Normalized defining polynomial
\( x^{22} - 3x^{20} + x^{18} + x^{16} + 6x^{14} - 8x^{12} + 5x^{6} - x^{2} - 1 \)
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: | \(70765994783241803539463274496\) \(\medspace = 2^{22}\cdot 167^{10}\) | 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: | \(20.48\) | 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: | \(2\), \(167\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{13}a^{20}-\frac{6}{13}a^{18}+\frac{6}{13}a^{16}-\frac{4}{13}a^{14}+\frac{5}{13}a^{12}+\frac{3}{13}a^{10}+\frac{4}{13}a^{8}+\frac{1}{13}a^{6}+\frac{2}{13}a^{4}-\frac{6}{13}a^{2}+\frac{4}{13}$, $\frac{1}{13}a^{21}-\frac{6}{13}a^{19}+\frac{6}{13}a^{17}-\frac{4}{13}a^{15}+\frac{5}{13}a^{13}+\frac{3}{13}a^{11}+\frac{4}{13}a^{9}+\frac{1}{13}a^{7}+\frac{2}{13}a^{5}-\frac{6}{13}a^{3}+\frac{4}{13}a$
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: | $a$, $\frac{8}{13}a^{20}-\frac{22}{13}a^{18}+\frac{9}{13}a^{16}-\frac{6}{13}a^{14}+\frac{53}{13}a^{12}-\frac{54}{13}a^{10}+\frac{19}{13}a^{8}-\frac{31}{13}a^{6}+\frac{42}{13}a^{4}-\frac{9}{13}a^{2}+\frac{6}{13}$, $\frac{8}{13}a^{20}-\frac{35}{13}a^{18}+\frac{35}{13}a^{16}+\frac{7}{13}a^{14}+\frac{40}{13}a^{12}-\frac{119}{13}a^{10}+\frac{58}{13}a^{8}+\frac{8}{13}a^{6}+\frac{29}{13}a^{4}-\frac{22}{13}a^{2}-\frac{20}{13}$, $\frac{6}{13}a^{21}-\frac{23}{13}a^{19}+\frac{23}{13}a^{17}+\frac{2}{13}a^{15}+\frac{17}{13}a^{13}-\frac{73}{13}a^{11}+\frac{50}{13}a^{9}+\frac{19}{13}a^{7}-\frac{1}{13}a^{5}-\frac{23}{13}a^{3}-\frac{15}{13}a$, $\frac{21}{13}a^{21}-\frac{61}{13}a^{19}+\frac{22}{13}a^{17}+\frac{7}{13}a^{15}+\frac{118}{13}a^{13}-\frac{145}{13}a^{11}+\frac{32}{13}a^{9}-\frac{18}{13}a^{7}+\frac{68}{13}a^{5}+\frac{17}{13}a^{3}+\frac{6}{13}a$, $\frac{8}{13}a^{21}-\frac{14}{13}a^{20}-\frac{35}{13}a^{19}+\frac{45}{13}a^{18}+\frac{35}{13}a^{17}-\frac{32}{13}a^{16}+\frac{7}{13}a^{15}+\frac{4}{13}a^{14}+\frac{40}{13}a^{13}-\frac{70}{13}a^{12}-\frac{119}{13}a^{11}+\frac{127}{13}a^{10}+\frac{58}{13}a^{9}-\frac{69}{13}a^{8}+\frac{8}{13}a^{7}+\frac{25}{13}a^{6}+\frac{29}{13}a^{5}-\frac{41}{13}a^{4}-\frac{22}{13}a^{3}+\frac{19}{13}a^{2}-\frac{20}{13}a-\frac{4}{13}$, $\frac{6}{13}a^{21}-\frac{5}{13}a^{20}-\frac{23}{13}a^{19}+\frac{17}{13}a^{18}+\frac{23}{13}a^{17}-\frac{4}{13}a^{16}+\frac{2}{13}a^{15}-\frac{19}{13}a^{14}+\frac{17}{13}a^{13}-\frac{25}{13}a^{12}-\frac{73}{13}a^{11}+\frac{50}{13}a^{10}+\frac{50}{13}a^{9}+\frac{19}{13}a^{8}+\frac{19}{13}a^{7}-\frac{31}{13}a^{6}-\frac{1}{13}a^{5}-\frac{23}{13}a^{4}-\frac{23}{13}a^{3}-\frac{9}{13}a^{2}-\frac{15}{13}a+\frac{6}{13}$, $\frac{8}{13}a^{21}+\frac{7}{13}a^{20}-\frac{35}{13}a^{19}-\frac{16}{13}a^{18}+\frac{35}{13}a^{17}-\frac{10}{13}a^{16}+\frac{7}{13}a^{15}+\frac{24}{13}a^{14}+\frac{40}{13}a^{13}+\frac{35}{13}a^{12}-\frac{119}{13}a^{11}-\frac{31}{13}a^{10}+\frac{58}{13}a^{9}-\frac{50}{13}a^{8}+\frac{8}{13}a^{7}+\frac{33}{13}a^{6}+\frac{29}{13}a^{5}+\frac{14}{13}a^{4}-\frac{35}{13}a^{3}+\frac{10}{13}a^{2}-\frac{7}{13}a-\frac{11}{13}$, $\frac{1}{13}a^{21}-\frac{6}{13}a^{19}+\frac{19}{13}a^{17}-\frac{30}{13}a^{15}+\frac{18}{13}a^{13}-\frac{23}{13}a^{11}+\frac{69}{13}a^{9}-\frac{77}{13}a^{7}+\frac{28}{13}a^{5}-\frac{6}{13}a^{3}+\frac{30}{13}a+1$, $\frac{2}{13}a^{21}+\frac{5}{13}a^{20}+\frac{1}{13}a^{19}-\frac{17}{13}a^{18}-\frac{27}{13}a^{17}+\frac{4}{13}a^{16}+\frac{31}{13}a^{15}+\frac{19}{13}a^{14}+\frac{10}{13}a^{13}+\frac{25}{13}a^{12}+\frac{19}{13}a^{11}-\frac{50}{13}a^{10}-\frac{83}{13}a^{9}-\frac{19}{13}a^{8}+\frac{54}{13}a^{7}+\frac{44}{13}a^{6}-\frac{9}{13}a^{5}+\frac{10}{13}a^{4}+\frac{27}{13}a^{3}-\frac{4}{13}a^{2}-\frac{5}{13}a-\frac{19}{13}$, $\frac{6}{13}a^{21}+\frac{6}{13}a^{20}-\frac{23}{13}a^{19}-\frac{23}{13}a^{18}+\frac{10}{13}a^{17}+\frac{23}{13}a^{16}+\frac{28}{13}a^{15}+\frac{2}{13}a^{14}+\frac{30}{13}a^{13}+\frac{17}{13}a^{12}-\frac{86}{13}a^{11}-\frac{73}{13}a^{10}-\frac{15}{13}a^{9}+\frac{50}{13}a^{8}+\frac{58}{13}a^{7}+\frac{19}{13}a^{6}+\frac{25}{13}a^{5}-\frac{1}{13}a^{4}-\frac{23}{13}a^{3}-\frac{23}{13}a^{2}-\frac{28}{13}a-\frac{2}{13}$ (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: | \( 232562.574566 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 232562.574566 \cdot 1}{2\cdot\sqrt{70765994783241803539463274496}}\cr\approx \mathstrut & 0.167670321786 \end{aligned}\] (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
11.1.129891985607.1 |
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: | 22.0.11817921128801381191090366840832.3 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.2.0.1}{2} }^{5}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{9}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{5}$ | ${\href{/padicField/19.11.0.1}{11} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{9}$ | ${\href{/padicField/29.11.0.1}{11} }^{2}$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.11.0.1}{11} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{5}$ | ${\href{/padicField/59.4.0.1}{4} }^{5}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 $22$ | $2$ | $11$ | $22$ | |||
\(167\) | $\Q_{167}$ | $x + 162$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{167}$ | $x + 162$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
167.2.1.2 | $x^{2} + 167$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
167.2.1.2 | $x^{2} + 167$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
167.2.1.2 | $x^{2} + 167$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
167.2.1.2 | $x^{2} + 167$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
167.2.1.2 | $x^{2} + 167$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
167.2.1.2 | $x^{2} + 167$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
167.4.2.1 | $x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
167.4.2.1 | $x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |