Normalized defining polynomial
\( x^{22} + 12 x^{20} + 64 x^{18} + 200 x^{16} + 406 x^{14} + 557 x^{12} + 507 x^{10} + 271 x^{8} + \cdots - 1 \)
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: | \(10994531599582075769676214829056\) \(\medspace = 2^{22}\cdot 1619043113033^{2}\) | 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: | \(25.76\) | 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: | \(2\), \(1619043113033\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{13}a^{20}+\frac{5}{13}a^{18}+\frac{3}{13}a^{16}-\frac{3}{13}a^{14}-\frac{2}{13}a^{12}-\frac{1}{13}a^{10}-\frac{6}{13}a^{8}+\frac{1}{13}a^{6}+\frac{5}{13}a^{4}-\frac{1}{13}a^{2}-\frac{2}{13}$, $\frac{1}{13}a^{21}+\frac{5}{13}a^{19}+\frac{3}{13}a^{17}-\frac{3}{13}a^{15}-\frac{2}{13}a^{13}-\frac{1}{13}a^{11}-\frac{6}{13}a^{9}+\frac{1}{13}a^{7}+\frac{5}{13}a^{5}-\frac{1}{13}a^{3}-\frac{2}{13}a$
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: | $a^{20}+11a^{18}+53a^{16}+147a^{14}+259a^{12}+298a^{10}+209a^{8}+62a^{6}-11a^{4}-7a^{2}-2$, $a$, $\frac{56}{13}a^{21}+\frac{670}{13}a^{19}+\frac{3548}{13}a^{17}+\frac{10947}{13}a^{15}+\frac{21767}{13}a^{13}+\frac{28895}{13}a^{11}+\frac{24832}{13}a^{9}+\frac{11574}{13}a^{7}+\frac{748}{13}a^{5}-\frac{1369}{13}a^{3}-\frac{294}{13}a$, $\frac{20}{13}a^{20}+\frac{256}{13}a^{18}+\frac{1438}{13}a^{16}+\frac{4672}{13}a^{14}+\frac{9723}{13}a^{12}+\frac{13461}{13}a^{10}+\frac{12113}{13}a^{8}+\frac{6026}{13}a^{6}+\frac{542}{13}a^{4}-\frac{696}{13}a^{2}-\frac{157}{13}$, $\frac{124}{13}a^{21}+\frac{1452}{13}a^{19}+\frac{7522}{13}a^{17}+\frac{22690}{13}a^{15}+\frac{44069}{13}a^{13}+\frac{57024}{13}a^{11}+\frac{47434}{13}a^{9}+\frac{20885}{13}a^{7}+\frac{776}{13}a^{5}-\frac{2438}{13}a^{3}-\frac{443}{13}a$, $\frac{47}{13}a^{20}+\frac{547}{13}a^{18}+\frac{2806}{13}a^{16}+\frac{8348}{13}a^{14}+\frac{15922}{13}a^{12}+\frac{20129}{13}a^{10}+\frac{16202}{13}a^{8}+\frac{6677}{13}a^{6}+\frac{27}{13}a^{4}-\frac{736}{13}a^{2}-\frac{146}{13}$, $\frac{69}{13}a^{21}+\frac{826}{13}a^{19}+\frac{4380}{13}a^{17}+\frac{13547}{13}a^{15}+\frac{27045}{13}a^{13}+\frac{36136}{13}a^{11}+\frac{31423}{13}a^{9}+\frac{15097}{13}a^{7}+\frac{1411}{13}a^{5}-\frac{1603}{13}a^{3}-\frac{398}{13}a$, $\frac{47}{13}a^{21}+\frac{547}{13}a^{19}+\frac{2806}{13}a^{17}+\frac{8348}{13}a^{15}+\frac{15922}{13}a^{13}+\frac{20129}{13}a^{11}+\frac{16202}{13}a^{9}+\frac{6677}{13}a^{7}+\frac{40}{13}a^{5}-\frac{697}{13}a^{3}-\frac{107}{13}a$, $\frac{34}{13}a^{21}+\frac{70}{13}a^{20}+\frac{404}{13}a^{19}+\frac{818}{13}a^{18}+\frac{2117}{13}a^{17}+\frac{4227}{13}a^{16}+\frac{6437}{13}a^{15}+\frac{12712}{13}a^{14}+\frac{12555}{13}a^{13}+\frac{24599}{13}a^{12}+\frac{16255}{13}a^{11}+\frac{31689}{13}a^{10}+\frac{13485}{13}a^{9}+\frac{26204}{13}a^{8}+\frac{5871}{13}a^{7}+\frac{11419}{13}a^{6}+\frac{183}{13}a^{5}+\frac{389}{13}a^{4}-\frac{606}{13}a^{3}-\frac{1292}{13}a^{2}-\frac{81}{13}a-\frac{231}{13}$, $\frac{40}{13}a^{21}+\frac{15}{13}a^{20}+\frac{473}{13}a^{19}+\frac{166}{13}a^{18}+\frac{2473}{13}a^{17}+\frac{799}{13}a^{16}+\frac{7524}{13}a^{15}+\frac{2191}{13}a^{14}+\frac{14727}{13}a^{13}+\frac{3753}{13}a^{12}+\frac{19187}{13}a^{11}+\frac{4067}{13}a^{10}+\frac{16062}{13}a^{9}+\frac{2458}{13}a^{8}+\frac{7099}{13}a^{7}+\frac{249}{13}a^{6}+\frac{213}{13}a^{5}-\frac{510}{13}a^{4}-\frac{885}{13}a^{3}-\frac{80}{13}a^{2}-\frac{171}{13}a+\frac{22}{13}$, $\frac{266}{13}a^{21}+\frac{51}{13}a^{20}+\frac{3241}{13}a^{19}+\frac{723}{13}a^{18}+\frac{17724}{13}a^{17}+\frac{4625}{13}a^{16}+\frac{57598}{13}a^{15}+\frac{17527}{13}a^{14}+\frac{123930}{13}a^{13}+\frac{43344}{13}a^{12}+\frac{184789}{13}a^{11}+\frac{72476}{13}a^{10}+\frac{190245}{13}a^{9}+\frac{81828}{13}a^{8}+\frac{127419}{13}a^{7}+\frac{60202}{13}a^{6}+\frac{49300}{13}a^{5}+\frac{26723}{13}a^{4}+\frac{9302}{13}a^{3}+\frac{6228}{13}a^{2}+\frac{625}{13}a+\frac{561}{13}$ (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: | \( 3341772.01853 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 3341772.01853 \cdot 1}{2\cdot\sqrt{10994531599582075769676214829056}}\cr\approx \mathstrut & 0.193293447300 \end{aligned}\] (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
11.7.1619043113033.1 |
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: | 22.2.840431756599608455702367012311719571812153210508326264722041853019088432659346485389647494483321252259172032729969587402093091559398721978368.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.7.0.1}{7} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.11.0.1}{11} }^{2}$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 $22$ | $2$ | $11$ | $22$ | |||
\(1619043113033\) | $\Q_{1619043113033}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1619043113033}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |