Normalized defining polynomial
\( x^{21} - 7 x^{20} - x^{19} + 114 x^{18} - 219 x^{17} - 472 x^{16} + 1808 x^{15} - 333 x^{14} - 4832 x^{13} + 5022 x^{12} + 3881 x^{11} - 8733 x^{10} + 1621 x^{9} + 5276 x^{8} - 3399 x^{7} + \cdots + 1 \)
Invariants
Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[21, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(122480818680759715483302268119489498578944\) \(\medspace = 2^{18}\cdot 13^{2}\cdot 73^{12}\cdot 347443^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(90.48\) | 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: | $2^{6/7}13^{2/3}73^{2/3}347443^{2/3}\approx 864554.2095127514$ | ||
Ramified primes: | \(2\), \(13\), \(73\), \(347443\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $20$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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: | $a-1$, $a$, $a^{19}-6a^{18}-7a^{17}+107a^{16}-112a^{15}-584a^{14}+1224a^{13}+891a^{12}-3941a^{11}+1081a^{10}+4962a^{9}-3771a^{8}-2150a^{7}+3126a^{6}-273a^{5}-917a^{4}+371a^{3}+38a^{2}-58a+13$, $1938a^{20}-12493a^{19}-8852a^{18}+216013a^{17}-304842a^{16}-1083196a^{15}+2903811a^{14}+960645a^{13}-8828668a^{12}+4847064a^{11}+10192466a^{10}-11277540a^{9}-3086153a^{8}+8503598a^{7}-1886941a^{6}-2281996a^{5}+1232378a^{4}+33456a^{3}-164471a^{2}+42956a-3548$, $19a^{20}-127a^{19}-57a^{18}+2135a^{17}-3500a^{16}-9842a^{15}+30977a^{14}+2202a^{13}-88309a^{12}+69367a^{11}+86593a^{10}-135890a^{9}-405a^{8}+91270a^{7}-41370a^{6}-17821a^{5}+18695a^{4}-2815a^{3}-1951a^{2}+892a-115$, $a^{20}-6a^{19}-7a^{18}+107a^{17}-112a^{16}-584a^{15}+1224a^{14}+891a^{13}-3941a^{12}+1081a^{11}+4962a^{10}-3771a^{9}-2150a^{8}+3126a^{7}-273a^{6}-917a^{5}+371a^{4}+38a^{3}-57a^{2}+11a-2$, $a^{19}-6a^{18}-7a^{17}+107a^{16}-112a^{15}-584a^{14}+1224a^{13}+891a^{12}-3941a^{11}+1081a^{10}+4962a^{9}-3771a^{8}-2150a^{7}+3126a^{6}-273a^{5}-917a^{4}+371a^{3}+38a^{2}-57a+11$, $19a^{20}-125a^{19}-69a^{18}+2121a^{17}-3286a^{16}-10066a^{15}+29809a^{14}+4650a^{13}-86527a^{12}+61485a^{11}+88755a^{10}-125966a^{9}-7947a^{8}+86970a^{7}-35118a^{6}-18367a^{5}+16861a^{4}-2073a^{3}-1875a^{2}+776a-91$, $1900a^{20}-12452a^{19}-7460a^{18}+213286a^{17}-320893a^{16}-1040293a^{15}+2971241a^{14}+694856a^{13}-8874121a^{12}+5579295a^{11}+9874944a^{10}-12189575a^{9}-2374176a^{8}+8976657a^{7}-2445773a^{6}-2324900a^{5}+1410391a^{4}-217a^{3}-181704a^{2}+49785a-4199$, $1938a^{20}-12741a^{19}-7364a^{18}+217807a^{17}-331668a^{16}-1056130a^{15}+3054209a^{14}+656334a^{13}-9085609a^{12}+5859640a^{11}+10019918a^{10}-12649671a^{9}-2252675a^{8}+9256972a^{7}-2636678a^{6}-2366258a^{5}+1483516a^{4}-14524a^{3}-188950a^{2}+53215a-4601$, $1746a^{20}-11199a^{19}-8302a^{18}+194148a^{17}-268672a^{16}-980941a^{15}+2581615a^{14}+927647a^{13}-7887533a^{12}+4154731a^{11}+9190653a^{10}-9866075a^{9}-2921425a^{8}+7492456a^{7}-1563545a^{6}-2031605a^{5}+1063401a^{4}+38396a^{3}-143586a^{2}+36729a-2980$, $1620a^{20}-10587a^{19}-6502a^{18}+181415a^{17}-270688a^{16}-886160a^{15}+2511878a^{14}+605081a^{13}-7493135a^{12}+4684233a^{11}+8292853a^{10}-10233537a^{9}-1909714a^{8}+7479910a^{7}-2134181a^{6}-1884418a^{5}+1206655a^{4}-27002a^{3}-152025a^{2}+45228a-4159$, $9733a^{20}-62718a^{19}-44610a^{18}+1084732a^{17}-1528286a^{16}-5443560a^{15}+14569559a^{14}+4859579a^{13}-44323574a^{12}+24233048a^{11}+51238363a^{10}-56502622a^{9}-15629011a^{8}+42651917a^{7}-9371985a^{6}-11472762a^{5}+6157202a^{4}+180623a^{3}-823765a^{2}+213694a-17521$, $1143a^{20}-7336a^{19}-5427a^{18}+127244a^{17}-176195a^{16}-643758a^{15}+1694172a^{14}+614300a^{13}-5187552a^{12}+2710196a^{11}+6082266a^{10}-6469828a^{9}-1998220a^{8}+4942805a^{7}-971349a^{6}-1361962a^{5}+683796a^{4}+35946a^{3}-93896a^{2}+22779a-1735$, $1667a^{20}-10818a^{19}-7188a^{18}+186361a^{17}-269958a^{16}-924484a^{15}+2542012a^{14}+741487a^{13}-7675635a^{12}+4456787a^{11}+8740760a^{10}-10101156a^{9}-2446240a^{8}+7550007a^{7}-1819755a^{6}-2003105a^{5}+1128116a^{4}+20320a^{3}-148657a^{2}+39294a-3250$, $3992a^{20}-26288a^{19}-14813a^{18}+448432a^{17}-688794a^{16}-2160803a^{15}+6311274a^{14}+1238318a^{13}-18668177a^{12}+12383724a^{11}+20277483a^{10}-26366756a^{9}-3998314a^{8}+19083289a^{7}-5896467a^{6}-4737817a^{5}+3191627a^{4}-97141a^{3}-398214a^{2}+118799a-10717$, $168a^{20}-1083a^{19}-759a^{18}+18678a^{17}-26500a^{16}-93013a^{15}+251060a^{14}+77947a^{13}-756513a^{12}+431818a^{11}+852176a^{10}-982145a^{9}-220863a^{8}+724443a^{7}-194990a^{6}-183048a^{5}+115501a^{4}-2616a^{3}-14733a^{2}+4393a-398$, $2368a^{20}-15323a^{19}-10428a^{18}+264148a^{17}-379177a^{16}-1313167a^{15}+3581437a^{14}+1077449a^{13}-10814445a^{12}+6220444a^{11}+12289836a^{10}-14148319a^{9}-3388535a^{8}+10549431a^{7}-2608740a^{6}-2770036a^{5}+1600785a^{4}+14098a^{3}-209342a^{2}+56782a-4794$, $1717a^{20}-11197a^{19}-7053a^{18}+192210a^{17}-284156a^{16}-943768a^{15}+2649375a^{14}+682022a^{13}-7937133a^{12}+4844171a^{11}+8876256a^{10}-10713004a^{9}-2206301a^{8}+7897719a^{7}-2124553a^{6}-2033546a^{5}+1239401a^{4}-7423a^{3}-159177a^{2}+44894a-3913$, $2575a^{20}-17165a^{19}-8314a^{18}+290814a^{17}-466773a^{16}-1371997a^{15}+4198417a^{14}+548237a^{13}-12269103a^{12}+8831008a^{11}+12973224a^{10}-18171319a^{9}-1931640a^{8}+12979845a^{7}-4402633a^{6}-3159960a^{5}+2264324a^{4}-91414a^{3}-278349a^{2}+83871a-7535$ (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: | \( 66161868019100 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{21}\cdot(2\pi)^{0}\cdot 66161868019100 \cdot 1}{2\cdot\sqrt{122480818680759715483302268119489498578944}}\cr\approx \mathstrut & 0.198231940175436 \end{aligned}\] (assuming GRH)
Galois group
$C_3^6.C_{21}:C_3$ (as 21T86):
A solvable group of order 45927 |
The 183 conjugacy class representatives for $C_3^6.C_{21}:C_3$ are not computed |
Character table for $C_3^6.C_{21}:C_3$ is not computed |
Intermediate fields
7.7.1817487424.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 21 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 | $21$ | ${\href{/padicField/5.9.0.1}{9} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | $21$ | ${\href{/padicField/11.9.0.1}{9} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }$ | R | ${\href{/padicField/17.7.0.1}{7} }^{3}$ | ${\href{/padicField/19.9.0.1}{9} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.9.0.1}{9} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.9.0.1}{9} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.9.0.1}{9} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }$ | $21$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.21.18.1 | $x^{21} + 7 x^{19} + 7 x^{18} + 21 x^{17} + 42 x^{16} + 62 x^{15} + 111 x^{14} + 98 x^{13} - 189 x^{12} - 189 x^{11} + 259 x^{10} + 1496 x^{9} + 2586 x^{8} + 925 x^{7} + 798 x^{6} - 1092 x^{5} + 1029 x^{4} - 174 x^{3} - 53 x^{2} - 313 x + 131$ | $7$ | $3$ | $18$ | 21T2 | $[\ ]_{7}^{3}$ |
\(13\) | 13.3.2.1 | $x^{3} + 26$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
\(73\) | 73.3.0.1 | $x^{3} + 2 x + 68$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
\(347443\) | Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $3$ | $1$ | $2$ | ||||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |