Normalized defining polynomial
\( x^{22} - 2 x^{20} + 6 x^{18} - 14 x^{16} + 22 x^{14} - 31 x^{12} + 38 x^{10} - 34 x^{8} + 27 x^{6} + \cdots - 1 \)
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: | \(147982356498821950109384704\) \(\medspace = 2^{22}\cdot 12917^{2}\cdot 459847^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.47\) | 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: | \(2\), \(12917\), \(459847\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{181}a^{20}+\frac{77}{181}a^{18}-\frac{65}{181}a^{16}-\frac{81}{181}a^{14}-\frac{42}{181}a^{12}+\frac{90}{181}a^{10}+\frac{89}{181}a^{8}-\frac{62}{181}a^{6}+\frac{16}{181}a^{4}-\frac{20}{181}a^{2}+\frac{55}{181}$, $\frac{1}{181}a^{21}+\frac{77}{181}a^{19}-\frac{65}{181}a^{17}-\frac{81}{181}a^{15}-\frac{42}{181}a^{13}+\frac{90}{181}a^{11}+\frac{89}{181}a^{9}-\frac{62}{181}a^{7}+\frac{16}{181}a^{5}-\frac{20}{181}a^{3}+\frac{55}{181}a$
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)
| |
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: | $\frac{346}{181}a^{20}-\frac{508}{181}a^{18}+\frac{1764}{181}a^{16}-\frac{3772}{181}a^{14}+\frac{5378}{181}a^{12}-\frac{7232}{181}a^{10}+\frac{8350}{181}a^{8}-\frac{6429}{181}a^{6}+\frac{4993}{181}a^{4}-\frac{2576}{181}a^{2}+\frac{568}{181}$, $\frac{313}{181}a^{20}-\frac{515}{181}a^{18}+\frac{1737}{181}a^{16}-\frac{3814}{181}a^{14}+\frac{5678}{181}a^{12}-\frac{8030}{181}a^{10}+\frac{9395}{181}a^{8}-\frac{7641}{181}a^{6}+\frac{6094}{181}a^{4}-\frac{3364}{181}a^{2}+\frac{744}{181}$, $a$, $\frac{269}{181}a^{20}-\frac{464}{181}a^{18}+\frac{1520}{181}a^{16}-\frac{3327}{181}a^{14}+\frac{4992}{181}a^{12}-\frac{6922}{181}a^{10}+\frac{7832}{181}a^{8}-\frac{6180}{181}a^{6}+\frac{4666}{181}a^{4}-\frac{2122}{181}a^{2}+\frac{315}{181}$, $\frac{421}{181}a^{21}-\frac{706}{181}a^{19}+\frac{2319}{181}a^{17}-\frac{5141}{181}a^{15}+\frac{7658}{181}a^{13}-\frac{10618}{181}a^{11}+\frac{12491}{181}a^{9}-\frac{10174}{181}a^{7}+\frac{7822}{181}a^{5}-\frac{4257}{181}a^{3}+\frac{1073}{181}a$, $\frac{31}{181}a^{21}-\frac{147}{181}a^{19}+\frac{338}{181}a^{17}-\frac{882}{181}a^{15}+\frac{1775}{181}a^{13}-\frac{2459}{181}a^{11}+\frac{3302}{181}a^{9}-\frac{3551}{181}a^{7}+\frac{2668}{181}a^{5}-\frac{1887}{181}a^{3}+\frac{800}{181}a+1$, $\frac{257}{181}a^{21}+\frac{18}{181}a^{20}-\frac{302}{181}a^{19}-\frac{62}{181}a^{18}+\frac{1214}{181}a^{17}+\frac{97}{181}a^{16}-\frac{2536}{181}a^{15}-\frac{372}{181}a^{14}+\frac{3324}{181}a^{13}+\frac{511}{181}a^{12}-\frac{4744}{181}a^{11}-\frac{733}{181}a^{10}+\frac{5497}{181}a^{9}+\frac{1059}{181}a^{8}-\frac{3988}{181}a^{7}-\frac{1116}{181}a^{6}+\frac{3388}{181}a^{5}+\frac{831}{181}a^{4}-\frac{1701}{181}a^{3}-\frac{903}{181}a^{2}+\frac{198}{181}a+\frac{266}{181}$, $\frac{421}{181}a^{21}-\frac{184}{181}a^{20}-\frac{706}{181}a^{19}+\frac{312}{181}a^{18}+\frac{2319}{181}a^{17}-\frac{1072}{181}a^{16}-\frac{5141}{181}a^{15}+\frac{2234}{181}a^{14}+\frac{7658}{181}a^{13}-\frac{3494}{181}a^{12}-\frac{10618}{181}a^{11}+\frac{4798}{181}a^{10}+\frac{12491}{181}a^{9}-\frac{5335}{181}a^{8}-\frac{10174}{181}a^{7}+\frac{4349}{181}a^{6}+\frac{7822}{181}a^{5}-\frac{3306}{181}a^{4}-\frac{4257}{181}a^{3}+\frac{1508}{181}a^{2}+\frac{892}{181}a-\frac{346}{181}$, $\frac{42}{181}a^{21}-\frac{62}{181}a^{20}-\frac{205}{181}a^{19}+\frac{113}{181}a^{18}+\frac{347}{181}a^{17}-\frac{314}{181}a^{16}-\frac{1049}{181}a^{15}+\frac{678}{181}a^{14}+\frac{1856}{181}a^{13}-\frac{1016}{181}a^{12}-\frac{2193}{181}a^{11}+\frac{1117}{181}a^{10}+\frac{2833}{181}a^{9}-\frac{1174}{181}a^{8}-\frac{2785}{181}a^{7}+\frac{948}{181}a^{6}+\frac{1577}{181}a^{5}-\frac{630}{181}a^{4}-\frac{1202}{181}a^{3}+\frac{154}{181}a^{2}+\frac{500}{181}a-\frac{152}{181}$, $\frac{305}{181}a^{21}-\frac{112}{181}a^{20}-\frac{407}{181}a^{19}+\frac{245}{181}a^{18}+\frac{1533}{181}a^{17}-\frac{684}{181}a^{16}-\frac{3166}{181}a^{15}+\frac{1651}{181}a^{14}+\frac{4385}{181}a^{13}-\frac{2536}{181}a^{12}-\frac{6035}{181}a^{11}+\frac{3495}{181}a^{10}+\frac{6692}{181}a^{9}-\frac{4176}{181}a^{8}-\frac{4792}{181}a^{7}+\frac{3324}{181}a^{6}+\frac{3794}{181}a^{5}-\frac{2335}{181}a^{4}-\frac{1756}{181}a^{3}+\frac{1335}{181}a^{2}+\frac{304}{181}a-\frac{187}{181}$, $\frac{136}{181}a^{21}-\frac{44}{181}a^{20}-\frac{207}{181}a^{19}+\frac{51}{181}a^{18}+\frac{753}{181}a^{17}-\frac{217}{181}a^{16}-\frac{1604}{181}a^{15}+\frac{487}{181}a^{14}+\frac{2433}{181}a^{13}-\frac{686}{181}a^{12}-\frac{3507}{181}a^{11}+\frac{1108}{181}a^{10}+\frac{4140}{181}a^{9}-\frac{1563}{181}a^{8}-\frac{3545}{181}a^{7}+\frac{1461}{181}a^{6}+\frac{2900}{181}a^{5}-\frac{1428}{181}a^{4}-\frac{1453}{181}a^{3}+\frac{1061}{181}a^{2}+\frac{421}{181}a-\frac{429}{181}$ (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: | \( 11319.5749854 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 11319.5749854 \cdot 1}{2\cdot\sqrt{147982356498821950109384704}}\cr\approx \mathstrut & 0.178465318173 \end{aligned}\] (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
11.1.5939843699.1 |
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: | 22.0.1362025042910688080257818810096245534886848009382349365076221097812844416934017717662457544046594347553335852138496.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.11.0.1}{11} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | $18{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | $18{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.10.14 | $x^{10} + 4 x^{9} - 6 x^{8} + 176 x^{7} + 848 x^{6} + 2256 x^{5} + 1216 x^{4} - 1088 x^{3} - 5392 x^{2} - 5120 x - 3616$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ |
2.12.12.22 | $x^{12} + 10 x^{11} - 16 x^{10} - 316 x^{9} - 364 x^{8} + 1040 x^{7} + 7648 x^{6} + 17632 x^{5} + 25008 x^{4} + 17056 x^{3} + 6016 x^{2} + 1344 x + 4544$ | $2$ | $6$ | $12$ | 12T134 | $[2, 2, 2, 2, 2, 2]^{6}$ | |
\(12917\) | $\Q_{12917}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{12917}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
\(459847\) | $\Q_{459847}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{459847}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |