\!\cdots\!61}a^{20}-\frac{30\!\cdots\!96}{48\!\cdots\!61}a^{18}-\frac{90\!\cdots\!35}{48\!\cdots\!61}a^{16}-\frac{61\!\cdots\!23}{16\!\cdots\!87}a^{14}+\frac{57\!\cdots\!90}{48\!\cdots\!61}a^{12}+\frac{72\!\cdots\!57}{16\!\cdots\!87}a^{10}-\frac{85\!\cdots\!45}{16\!\cdots\!87}a^{8}-\frac{12\!\cdots\!50}{48\!\cdots\!61}a^{6}+\frac{86\!\cdots\!39}{48\!\cdots\!61}a^{4}+\frac{73\!\cdots\!47}{54\!\cdots\!29}a^{2}-\frac{35\!\cdots\!94}{60\!\cdots\!81}$, $\frac{12\!\cdots\!50}{16\!\cdots\!87}a^{21}-\frac{28\!\cdots\!35}{48\!\cdots\!61}a^{20}-\frac{17\!\cdots\!18}{54\!\cdots\!29}a^{19}+\frac{11\!\cdots\!58}{48\!\cdots\!61}a^{18}-\frac{10\!\cdots\!98}{54\!\cdots\!29}a^{17}+\frac{71\!\cdots\!77}{48\!\cdots\!61}a^{16}-\frac{77\!\cdots\!72}{16\!\cdots\!87}a^{15}+\frac{59\!\cdots\!77}{16\!\cdots\!87}a^{14}+\frac{19\!\cdots\!82}{16\!\cdots\!87}a^{13}-\frac{44\!\cdots\!63}{48\!\cdots\!61}a^{12}+\frac{84\!\cdots\!23}{16\!\cdots\!87}a^{11}-\frac{63\!\cdots\!04}{16\!\cdots\!87}a^{10}-\frac{63\!\cdots\!13}{18\!\cdots\!43}a^{9}+\frac{42\!\cdots\!22}{16\!\cdots\!87}a^{8}-\frac{78\!\cdots\!81}{16\!\cdots\!87}a^{7}+\frac{18\!\cdots\!83}{48\!\cdots\!61}a^{6}+\frac{17\!\cdots\!37}{54\!\cdots\!29}a^{5}-\frac{10\!\cdots\!45}{48\!\cdots\!61}a^{4}+\frac{11\!\cdots\!87}{16\!\cdots\!87}a^{3}-\frac{29\!\cdots\!92}{54\!\cdots\!29}a^{2}+\frac{52\!\cdots\!32}{60\!\cdots\!81}a-\frac{42\!\cdots\!38}{60\!\cdots\!81}$, $\frac{57\!\cdots\!10}{14\!\cdots\!83}a^{21}+\frac{25\!\cdots\!77}{48\!\cdots\!61}a^{20}-\frac{22\!\cdots\!87}{14\!\cdots\!83}a^{19}-\frac{12\!\cdots\!14}{48\!\cdots\!61}a^{18}-\frac{14\!\cdots\!66}{14\!\cdots\!83}a^{17}-\frac{62\!\cdots\!18}{48\!\cdots\!61}a^{16}-\frac{11\!\cdots\!39}{48\!\cdots\!61}a^{15}-\frac{49\!\cdots\!19}{16\!\cdots\!87}a^{14}+\frac{90\!\cdots\!38}{14\!\cdots\!83}a^{13}+\frac{39\!\cdots\!19}{48\!\cdots\!61}a^{12}+\frac{12\!\cdots\!83}{48\!\cdots\!61}a^{11}+\frac{54\!\cdots\!66}{16\!\cdots\!87}a^{10}-\frac{83\!\cdots\!03}{48\!\cdots\!61}a^{9}-\frac{42\!\cdots\!38}{16\!\cdots\!87}a^{8}-\frac{37\!\cdots\!28}{14\!\cdots\!83}a^{7}-\frac{14\!\cdots\!88}{48\!\cdots\!61}a^{6}+\frac{41\!\cdots\!58}{14\!\cdots\!83}a^{5}+\frac{19\!\cdots\!26}{48\!\cdots\!61}a^{4}+\frac{53\!\cdots\!44}{16\!\cdots\!87}a^{3}+\frac{27\!\cdots\!74}{54\!\cdots\!29}a^{2}+\frac{13\!\cdots\!17}{60\!\cdots\!81}a+\frac{38\!\cdots\!89}{60\!\cdots\!81}$, $\frac{37\!\cdots\!00}{14\!\cdots\!83}a^{21}-\frac{11\!\cdots\!24}{48\!\cdots\!61}a^{20}-\frac{57\!\cdots\!67}{14\!\cdots\!83}a^{19}-\frac{13\!\cdots\!59}{48\!\cdots\!61}a^{18}-\frac{97\!\cdots\!22}{14\!\cdots\!83}a^{17}+\frac{28\!\cdots\!64}{48\!\cdots\!61}a^{16}+\frac{38\!\cdots\!95}{48\!\cdots\!61}a^{15}+\frac{38\!\cdots\!18}{16\!\cdots\!87}a^{14}+\frac{10\!\cdots\!60}{14\!\cdots\!83}a^{13}-\frac{16\!\cdots\!66}{48\!\cdots\!61}a^{12}-\frac{19\!\cdots\!24}{48\!\cdots\!61}a^{11}-\frac{34\!\cdots\!64}{16\!\cdots\!87}a^{10}-\frac{15\!\cdots\!58}{48\!\cdots\!61}a^{9}-\frac{31\!\cdots\!37}{16\!\cdots\!87}a^{8}-\frac{21\!\cdots\!24}{14\!\cdots\!83}a^{7}+\frac{43\!\cdots\!95}{48\!\cdots\!61}a^{6}+\frac{14\!\cdots\!60}{14\!\cdots\!83}a^{5}+\frac{33\!\cdots\!60}{48\!\cdots\!61}a^{4}+\frac{30\!\cdots\!23}{54\!\cdots\!29}a^{3}+\frac{13\!\cdots\!41}{54\!\cdots\!29}a^{2}+\frac{11\!\cdots\!56}{18\!\cdots\!43}a+\frac{14\!\cdots\!00}{60\!\cdots\!81}$, $\frac{56\!\cdots\!52}{14\!\cdots\!83}a^{21}+\frac{19\!\cdots\!20}{48\!\cdots\!61}a^{20}+\frac{43\!\cdots\!39}{14\!\cdots\!83}a^{19}+\frac{17\!\cdots\!45}{48\!\cdots\!61}a^{18}-\frac{14\!\cdots\!97}{14\!\cdots\!83}a^{17}-\frac{49\!\cdots\!03}{48\!\cdots\!61}a^{16}-\frac{14\!\cdots\!34}{48\!\cdots\!61}a^{15}-\frac{48\!\cdots\!92}{16\!\cdots\!87}a^{14}+\frac{86\!\cdots\!13}{14\!\cdots\!83}a^{13}+\frac{30\!\cdots\!82}{48\!\cdots\!61}a^{12}+\frac{14\!\cdots\!43}{48\!\cdots\!61}a^{11}+\frac{48\!\cdots\!90}{16\!\cdots\!87}a^{10}-\frac{74\!\cdots\!77}{48\!\cdots\!61}a^{9}-\frac{19\!\cdots\!60}{16\!\cdots\!87}a^{8}-\frac{22\!\cdots\!95}{14\!\cdots\!83}a^{7}-\frac{78\!\cdots\!67}{48\!\cdots\!61}a^{6}-\frac{10\!\cdots\!40}{14\!\cdots\!83}a^{5}-\frac{51\!\cdots\!54}{48\!\cdots\!61}a^{4}+\frac{30\!\cdots\!63}{16\!\cdots\!87}a^{3}+\frac{97\!\cdots\!81}{54\!\cdots\!29}a^{2}+\frac{48\!\cdots\!57}{18\!\cdots\!43}a+\frac{16\!\cdots\!57}{60\!\cdots\!81}$, $\frac{15\!\cdots\!12}{14\!\cdots\!83}a^{21}-\frac{34\!\cdots\!39}{48\!\cdots\!61}a^{20}-\frac{86\!\cdots\!26}{14\!\cdots\!83}a^{19}+\frac{19\!\cdots\!22}{48\!\cdots\!61}a^{18}-\frac{39\!\cdots\!91}{14\!\cdots\!83}a^{17}+\frac{87\!\cdots\!23}{48\!\cdots\!61}a^{16}-\frac{30\!\cdots\!36}{48\!\cdots\!61}a^{15}+\frac{68\!\cdots\!66}{16\!\cdots\!87}a^{14}+\frac{24\!\cdots\!87}{14\!\cdots\!83}a^{13}-\frac{55\!\cdots\!74}{48\!\cdots\!61}a^{12}+\frac{33\!\cdots\!69}{48\!\cdots\!61}a^{11}-\frac{75\!\cdots\!85}{16\!\cdots\!87}a^{10}-\frac{28\!\cdots\!82}{48\!\cdots\!61}a^{9}+\frac{64\!\cdots\!84}{16\!\cdots\!87}a^{8}-\frac{95\!\cdots\!46}{14\!\cdots\!83}a^{7}+\frac{21\!\cdots\!31}{48\!\cdots\!61}a^{6}+\frac{21\!\cdots\!04}{14\!\cdots\!83}a^{5}-\frac{48\!\cdots\!35}{48\!\cdots\!61}a^{4}+\frac{77\!\cdots\!09}{54\!\cdots\!29}a^{3}-\frac{51\!\cdots\!26}{54\!\cdots\!29}a^{2}+\frac{31\!\cdots\!61}{18\!\cdots\!43}a-\frac{70\!\cdots\!97}{60\!\cdots\!81}$ (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: | \( 962382965570000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{6}\cdot 962382965570000 \cdot 1}{2\cdot\sqrt{4129233136056857981979443884256982828952059904}}\cr\approx \mathstrut & 0.471804845140777 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.\PSL(2,11)$ (as 22T39):
A non-solvable group of order 675840 |
The 56 conjugacy class representatives for $C_2^{10}.\PSL(2,11)$ are not computed |
Character table for $C_2^{10}.\PSL(2,11)$ is not computed |
Intermediate fields
11.11.31376518243389673201.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 22 sibling: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.11.0.1}{11} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$ |
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 $22$ | $2$ | $11$ | $22$ | |||
\(74843\) | $\Q_{74843}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{74843}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ |