Normalized defining polynomial
\( x^{22} + 25 x^{20} + 250 x^{18} + 1243 x^{16} + 2879 x^{14} + 334 x^{12} - 14126 x^{10} - 33691 x^{8} + \cdots - 289 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(56501459388151144478039723653407440896\) \(\medspace = 2^{22}\cdot 1297^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(52.00\) | 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: | \(2\), \(1297\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5}a^{14}+\frac{1}{5}a^{12}-\frac{2}{5}a^{10}-\frac{1}{5}a^{8}-\frac{1}{5}a^{6}-\frac{2}{5}a^{4}+\frac{1}{5}a^{2}+\frac{1}{5}$, $\frac{1}{5}a^{15}+\frac{1}{5}a^{13}-\frac{2}{5}a^{11}-\frac{1}{5}a^{9}-\frac{1}{5}a^{7}-\frac{2}{5}a^{5}+\frac{1}{5}a^{3}+\frac{1}{5}a$, $\frac{1}{5}a^{16}+\frac{2}{5}a^{12}+\frac{1}{5}a^{10}-\frac{1}{5}a^{6}-\frac{2}{5}a^{4}-\frac{1}{5}$, $\frac{1}{5}a^{17}+\frac{2}{5}a^{13}+\frac{1}{5}a^{11}-\frac{1}{5}a^{7}-\frac{2}{5}a^{5}-\frac{1}{5}a$, $\frac{1}{5}a^{18}-\frac{1}{5}a^{12}-\frac{1}{5}a^{10}+\frac{1}{5}a^{8}-\frac{1}{5}a^{4}+\frac{2}{5}a^{2}-\frac{2}{5}$, $\frac{1}{5}a^{19}-\frac{1}{5}a^{13}-\frac{1}{5}a^{11}+\frac{1}{5}a^{9}-\frac{1}{5}a^{5}+\frac{2}{5}a^{3}-\frac{2}{5}a$, $\frac{1}{159896605}a^{20}+\frac{11279044}{159896605}a^{18}-\frac{1540162}{159896605}a^{16}+\frac{2479896}{159896605}a^{14}-\frac{78904352}{159896605}a^{12}+\frac{1015516}{5157955}a^{10}+\frac{4343017}{159896605}a^{8}+\frac{70657104}{159896605}a^{6}+\frac{38849433}{159896605}a^{4}-\frac{870357}{14536055}a^{2}+\frac{36699661}{159896605}$, $\frac{1}{2718242285}a^{21}-\frac{212576203}{2718242285}a^{19}-\frac{33519483}{2718242285}a^{17}-\frac{125437388}{2718242285}a^{15}+\frac{272868179}{2718242285}a^{13}+\frac{6186331}{17537047}a^{11}-\frac{571284761}{2718242285}a^{9}+\frac{550346919}{2718242285}a^{7}+\frac{116499578}{543648457}a^{5}+\frac{18432079}{49422587}a^{3}-\frac{570907438}{2718242285}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$ (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: | $\frac{22884694}{2718242285}a^{21}+\frac{634350729}{2718242285}a^{19}+\frac{7035927626}{2718242285}a^{17}+\frac{38861684683}{2718242285}a^{15}+\frac{101518562241}{2718242285}a^{13}+\frac{1110076012}{87685235}a^{11}-\frac{473882170643}{2718242285}a^{9}-\frac{1180019431352}{2718242285}a^{7}-\frac{1165331113627}{2718242285}a^{5}-\frac{41717162518}{247112935}a^{3}-\frac{38810453412}{2718242285}a$, $\frac{6600328}{543648457}a^{21}+\frac{860651733}{2718242285}a^{19}+\frac{8727327711}{2718242285}a^{17}+\frac{41785161133}{2718242285}a^{15}+\frac{80448533452}{2718242285}a^{13}-\frac{2035862338}{87685235}a^{11}-\frac{107034021902}{543648457}a^{9}-\frac{858155966634}{2718242285}a^{7}-\frac{541869959921}{2718242285}a^{5}-\frac{9865666146}{247112935}a^{3}-\frac{753774144}{2718242285}a$, $\frac{14445657}{2718242285}a^{21}+\frac{287300242}{2718242285}a^{19}+\frac{409581246}{543648457}a^{17}+\frac{1076892717}{543648457}a^{15}-\frac{3791064172}{2718242285}a^{13}-\frac{1445997984}{87685235}a^{11}-\frac{12421079415}{543648457}a^{9}+\frac{32858173981}{2718242285}a^{7}+\frac{131019121803}{2718242285}a^{5}+\frac{8181572547}{247112935}a^{3}+\frac{17870132433}{2718242285}a$, $\frac{8954942}{159896605}a^{20}+\frac{209058184}{159896605}a^{18}+\frac{1898939536}{159896605}a^{16}+\frac{8111670178}{159896605}a^{14}+\frac{13272688656}{159896605}a^{12}-\frac{519311601}{5157955}a^{10}-\frac{98202513001}{159896605}a^{8}-\frac{151239433888}{159896605}a^{6}-\frac{101609776582}{159896605}a^{4}-\frac{2516892111}{14536055}a^{2}-\frac{2102150669}{159896605}$, $\frac{882702}{14536055}a^{20}+\frac{4051353}{2907211}a^{18}+\frac{36212742}{2907211}a^{16}+\frac{763068136}{14536055}a^{14}+\frac{1234088256}{14536055}a^{12}-\frac{49510594}{468905}a^{10}-\frac{9298346783}{14536055}a^{8}-\frac{2853914099}{2907211}a^{6}-\frac{9253948907}{14536055}a^{4}-\frac{2226293856}{14536055}a^{2}-\frac{156029657}{14536055}$, $\frac{2655223}{159896605}a^{20}+\frac{62723522}{159896605}a^{18}+\frac{585651811}{159896605}a^{16}+\frac{2661069267}{159896605}a^{14}+\frac{5246411807}{159896605}a^{12}-\frac{70780288}{5157955}a^{10}-\frac{31372280068}{159896605}a^{8}-\frac{62068476284}{159896605}a^{6}-\frac{55417032458}{159896605}a^{4}-\frac{1971567182}{14536055}a^{2}-\frac{467699236}{31979321}$, $\frac{21299315}{543648457}a^{21}+\frac{2488351324}{2718242285}a^{19}+\frac{22632802439}{2718242285}a^{17}+\frac{96776786071}{2718242285}a^{15}+\frac{31468438885}{543648457}a^{13}-\frac{6518581017}{87685235}a^{11}-\frac{1194176071192}{2718242285}a^{9}-\frac{355276750143}{543648457}a^{7}-\frac{1073808356899}{2718242285}a^{5}-\frac{18578105921}{247112935}a^{3}-\frac{4841777706}{2718242285}a$, $\frac{11292051}{159896605}a^{20}+\frac{265836248}{159896605}a^{18}+\frac{2444529453}{159896605}a^{16}+\frac{10653055766}{159896605}a^{14}+\frac{18305224444}{159896605}a^{12}-\frac{607542973}{5157955}a^{10}-\frac{129866491054}{159896605}a^{8}-\frac{208155683966}{159896605}a^{6}-\frac{144662823441}{159896605}a^{4}-\frac{3783314329}{14536055}a^{2}-\frac{4234012807}{159896605}$, $\frac{3154084}{2718242285}a^{21}+\frac{72300419}{2718242285}a^{19}+\frac{118812535}{543648457}a^{17}+\frac{1737034529}{2718242285}a^{15}-\frac{469145807}{543648457}a^{13}-\frac{763611581}{87685235}a^{11}-\frac{33424171404}{2718242285}a^{9}+\frac{44824656348}{2718242285}a^{7}+\frac{149865339903}{2718242285}a^{5}+\frac{9551602273}{247112935}a^{3}+\frac{2025933983}{543648457}a$, $\frac{10832731}{159896605}a^{20}+\frac{257327489}{159896605}a^{18}+\frac{478644162}{31979321}a^{16}+\frac{10594701103}{159896605}a^{14}+\frac{18807488994}{159896605}a^{12}-\frac{556522801}{5157955}a^{10}-\frac{26004578288}{31979321}a^{8}-\frac{42713456329}{31979321}a^{6}-\frac{30017534571}{31979321}a^{4}-\frac{756885782}{2907211}a^{2}-\frac{3127300214}{159896605}$, $\frac{16\!\cdots\!78}{2718242285}a^{21}+\frac{14\!\cdots\!36}{159896605}a^{20}+\frac{91\!\cdots\!43}{543648457}a^{19}+\frac{80\!\cdots\!01}{31979321}a^{18}+\frac{10\!\cdots\!53}{543648457}a^{17}+\frac{92\!\cdots\!54}{31979321}a^{16}+\frac{32\!\cdots\!04}{2718242285}a^{15}+\frac{57\!\cdots\!77}{31979321}a^{14}+\frac{12\!\cdots\!29}{2718242285}a^{13}+\frac{21\!\cdots\!72}{31979321}a^{12}+\frac{90\!\cdots\!14}{87685235}a^{11}+\frac{79\!\cdots\!04}{5157955}a^{10}+\frac{39\!\cdots\!68}{2718242285}a^{9}+\frac{35\!\cdots\!84}{159896605}a^{8}+\frac{66\!\cdots\!53}{543648457}a^{7}+\frac{29\!\cdots\!68}{159896605}a^{6}+\frac{15\!\cdots\!42}{2718242285}a^{5}+\frac{26\!\cdots\!48}{31979321}a^{4}+\frac{28\!\cdots\!21}{247112935}a^{3}+\frac{25\!\cdots\!04}{14536055}a^{2}+\frac{21\!\cdots\!47}{2718242285}a+\frac{18\!\cdots\!21}{159896605}$ (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: | \( 4115020682.75 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 4115020682.75 \cdot 3}{2\cdot\sqrt{56501459388151144478039723653407440896}}\cr\approx \mathstrut & 0.314986632624 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.D_{11}$ (as 22T30):
A solvable group of order 22528 |
The 100 conjugacy class representatives for $C_2^{10}.D_{11}$ |
Character table for $C_2^{10}.D_{11}$ |
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: | 22.14.73282392826432034388017521578469450842112.4 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.11.0.1}{11} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{5}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{9}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.11.0.1}{11} }^{2}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}$ |
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 $22$ | $2$ | $11$ | $22$ | |||
\(1297\) | $\Q_{1297}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
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 $4$ | $2$ | $2$ | $2$ |