Normalized defining polynomial
\( x^{22} + 2x^{20} - x^{18} + 2x^{16} - 18x^{12} - 24x^{10} + 3x^{8} + 36x^{6} + 27x^{4} + 4x^{2} - 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(104070916689387515610916716544\) \(\medspace = 2^{22}\cdot 19457^{2}\cdot 8095783^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(20.84\) | 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))
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Galois root discriminant: | not computed | ||
Ramified primes: | \(2\), \(19457\), \(8095783\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{131749}a^{20}-\frac{5480}{131749}a^{18}+\frac{2587}{131749}a^{16}+\frac{46960}{131749}a^{14}+\frac{2826}{131749}a^{12}+\frac{54232}{131749}a^{10}+\frac{57645}{131749}a^{8}+\frac{55964}{131749}a^{6}+\frac{48809}{131749}a^{4}+\frac{11308}{131749}a^{2}+\frac{63327}{131749}$, $\frac{1}{131749}a^{21}-\frac{5480}{131749}a^{19}+\frac{2587}{131749}a^{17}+\frac{46960}{131749}a^{15}+\frac{2826}{131749}a^{13}+\frac{54232}{131749}a^{11}+\frac{57645}{131749}a^{9}+\frac{55964}{131749}a^{7}+\frac{48809}{131749}a^{5}+\frac{11308}{131749}a^{3}+\frac{63327}{131749}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | 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)
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Fundamental units: | $\frac{9568}{131749}a^{20}+\frac{3462}{131749}a^{18}-\frac{16396}{131749}a^{16}+\frac{49190}{131749}a^{14}-\frac{101126}{131749}a^{12}-\frac{67535}{131749}a^{10}-\frac{85703}{131749}a^{8}+\frac{35616}{131749}a^{6}+\frac{217805}{131749}a^{4}+\frac{160764}{131749}a^{2}+\frac{130834}{131749}$, $\frac{227019}{131749}a^{20}+\frac{305185}{131749}a^{18}-\frac{434136}{131749}a^{16}+\frac{737152}{131749}a^{14}-\frac{457183}{131749}a^{12}-\frac{3806865}{131749}a^{10}-\frac{2884644}{131749}a^{8}+\frac{2706728}{131749}a^{6}+\frac{6408176}{131749}a^{4}+\frac{1714324}{131749}a^{2}-\frac{413914}{131749}$, $a$, $\frac{641126}{131749}a^{20}+\frac{902346}{131749}a^{18}-\frac{1180940}{131749}a^{16}+\frac{1971715}{131749}a^{14}-\frac{1175913}{131749}a^{12}-\frac{10873278}{131749}a^{10}-\frac{8953146}{131749}a^{8}+\frac{7225995}{131749}a^{6}+\frac{18931808}{131749}a^{4}+\frac{6161039}{131749}a^{2}-\frac{1161873}{131749}$, $\frac{251539}{131749}a^{21}+\frac{319565}{131749}a^{19}-\frac{503914}{131749}a^{17}+\frac{841841}{131749}a^{15}-\frac{595386}{131749}a^{13}-\frac{4176129}{131749}a^{11}-\frac{2964265}{131749}a^{9}+\frac{3173420}{131749}a^{7}+\frac{7055685}{131749}a^{5}+\frac{1654839}{131749}a^{3}-\frac{561337}{131749}a$, $\frac{235865}{131749}a^{21}+\frac{312737}{131749}a^{19}-\frac{473860}{131749}a^{17}+\frac{740715}{131749}a^{15}-\frac{490697}{131749}a^{13}-\frac{4032200}{131749}a^{11}-\frac{2957353}{131749}a^{9}+\frac{3046777}{131749}a^{7}+\frac{6958113}{131749}a^{5}+\frac{1747401}{131749}a^{3}-\frac{683518}{131749}a$, $\frac{551453}{131749}a^{20}+\frac{754867}{131749}a^{18}-\frac{1023253}{131749}a^{16}+\frac{1757424}{131749}a^{14}-\frac{1106735}{131749}a^{12}-\frac{9187589}{131749}a^{10}-\frac{7400228}{131749}a^{8}+\frac{6295139}{131749}a^{6}+\frac{15885653}{131749}a^{4}+\frac{4893318}{131749}a^{2}-\frac{975048}{131749}$, $\frac{161540}{131749}a^{21}-\frac{130438}{131749}a^{20}+\frac{245829}{131749}a^{19}-\frac{201583}{131749}a^{18}-\frac{267346}{131749}a^{17}+\frac{229581}{131749}a^{16}+\frac{469725}{131749}a^{15}-\frac{357470}{131749}a^{14}-\frac{261743}{131749}a^{13}+\frac{147663}{131749}a^{12}-\frac{2779204}{131749}a^{11}+\frac{2325174}{131749}a^{10}-\frac{2549251}{131749}a^{9}+\frac{2056653}{131749}a^{