Normalized defining polynomial
\( x^{22} - 9 x^{21} - 40 x^{20} + 537 x^{19} + 295 x^{18} - 13162 x^{17} + 9563 x^{16} + 170660 x^{15} - 231298 x^{14} - 1254848 x^{13} + 2189761 x^{12} + 5193994 x^{11} - 10720346 x^{10} - 11165111 x^{9} + 27735943 x^{8} + 9739465 x^{7} - 35394797 x^{6} + 189445 x^{5} + 19032841 x^{4} - 2534661 x^{3} - 3171418 x^{2} + 176355 x + 33811 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[22, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(601130775140836298755595442714814879781421=3^{11}\cdot 7^{11}\cdot 23^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $79.26$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $3, 7, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(483=3\cdot 7\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{483}(64,·)$, $\chi_{483}(1,·)$, $\chi_{483}(461,·)$, $\chi_{483}(335,·)$, $\chi_{483}(400,·)$, $\chi_{483}(209,·)$, $\chi_{483}(146,·)$, $\chi_{483}(211,·)$, $\chi_{483}(85,·)$, $\chi_{483}(463,·)$, $\chi_{483}(358,·)$, $\chi_{483}(167,·)$, $\chi_{483}(104,·)$, $\chi_{483}(41,·)$, $\chi_{483}(232,·)$, $\chi_{483}(190,·)$, $\chi_{483}(169,·)$, $\chi_{483}(440,·)$, $\chi_{483}(377,·)$, $\chi_{483}(188,·)$, $\chi_{483}(62,·)$, $\chi_{483}(127,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{13019} a^{20} - \frac{3305}{13019} a^{19} + \frac{2395}{13019} a^{18} - \frac{2337}{13019} a^{17} + \frac{1543}{13019} a^{16} - \frac{1335}{13019} a^{15} - \frac{3866}{13019} a^{14} - \frac{4483}{13019} a^{13} + \frac{3400}{13019} a^{12} - \frac{2445}{13019} a^{11} + \frac{3670}{13019} a^{10} + \frac{1556}{13019} a^{9} + \frac{4628}{13019} a^{8} - \frac{2956}{13019} a^{7} + \frac{3735}{13019} a^{6} + \frac{344}{13019} a^{5} + \frac{2301}{13019} a^{4} + \frac{1053}{13019} a^{3} + \frac{557}{13019} a^{2} + \frac{3615}{13019} a + \frac{4948}{13019}$, $\frac{1}{181750129770760123313809691093406224897406317373526549600840147} a^{21} - \frac{6646583075183165177247966945359191097216182749032077964638}{181750129770760123313809691093406224897406317373526549600840147} a^{20} + \frac{81131286436405733062908355798949418384802188023674226871020106}{181750129770760123313809691093406224897406317373526549600840147} a^{19} - \frac{53693647868586322048504241293165321389656552907966040601156157}{181750129770760123313809691093406224897406317373526549600840147} a^{18} - \frac{12238984684616774171029207781932638182369103659736905652372648}{181750129770760123313809691093406224897406317373526549600840147} a^{17} - \frac{34878611231721957831586626008987150238886609021419313981225595}{181750129770760123313809691093406224897406317373526549600840147} a^{16} - \frac{74394634960738979991680481152309911876281995844898567162171205}{181750129770760123313809691093406224897406317373526549600840147} a^{15} - \frac{35522536708830155452970985653554646884389145869672790417508827}{181750129770760123313809691093406224897406317373526549600840147} a^{14} - \frac{84231202358107163181433229312883182978546678196350117455570111}{181750129770760123313809691093406224897406317373526549600840147} a^{13} - \frac{13786520336371966680958761768479392415950108617820527130576188}{181750129770760123313809691093406224897406317373526549600840147} a^{12} - \frac{29568208910156100128089842859573068062278194678314933878397706}{181750129770760123313809691093406224897406317373526549600840147} a^{11} - \frac{7633500183870065341778369441837989182349015104305213169318732}{181750129770760123313809691093406224897406317373526549600840147} a^{10} - \frac{67392790315799848654119444543397765298458259683950136250044311}{181750129770760123313809691093406224897406317373526549600840147} a^{9} - \frac{73645745397048248141532063044322764210359827403688710148539744}{181750129770760123313809691093406224897406317373526549600840147} a^{8} + \frac{87307533668004385603172397533256493084511136367952816354470958}{181750129770760123313809691093406224897406317373526549600840147} a^{7} + \frac{81741479608815464380063831952188739544750708805991279998691785}{181750129770760123313809691093406224897406317373526549600840147} a^{6} + \frac{33040340528071411432963346419218526058299056450131864869999762}{181750129770760123313809691093406224897406317373526549600840147} a^{5} - \frac{59731532737373379351272306522915600173758821357234991542475685}{181750129770760123313809691093406224897406317373526549600840147} a^{4} + \frac{70773697461484654687722759150445832459976283592859368720198835}{181750129770760123313809691093406224897406317373526549600840147} a^{3} + \frac{31012941898802304917288135168963864371460190496082341104245870}{181750129770760123313809691093406224897406317373526549600840147} a^{2} + \frac{47171105621782718526199458728891943236368234524247719239414206}{181750129770760123313809691093406224897406317373526549600840147} a + \frac{64397231864856574283667891993889054393015516659078971181141166}{181750129770760123313809691093406224897406317373526549600840147}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $21$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 60706240430800 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 22 |
| The 22 conjugacy class representatives for $C_{22}$ |
| Character table for $C_{22}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{21}) \), \(\Q(\zeta_{23})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $22$ | R | ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ | R | $22$ | $22$ | ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ | $22$ | R | $22$ | $22$ | ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ | $22$ | ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 7 | Data not computed | ||||||
| 23 | Data not computed | ||||||