Normalized defining polynomial
\( x^{20} + 61 x^{18} + 1518 x^{16} + 20116 x^{14} + 172707 x^{12} + 1240037 x^{10} + 6798195 x^{8} + \cdots + 707506801 \)
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: | \(758173088478631191940755669336601299779584\) \(\medspace = 2^{20}\cdot 11^{16}\cdot 1583^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(124.16\) | 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: | $2\cdot 11^{4/5}1583^{1/2}\approx 541.8568914926865$ | ||
Ramified primes: | \(2\), \(11\), \(1583\) | 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 a CM field. | |||
Reflex fields: | unavailable$^{512}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{98\!\cdots\!21}a^{18}-\frac{22\!\cdots\!42}{98\!\cdots\!21}a^{16}-\frac{42\!\cdots\!29}{98\!\cdots\!21}a^{14}-\frac{46\!\cdots\!42}{98\!\cdots\!21}a^{12}+\frac{48\!\cdots\!63}{98\!\cdots\!21}a^{10}+\frac{13\!\cdots\!38}{98\!\cdots\!21}a^{8}-\frac{43\!\cdots\!75}{98\!\cdots\!21}a^{6}+\frac{33\!\cdots\!58}{98\!\cdots\!21}a^{4}-\frac{21\!\cdots\!95}{98\!\cdots\!21}a^{2}+\frac{94\!\cdots\!68}{98\!\cdots\!21}$, $\frac{1}{26\!\cdots\!79}a^{19}-\frac{92\!\cdots\!56}{26\!\cdots\!79}a^{17}-\frac{10\!\cdots\!88}{26\!\cdots\!79}a^{15}+\frac{52\!\cdots\!99}{26\!\cdots\!79}a^{13}-\frac{15\!\cdots\!84}{26\!\cdots\!79}a^{11}-\frac{11\!\cdots\!59}{26\!\cdots\!79}a^{9}+\frac{50\!\cdots\!82}{26\!\cdots\!79}a^{7}-\frac{10\!\cdots\!95}{26\!\cdots\!79}a^{5}-\frac{10\!\cdots\!15}{26\!\cdots\!79}a^{3}+\frac{83\!\cdots\!02}{26\!\cdots\!79}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{275448}$, which has order $1101792$ (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: | $\frac{14\!\cdots\!18}{98\!\cdots\!21}a^{18}-\frac{98\!\cdots\!99}{98\!\cdots\!21}a^{16}-\frac{27\!\cdots\!10}{98\!\cdots\!21}a^{14}-\frac{39\!\cdots\!32}{98\!\cdots\!21}a^{12}-\frac{30\!\cdots\!61}{98\!\cdots\!21}a^{10}-\frac{12\!\cdots\!10}{98\!\cdots\!21}a^{8}-\frac{36\!\cdots\!58}{98\!\cdots\!21}a^{6}-\frac{31\!\cdots\!17}{98\!\cdots\!21}a^{4}-\frac{14\!\cdots\!11}{98\!\cdots\!21}a^{2}-\frac{79\!\cdots\!40}{98\!\cdots\!21}$, $\frac{34\!\cdots\!14}{98\!\cdots\!21}a^{18}+\frac{22\!\cdots\!40}{98\!\cdots\!21}a^{16}+\frac{57\!\cdots\!15}{98\!\cdots\!21}a^{14}+\frac{75\!\cdots\!67}{98\!\cdots\!21}a^{12}+\frac{53\!\cdots\!04}{98\!\cdots\!21}a^{10}+\frac{21\!\cdots\!13}{98\!\cdots\!21}a^{8}+\frac{42\!\cdots\!14}{98\!\cdots\!21}a^{6}-\frac{16\!\cdots\!91}{98\!\cdots\!21}a^{4}+\frac{18\!\cdots\!75}{98\!\cdots\!21}a^{2}-\frac{93\!\cdots\!56}{98\!\cdots\!21}$, $\frac{63\!\cdots\!48}{98\!\cdots\!21}a^{18}-\frac{34\!\cdots\!29}{98\!\cdots\!21}a^{16}-\frac{70\!\cdots\!86}{98\!\cdots\!21}a^{14}-\frac{60\!\cdots\!57}{98\!\cdots\!21}a^{12}-\frac{17\!\cdots\!01}{98\!\cdots\!21}a^{10}-\frac{17\!\cdots\!94}{98\!\cdots\!21}a^{8}+\frac{11\!\cdots\!20}{98\!\cdots\!21}a^{6}+\frac{16\!\cdots\!59}{98\!\cdots\!21}a^{4}+\frac{31\!\cdots\!44}{98\!\cdots\!21}a^{2}+\frac{42\!\cdots\!47}{98\!\cdots\!21}$, $\frac{67\!\cdots\!77}{98\!\cdots\!21}a^{18}+\frac{38\!\cdots\!32}{98\!\cdots\!21}a^{16}+\frac{86\!\cdots\!93}{98\!\cdots\!21}a^{14}+\frac{92\!\cdots\!52}{98\!\cdots\!21}a^{12}+\frac{50\!\cdots\!37}{98\!\cdots\!21}a^{10}+\frac{17\!\cdots\!91}{98\!\cdots\!21}a^{8}-\frac{32\!\cdots\!97}{98\!\cdots\!21}a^{6}-\frac{10\!\cdots\!27}{98\!\cdots\!21}a^{4}-\frac{50\!\cdots\!47}{98\!\cdots\!21}a^{2}-\frac{24\!\cdots\!06}{98\!\cdots\!21}$, $\frac{68\!\cdots\!83}{26\!\cdots\!79}a^{19}+\frac{10\!\cdots\!89}{98\!\cdots\!21}a^{18}+\frac{39\!\cdots\!85}{26\!\cdots\!79}a^{17}+\frac{60\!\cdots\!96}{98\!\cdots\!21}a^{16}+\frac{94\!\cdots\!69}{26\!\cdots\!79}a^{15}+\frac{11\!\cdots\!03}{98\!\cdots\!21}a^{14}+\frac{11\!\cdots\!78}{26\!\cdots\!79}a^{13}+\frac{76\!\cdots\!37}{98\!\cdots\!21}a^{12}+\frac{10\!\cdots\!15}{26\!\cdots\!79}a^{11}-\frac{25\!\cdots\!75}{98\!\cdots\!21}a^{10}+\frac{80\!\cdots\!12}{26\!\cdots\!79}a^{9}-\frac{44\!\cdots\!87}{98\!\cdots\!21}a^{8}+\frac{44\!\cdots\!11}{26\!\cdots\!79}a^{7}-\frac{44\!\cdots\!65}{98\!\cdots\!21}a^{6}+\frac{18\!\cdots\!02}{26\!\cdots\!79}a^{5}-\frac{56\!\cdots\!15}{98\!\cdots\!21}a^{4}+\frac{12\!\cdots\!62}{26\!\cdots\!79}a^{3}-\frac{12\!\cdots\!86}{98\!\cdots\!21}a^{2}-\frac{19\!\cdots\!76}{26\!\cdots\!79}a+\frac{18\!\cdots\!62}{98\!\cdots\!21}$, $\frac{28\!\cdots\!66}{26\!\cdots\!79}a^{19}-\frac{11\!\cdots\!58}{98\!\cdots\!21}a^{18}-\frac{17\!\cdots\!84}{26\!\cdots\!79}a^{17}-\frac{65\!\cdots\!23}{98\!\cdots\!21}a^{16}-\frac{42\!\cdots\!32}{26\!\cdots\!79}a^{15}-\frac{14\!\cdots\!81}{98\!\cdots\!21}a^{14}-\frac{54\!\cdots\!56}{26\!\cdots\!79}a^{13}-\frac{14\!\cdots\!69}{98\!\cdots\!21}a^{12}-\frac{43\!\cdots\!48}{26\!\cdots\!79}a^{11}-\frac{77\!\cdots\!31}{98\!\cdots\!21}a^{10}-\frac{27\!\cdots\!95}{26\!\cdots\!79}a^{9}-\frac{25\!\cdots\!79}{98\!\cdots\!21}a^{8}-\frac{12\!\cdots\!54}{26\!\cdots\!79}a^{7}+\frac{72\!\cdots\!50}{98\!\cdots\!21}a^{6}-\frac{22\!\cdots\!57}{26\!\cdots\!79}a^{5}+\frac{18\!\cdots\!46}{98\!\cdots\!21}a^{4}-\frac{20\!\cdots\!11}{26\!\cdots\!79}a^{3}+\frac{14\!\cdots\!58}{98\!\cdots\!21}a^{2}+\frac{32\!\cdots\!55}{26\!\cdots\!79}a+\frac{40\!\cdots\!37}{98\!\cdots\!21}$, $\frac{16\!\cdots\!55}{26\!\cdots\!79}a^{19}+\frac{74\!\cdots\!37}{98\!\cdots\!21}a^{18}+\frac{97\!\cdots\!49}{26\!\cdots\!79}a^{17}+\frac{40\!\cdots\!25}{98\!\cdots\!21}a^{16}+\frac{22\!\cdots\!51}{26\!\cdots\!79}a^{15}+\frac{82\!\cdots\!89}{98\!\cdots\!21}a^{14}+\frac{26\!\cdots\!99}{26\!\cdots\!79}a^{13}+\frac{68\!\cdots\!94}{98\!\cdots\!21}a^{12}+\frac{20\!\cdots\!95}{26\!\cdots\!79}a^{11}+\frac{15\!\cdots\!26}{98\!\cdots\!21}a^{10}+\frac{13\!\cdots\!34}{26\!\cdots\!79}a^{9}-\frac{42\!\cdots\!93}{98\!\cdots\!21}a^{8}+\frac{73\!\cdots\!50}{26\!\cdots\!79}a^{7}-\frac{15\!\cdots\!85}{98\!\cdots\!21}a^{6}+\frac{16\!\cdots\!53}{26\!\cdots\!79}a^{5}-\frac{21\!\cdots\!74}{98\!\cdots\!21}a^{4}+\frac{16\!\cdots\!58}{26\!\cdots\!79}a^{3}-\frac{44\!\cdots\!30}{98\!\cdots\!21}a^{2}-\frac{28\!\cdots\!90}{26\!\cdots\!79}a-\frac{43\!\cdots\!27}{98\!\cdots\!21}$, $\frac{41\!\cdots\!15}{26\!\cdots\!79}a^{19}-\frac{31\!\cdots\!12}{98\!\cdots\!21}a^{18}+\frac{29\!\cdots\!03}{26\!\cdots\!79}a^{17}-\frac{11\!\cdots\!44}{98\!\cdots\!21}a^{16}+\frac{84\!\cdots\!22}{26\!\cdots\!79}a^{15}-\frac{84\!\cdots\!41}{98\!\cdots\!21}a^{14}+\frac{13\!\cdots\!62}{26\!\cdots\!79}a^{13}+\frac{41\!\cdots\!97}{98\!\cdots\!21}a^{12}+\frac{13\!\cdots\!36}{26\!\cdots\!79}a^{11}+\frac{58\!\cdots\!38}{98\!\cdots\!21}a^{10}+\frac{92\!\cdots\!74}{26\!\cdots\!79}a^{9}+\frac{30\!\cdots\!57}{98\!\cdots\!21}a^{8}+\frac{42\!\cdots\!32}{26\!\cdots\!79}a^{7}+\frac{19\!\cdots\!36}{98\!\cdots\!21}a^{6}+\frac{16\!\cdots\!37}{26\!\cdots\!79}a^{5}+\frac{15\!\cdots\!81}{98\!\cdots\!21}a^{4}+\frac{86\!\cdots\!10}{26\!\cdots\!79}a^{3}+\frac{64\!\cdots\!19}{98\!\cdots\!21}a^{2}-\frac{11\!\cdots\!22}{26\!\cdots\!79}a+\frac{19\!\cdots\!44}{98\!\cdots\!21}$, $\frac{34\!\cdots\!10}{26\!\cdots\!79}a^{19}-\frac{84\!\cdots\!73}{98\!\cdots\!21}a^{18}-\frac{26\!\cdots\!32}{26\!\cdots\!79}a^{17}-\frac{46\!\cdots\!86}{98\!\cdots\!21}a^{16}-\frac{98\!\cdots\!61}{26\!\cdots\!79}a^{15}-\frac{10\!\cdots\!73}{98\!\cdots\!21}a^{14}-\frac{18\!\cdots\!16}{26\!\cdots\!79}a^{13}-\frac{10\!\cdots\!97}{98\!\cdots\!21}a^{12}-\frac{13\!\cdots\!22}{26\!\cdots\!79}a^{11}-\frac{57\!\cdots\!63}{98\!\cdots\!21}a^{10}+\frac{20\!\cdots\!13}{26\!\cdots\!79}a^{9}-\frac{20\!\cdots\!63}{98\!\cdots\!21}a^{8}+\frac{81\!\cdots\!03}{26\!\cdots\!79}a^{7}+\frac{33\!\cdots\!41}{98\!\cdots\!21}a^{6}+\frac{69\!\cdots\!49}{26\!\cdots\!79}a^{5}+\frac{10\!\cdots\!23}{98\!\cdots\!21}a^{4}+\frac{53\!\cdots\!39}{26\!\cdots\!79}a^{3}+\frac{52\!\cdots\!36}{98\!\cdots\!21}a^{2}-\frac{95\!\cdots\!17}{26\!\cdots\!79}a+\frac{12\!\cdots\!40}{98\!\cdots\!21}$ (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: | \( 4115403.0666462025 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 4115403.0666462025 \cdot 1101792}{2\cdot\sqrt{758173088478631191940755669336601299779584}}\cr\approx \mathstrut & 0.249687325207045 \end{aligned}\] (assuming GRH)
Galois group
$C_2^5:C_{10}$ (as 20T72):
A solvable group of order 320 |
The 32 conjugacy class representatives for $C_2^5:C_{10}$ |
Character table for $C_2^5:C_{10}$ |
Intermediate fields
\(\Q(\zeta_{11})^+\), deg 10, 10.0.850323586567180847.1, 10.10.347474031229952.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 20 siblings: | data not computed |
Degree 40 siblings: | 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 | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{8}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 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 | $x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 592 x^{5} + 656 x^{4} + 384 x^{3} - 112 x^{2} - 352 x - 1248$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
\(11\) | 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
\(1583\) | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
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 $4$ | $2$ | $2$ | $2$ |