8}+\frac{1389168}{131749}a^{7}-\frac{1464628}{131749}a^{6}+\frac{4961668}{131749}a^{5}-\frac{3993885}{131749}a^{4}+\frac{2365917}{131749}a^{3}-\frac{1643837}{131749}a^{2}+\frac{60726}{131749}a+\frac{151576}{131749}$, $\frac{529400}{131749}a^{21}+\frac{382892}{131749}a^{20}+\frac{791474}{131749}a^{19}+\frac{513410}{131749}a^{18}-\frac{895298}{131749}a^{17}-\frac{737872}{131749}a^{16}+\frac{1563935}{131749}a^{15}+\frac{1217537}{131749}a^{14}-\frac{847738}{131749}a^{13}-\frac{792239}{131749}a^{12}-\frac{9016714}{131749}a^{11}-\frac{6416647}{131749}a^{10}-\frac{8189806}{131749}a^{9}-\frac{4975343}{131749}a^{8}+\frac{5159938}{131749}a^{7}+\frac{4726496}{131749}a^{6}+\frac{15890106}{131749}a^{5}+\frac{11046894}{131749}a^{4}+\frac{6631588}{131749}a^{3}+\frac{2973827}{131749}a^{2}-\frac{63736}{131749}a-\frac{870017}{131749}$, $\frac{123365}{131749}a^{21}-\frac{254199}{131749}a^{20}+\frac{95668}{131749}a^{19}-\frac{366904}{131749}a^{18}-\frac{346070}{131749}a^{17}+\frac{473442}{131749}a^{16}+\frac{480368}{131749}a^{15}-\frac{725640}{131749}a^{14}-\frac{505360}{131749}a^{13}+\frac{456170}{131749}a^{12}-\frac{1991524}{131749}a^{11}+\frac{4447662}{131749}a^{10}-\frac{567344}{131749}a^{9}+\frac{3774895}{131749}a^{8}+\frac{2590993}{131749}a^{7}-\frac{2897792}{131749}a^{6}+\frac{3159714}{131749}a^{5}-\frac{7905354}{131749}a^{4}-\frac{737486}{131749}a^{3}-\frac{3012837}{131749}a^{2}-\frac{907341}{131749}a+\frac{223241}{131749}$, $\frac{656404}{131749}a^{21}+\frac{566405}{131749}a^{20}+\frac{839521}{131749}a^{19}+\frac{765785}{131749}a^{18}-\frac{1313203}{131749}a^{17}-\frac{1076635}{131749}a^{16}+\frac{2185039}{131749}a^{15}+\frac{1812923}{131749}a^{14}-\frac{1477455}{131749}a^{13}-\frac{1143812}{131749}a^{12}-\frac{10886243}{131749}a^{11}-\frac{9489318}{131749}a^{10}-\frac{7809160}{131749}a^{9}-\frac{7394146}{131749}a^{8}+\frac{8510467}{131749}a^{7}+\frac{6726215}{131749}a^{6}+\frac{18277625}{131749}a^{5}+\frac{16183608}{131749}a^{4}+\frac{3830242}{131749}a^{3}+\frac{4541320}{131749}a^{2}-\frac{1875368}{131749}a-\frac{1253305}{131749}$ (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)]
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Regulator: | \( 243223.959653 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 243223.959653 \cdot 1}{2\cdot\sqrt{104070916689387515610916716544}}\cr\approx \mathstrut & 0.144600682806 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.S_{11}$ (as 22T51):
A non-solvable group of order 40874803200 |
The 376 conjugacy class representatives for $C_2^{10}.S_{11}$ |
Character table for $C_2^{10}.S_{11}$ |
Intermediate fields
11.5.157519649831.1 |
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 |
Degree 44 sibling: | data not computed |
Minimal sibling: | 22.8.6213581786996136997545952595085659731356637711243979839317721535086212076464978309381089131358842015486585577154540390549316173824.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.11.0.1}{11} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | $20{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.11.0.1}{11} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.9.0.1}{9} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | ${\href{/padicField/43.9.0.1}{9} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | $18{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 $22$ | $2$ | $11$ | $22$ | |||
\(19457\) | $\Q_{19457}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{19457}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(8095783\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |