Normalized defining polynomial
\( x^{20} - 4 x^{19} + 4 x^{18} - 12 x^{17} + 55 x^{16} - 98 x^{15} + 194 x^{14} - 348 x^{13} + 615 x^{12} + \cdots + 1 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(5969915757478328440239161344\) \(\medspace = 2^{30}\cdot 11^{18}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(24.48\) | 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: | $2^{31/16}11^{9/10}\approx 33.15118318510912$ | ||
Ramified primes: | \(2\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{69\!\cdots\!39}a^{19}-\frac{63\!\cdots\!17}{69\!\cdots\!39}a^{18}+\frac{21\!\cdots\!71}{69\!\cdots\!39}a^{17}-\frac{84\!\cdots\!22}{69\!\cdots\!39}a^{16}+\frac{34\!\cdots\!11}{69\!\cdots\!39}a^{15}+\frac{57\!\cdots\!00}{69\!\cdots\!39}a^{14}-\frac{34\!\cdots\!80}{69\!\cdots\!39}a^{13}+\frac{87\!\cdots\!51}{69\!\cdots\!39}a^{12}-\frac{14\!\cdots\!23}{69\!\cdots\!39}a^{11}+\frac{22\!\cdots\!82}{69\!\cdots\!39}a^{10}-\frac{84\!\cdots\!21}{69\!\cdots\!39}a^{9}-\frac{29\!\cdots\!01}{69\!\cdots\!39}a^{8}+\frac{17\!\cdots\!38}{69\!\cdots\!39}a^{7}-\frac{23\!\cdots\!23}{69\!\cdots\!39}a^{6}+\frac{31\!\cdots\!79}{69\!\cdots\!39}a^{5}-\frac{23\!\cdots\!00}{69\!\cdots\!39}a^{4}-\frac{15\!\cdots\!36}{69\!\cdots\!39}a^{3}+\frac{20\!\cdots\!95}{69\!\cdots\!39}a^{2}+\frac{33\!\cdots\!80}{69\!\cdots\!39}a+\frac{26\!\cdots\!70}{69\!\cdots\!39}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | 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: | $\frac{72\!\cdots\!21}{69\!\cdots\!39}a^{19}-\frac{27\!\cdots\!95}{69\!\cdots\!39}a^{18}+\frac{21\!\cdots\!75}{69\!\cdots\!39}a^{17}-\frac{82\!\cdots\!19}{69\!\cdots\!39}a^{16}+\frac{37\!\cdots\!25}{69\!\cdots\!39}a^{15}-\frac{61\!\cdots\!27}{69\!\cdots\!39}a^{14}+\frac{12\!\cdots\!32}{69\!\cdots\!39}a^{13}-\frac{21\!\cdots\!66}{69\!\cdots\!39}a^{12}+\frac{38\!\cdots\!94}{69\!\cdots\!39}a^{11}-\frac{76\!\cdots\!10}{69\!\cdots\!39}a^{10}+\frac{12\!\cdots\!46}{69\!\cdots\!39}a^{9}-\frac{24\!\cdots\!66}{69\!\cdots\!39}a^{8}+\frac{20\!\cdots\!87}{69\!\cdots\!39}a^{7}-\frac{24\!\cdots\!79}{69\!\cdots\!39}a^{6}+\frac{45\!\cdots\!70}{69\!\cdots\!39}a^{5}-\frac{81\!\cdots\!73}{69\!\cdots\!39}a^{4}+\frac{63\!\cdots\!23}{69\!\cdots\!39}a^{3}-\frac{14\!\cdots\!40}{69\!\cdots\!39}a^{2}-\frac{28\!\cdots\!25}{69\!\cdots\!39}a+\frac{40\!\cdots\!46}{69\!\cdots\!39}$, $\frac{69\!\cdots\!60}{69\!\cdots\!39}a^{19}-\frac{27\!\cdots\!69}{69\!\cdots\!39}a^{18}+\frac{26\!\cdots\!99}{69\!\cdots\!39}a^{17}-\frac{80\!\cdots\!15}{69\!\cdots\!39}a^{16}+\frac{38\!\cdots\!25}{69\!\cdots\!39}a^{15}-\frac{66\!\cdots\!95}{69\!\cdots\!39}a^{14}+\frac{12\!\cdots\!76}{69\!\cdots\!39}a^{13}-\frac{23\!\cdots\!29}{69\!\cdots\!39}a^{12}+\frac{40\!\cdots\!24}{69\!\cdots\!39}a^{11}-\frac{79\!\cdots\!66}{69\!\cdots\!39}a^{10}+\frac{13\!\cdots\!47}{69\!\cdots\!39}a^{9}-\frac{25\!\cdots\!46}{69\!\cdots\!39}a^{8}+\frac{23\!\cdots\!38}{69\!\cdots\!39}a^{7}-\frac{25\!\cdots\!11}{69\!\cdots\!39}a^{6}+\frac{48\!\cdots\!31}{69\!\cdots\!39}a^{5}-\frac{85\!\cdots\!30}{69\!\cdots\!39}a^{4}+\frac{74\!\cdots\!94}{69\!\cdots\!39}a^{3}-\frac{19\!\cdots\!53}{69\!\cdots\!39}a^{2}-\frac{35\!\cdots\!29}{69\!\cdots\!39}a+\frac{53\!\cdots\!03}{69\!\cdots\!39}$, $\frac{69\!\cdots\!58}{69\!\cdots\!39}a^{19}-\frac{27\!\cdots\!78}{69\!\cdots\!39}a^{18}+\frac{24\!\cdots\!51}{69\!\cdots\!39}a^{17}-\frac{80\!\cdots\!77}{69\!\cdots\!39}a^{16}+\frac{37\!\cdots\!81}{69\!\cdots\!39}a^{15}-\frac{64\!\cdots\!39}{69\!\cdots\!39}a^{14}+\frac{12\!\cdots\!69}{69\!\cdots\!39}a^{13}-\frac{22\!\cdots\!37}{69\!\cdots\!39}a^{12}+\frac{40\!\cdots\!23}{69\!\cdots\!39}a^{11}-\frac{78\!\cdots\!51}{69\!\cdots\!39}a^{10}+\frac{12\!\cdots\!32}{69\!\cdots\!39}a^{9}-\frac{25\!\cdots\!93}{69\!\cdots\!39}a^{8}+\frac{22\!\cdots\!36}{69\!\cdots\!39}a^{7}-\frac{25\!\cdots\!13}{69\!\cdots\!39}a^{6}+\frac{46\!\cdots\!54}{69\!\cdots\!39}a^{5}-\frac{84\!\cdots\!19}{69\!\cdots\!39}a^{4}+\frac{71\!\cdots\!65}{69\!\cdots\!39}a^{3}-\frac{17\!\cdots\!51}{69\!\cdots\!39}a^{2}-\frac{32\!\cdots\!01}{69\!\cdots\!39}a+\frac{49\!\cdots\!01}{69\!\cdots\!39}$, $\frac{62\!\cdots\!83}{69\!\cdots\!39}a^{19}-\frac{22\!\cdots\!93}{69\!\cdots\!39}a^{18}+\frac{16\!\cdots\!17}{69\!\cdots\!39}a^{17}-\frac{69\!\cdots\!00}{69\!\cdots\!39}a^{16}+\frac{31\!\cdots\!51}{69\!\cdots\!39}a^{15}-\frac{49\!\cdots\!56}{69\!\cdots\!39}a^{14}+\frac{10\!\cdots\!05}{69\!\cdots\!39}a^{13}-\frac{18\!\cdots\!52}{69\!\cdots\!39}a^{12}+\frac{32\!\cdots\!78}{69\!\cdots\!39}a^{11}-\frac{62\!\cdots\!30}{69\!\cdots\!39}a^{10}+\frac{10\!\cdots\!30}{69\!\cdots\!39}a^{9}-\frac{20\!\cdots\!53}{69\!\cdots\!39}a^{8}+\frac{15\!\cdots\!94}{69\!\cdots\!39}a^{7}-\frac{20\!\cdots\!57}{69\!\cdots\!39}a^{6}+\frac{37\!\cdots\!91}{69\!\cdots\!39}a^{5}-\frac{67\!\cdots\!67}{69\!\cdots\!39}a^{4}+\frac{49\!\cdots\!51}{69\!\cdots\!39}a^{3}-\frac{10\!\cdots\!82}{69\!\cdots\!39}a^{2}-\frac{20\!\cdots\!98}{69\!\cdots\!39}a+\frac{29\!\cdots\!03}{69\!\cdots\!39}$, $\frac{17\!\cdots\!53}{69\!\cdots\!39}a^{19}-\frac{75\!\cdots\!79}{69\!\cdots\!39}a^{18}+\frac{87\!\cdots\!82}{69\!\cdots\!39}a^{17}-\frac{22\!\cdots\!32}{69\!\cdots\!39}a^{16}+\frac{10\!\cdots\!25}{69\!\cdots\!39}a^{15}-\frac{19\!\cdots\!41}{69\!\cdots\!39}a^{14}+\frac{38\!\cdots\!45}{69\!\cdots\!39}a^{13}-\frac{69\!\cdots\!59}{69\!\cdots\!39}a^{12}+\frac{12\!\cdots\!50}{69\!\cdots\!39}a^{11}-\frac{23\!\cdots\!16}{69\!\cdots\!39}a^{10}+\frac{39\!\cdots\!19}{69\!\cdots\!39}a^{9}-\frac{75\!\cdots\!77}{69\!\cdots\!39}a^{8}+\frac{80\!\cdots\!04}{69\!\cdots\!39}a^{7}-\frac{86\!\cdots\!59}{69\!\cdots\!39}a^{6}+\frac{14\!\cdots\!37}{69\!\cdots\!39}a^{5}-\frac{25\!\cdots\!24}{69\!\cdots\!39}a^{4}+\frac{26\!\cdots\!65}{69\!\cdots\!39}a^{3}-\frac{11\!\cdots\!77}{69\!\cdots\!39}a^{2}+\frac{15\!\cdots\!38}{69\!\cdots\!39}a+\frac{37\!\cdots\!76}{69\!\cdots\!39}$, $\frac{77\!\cdots\!73}{69\!\cdots\!39}a^{19}-\frac{23\!\cdots\!98}{69\!\cdots\!39}a^{18}+\frac{28\!\cdots\!96}{69\!\cdots\!39}a^{17}-\frac{70\!\cdots\!85}{69\!\cdots\!39}a^{16}+\frac{34\!\cdots\!98}{69\!\cdots\!39}a^{15}-\frac{36\!\cdots\!11}{69\!\cdots\!39}a^{14}+\frac{87\!\cdots\!04}{69\!\cdots\!39}a^{13}-\frac{13\!\cdots\!38}{69\!\cdots\!39}a^{12}+\frac{24\!\cdots\!90}{69\!\cdots\!39}a^{11}-\frac{51\!\cdots\!35}{69\!\cdots\!39}a^{10}+\frac{73\!\cdots\!71}{69\!\cdots\!39}a^{9}-\frac{16\!\cdots\!49}{69\!\cdots\!39}a^{8}+\frac{27\!\cdots\!73}{69\!\cdots\!39}a^{7}-\frac{10\!\cdots\!35}{69\!\cdots\!39}a^{6}+\frac{29\!\cdots\!71}{69\!\cdots\!39}a^{5}-\frac{52\!\cdots\!35}{69\!\cdots\!39}a^{4}+\frac{59\!\cdots\!71}{69\!\cdots\!39}a^{3}+\frac{34\!\cdots\!83}{69\!\cdots\!39}a^{2}-\frac{13\!\cdots\!37}{69\!\cdots\!39}a-\frac{15\!\cdots\!56}{69\!\cdots\!39}$, $\frac{93\!\cdots\!22}{69\!\cdots\!39}a^{19}-\frac{40\!\cdots\!77}{69\!\cdots\!39}a^{18}+\frac{51\!\cdots\!60}{69\!\cdots\!39}a^{17}-\frac{12\!\cdots\!76}{69\!\cdots\!39}a^{16}+\frac{55\!\cdots\!89}{69\!\cdots\!39}a^{15}-\frac{11\!\cdots\!89}{69\!\cdots\!39}a^{14}+\frac{21\!\cdots\!11}{69\!\cdots\!39}a^{13}-\frac{39\!\cdots\!17}{69\!\cdots\!39}a^{12}+\frac{69\!\cdots\!93}{69\!\cdots\!39}a^{11}-\frac{13\!\cdots\!41}{69\!\cdots\!39}a^{10}+\frac{22\!\cdots\!24}{69\!\cdots\!39}a^{9}-\frac{42\!\cdots\!69}{69\!\cdots\!39}a^{8}+\frac{47\!\cdots\!65}{69\!\cdots\!39}a^{7}-\frac{49\!\cdots\!97}{69\!\cdots\!39}a^{6}+\frac{79\!\cdots\!83}{69\!\cdots\!39}a^{5}-\frac{14\!\cdots\!06}{69\!\cdots\!39}a^{4}+\frac{15\!\cdots\!68}{69\!\cdots\!39}a^{3}-\frac{76\!\cdots\!11}{69\!\cdots\!39}a^{2}+\frac{10\!\cdots\!78}{69\!\cdots\!39}a+\frac{20\!\cdots\!16}{69\!\cdots\!39}$, $\frac{40\!\cdots\!10}{69\!\cdots\!39}a^{19}-\frac{15\!\cdots\!02}{69\!\cdots\!39}a^{18}+\frac{12\!\cdots\!66}{69\!\cdots\!39}a^{17}-\frac{45\!\cdots\!63}{69\!\cdots\!39}a^{16}+\frac{21\!\cdots\!81}{69\!\cdots\!39}a^{15}-\frac{34\!\cdots\!29}{69\!\cdots\!39}a^{14}+\frac{69\!\cdots\!70}{69\!\cdots\!39}a^{13}-\frac{12\!\cdots\!01}{69\!\cdots\!39}a^{12}+\frac{21\!\cdots\!40}{69\!\cdots\!39}a^{11}-\frac{42\!\cdots\!86}{69\!\cdots\!39}a^{10}+\frac{68\!\cdots\!98}{69\!\cdots\!39}a^{9}-\frac{13\!\cdots\!77}{69\!\cdots\!39}a^{8}+\frac{11\!\cdots\!02}{69\!\cdots\!39}a^{7}-\frac{13\!\cdots\!28}{69\!\cdots\!39}a^{6}+\frac{25\!\cdots\!25}{69\!\cdots\!39}a^{5}-\frac{45\!\cdots\!48}{69\!\cdots\!39}a^{4}+\frac{35\!\cdots\!73}{69\!\cdots\!39}a^{3}-\frac{72\!\cdots\!61}{69\!\cdots\!39}a^{2}-\frac{18\!\cdots\!15}{69\!\cdots\!39}a-\frac{34\!\cdots\!02}{69\!\cdots\!39}$, $\frac{84\!\cdots\!25}{69\!\cdots\!39}a^{19}-\frac{31\!\cdots\!09}{69\!\cdots\!39}a^{18}+\frac{29\!\cdots\!91}{69\!\cdots\!39}a^{17}-\frac{10\!\cdots\!75}{69\!\cdots\!39}a^{16}+\frac{44\!\cdots\!27}{69\!\cdots\!39}a^{15}-\frac{75\!\cdots\!42}{69\!\cdots\!39}a^{14}+\frac{16\!\cdots\!41}{69\!\cdots\!39}a^{13}-\frac{27\!\cdots\!31}{69\!\cdots\!39}a^{12}+\frac{49\!\cdots\!98}{69\!\cdots\!39}a^{11}-\frac{96\!\cdots\!71}{69\!\cdots\!39}a^{10}+\frac{15\!\cdots\!08}{69\!\cdots\!39}a^{9}-\frac{31\!\cdots\!56}{69\!\cdots\!39}a^{8}+\frac{27\!\cdots\!70}{69\!\cdots\!39}a^{7}-\frac{36\!\cdots\!38}{69\!\cdots\!39}a^{6}+\frac{58\!\cdots\!05}{69\!\cdots\!39}a^{5}-\frac{10\!\cdots\!62}{69\!\cdots\!39}a^{4}+\frac{90\!\cdots\!93}{69\!\cdots\!39}a^{3}-\frac{41\!\cdots\!86}{69\!\cdots\!39}a^{2}+\frac{85\!\cdots\!20}{69\!\cdots\!39}a-\frac{25\!\cdots\!95}{69\!\cdots\!39}$, $\frac{73\!\cdots\!62}{30\!\cdots\!93}a^{19}-\frac{28\!\cdots\!76}{30\!\cdots\!93}a^{18}+\frac{27\!\cdots\!31}{30\!\cdots\!93}a^{17}-\frac{86\!\cdots\!96}{30\!\cdots\!93}a^{16}+\frac{39\!\cdots\!62}{30\!\cdots\!93}a^{15}-\frac{69\!\cdots\!74}{30\!\cdots\!93}a^{14}+\frac{13\!\cdots\!53}{30\!\cdots\!93}a^{13}-\frac{24\!\cdots\!99}{30\!\cdots\!93}a^{12}+\frac{43\!\cdots\!97}{30\!\cdots\!93}a^{11}-\frac{84\!\cdots\!02}{30\!\cdots\!93}a^{10}+\frac{13\!\cdots\!59}{30\!\cdots\!93}a^{9}-\frac{27\!\cdots\!47}{30\!\cdots\!93}a^{8}+\frac{25\!\cdots\!01}{30\!\cdots\!93}a^{7}-\frac{28\!\cdots\!04}{30\!\cdots\!93}a^{6}+\frac{51\!\cdots\!92}{30\!\cdots\!93}a^{5}-\frac{91\!\cdots\!67}{30\!\cdots\!93}a^{4}+\frac{81\!\cdots\!06}{30\!\cdots\!93}a^{3}-\frac{26\!\cdots\!02}{30\!\cdots\!93}a^{2}+\frac{52\!\cdots\!83}{30\!\cdots\!93}a+\frac{32\!\cdots\!85}{30\!\cdots\!93}$, $\frac{14\!\cdots\!38}{30\!\cdots\!93}a^{19}-\frac{56\!\cdots\!50}{30\!\cdots\!93}a^{18}+\frac{51\!\cdots\!00}{30\!\cdots\!93}a^{17}-\frac{17\!\cdots\!38}{30\!\cdots\!93}a^{16}+\frac{78\!\cdots\!62}{30\!\cdots\!93}a^{15}-\frac{13\!\cdots\!80}{30\!\cdots\!93}a^{14}+\frac{26\!\cdots\!05}{30\!\cdots\!93}a^{13}-\frac{47\!\cdots\!95}{30\!\cdots\!93}a^{12}+\frac{84\!\cdots\!23}{30\!\cdots\!93}a^{11}-\frac{16\!\cdots\!84}{30\!\cdots\!93}a^{10}+\frac{27\!\cdots\!51}{30\!\cdots\!93}a^{9}-\frac{52\!\cdots\!17}{30\!\cdots\!93}a^{8}+\frac{47\!\cdots\!34}{30\!\cdots\!93}a^{7}-\frac{55\!\cdots\!60}{30\!\cdots\!93}a^{6}+\frac{99\!\cdots\!37}{30\!\cdots\!93}a^{5}-\frac{17\!\cdots\!52}{30\!\cdots\!93}a^{4}+\frac{15\!\cdots\!55}{30\!\cdots\!93}a^{3}-\frac{47\!\cdots\!26}{30\!\cdots\!93}a^{2}+\frac{89\!\cdots\!12}{30\!\cdots\!93}a+\frac{11\!\cdots\!82}{30\!\cdots\!93}$ (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: | \( 1092701.97171 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 1092701.97171 \cdot 1}{2\cdot\sqrt{5969915757478328440239161344}}\cr\approx \mathstrut & 0.274818876111 \end{aligned}\] (assuming GRH)
Galois group
$C_2\wr C_5$ (as 20T46):
A solvable group of order 160 |
The 16 conjugacy class representatives for $C_2\wr C_5$ |
Character table for $C_2\wr C_5$ |
Intermediate fields
\(\Q(\zeta_{11})^+\), 10.6.2414538435584.2, 10.6.2414538435584.1, 10.2.219503494144.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 10 siblings: | data not computed |
Degree 20 siblings: | data not computed |
Degree 32 sibling: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | 10.2.2414538435584.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{8}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 $20$ | $4$ | $5$ | $30$ | |||
\(11\) | 11.10.9.1 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
11.10.9.1 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |