Normalized defining polynomial
\( x^{22} - 4 x^{20} - 4 x^{18} + 31 x^{16} - 8 x^{14} - 81 x^{12} + 48 x^{10} + 88 x^{8} - 60 x^{6} - 35 x^{4} + 22 x^{2} + 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[12, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-12701184200335173359101913935642624\) \(\medspace = -\,2^{22}\cdot 55029067682009^{2}\) | 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: | \(35.50\) | 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: | \(2\), \(55029067682009\) | 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: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $16$ | 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: | $a$, $a^{16}-2a^{14}-7a^{12}+13a^{10}+15a^{8}-24a^{6}-11a^{4}+13a^{2}+1$, $a^{17}-3a^{15}-5a^{13}+20a^{11}+2a^{9}-39a^{7}+13a^{5}+23a^{3}-11a$, $a^{18}-3a^{16}-5a^{14}+19a^{12}+4a^{10}-34a^{8}+4a^{6}+18a^{4}-4a^{2}$, $a^{21}-3a^{19}-6a^{17}+22a^{15}+9a^{13}-53a^{11}-a^{9}+53a^{7}-4a^{5}-20a^{3}+a$, $a^{2}-1$, $a^{4}-2$, $a^{21}-2a^{19}-9a^{17}+16a^{15}+30a^{13}-42a^{11}-47a^{9}+41a^{7}+33a^{5}-11a^{3}-6a$, $a^{21}-3a^{19}-6a^{17}+21a^{15}+11a^{13}-46a^{11}-13a^{9}+38a^{7}+14a^{5}-10a^{3}-4a$, $2a^{20}-5a^{18}-16a^{16}+40a^{14}+45a^{12}-107a^{10}-56a^{8}+113a^{6}+27a^{4}-39a^{2}-1$, $a+1$, $2a^{21}-a^{20}-5a^{19}+3a^{18}-16a^{17}+6a^{16}+41a^{15}-22a^{14}+44a^{13}-9a^{12}-114a^{11}+52a^{10}-51a^{9}+2a^{8}+127a^{7}-47a^{6}+21a^{5}-46a^{3}+12a^{2}+a+1$, $a^{20}+a^{19}-3a^{18}-3a^{17}-7a^{16}-6a^{15}+24a^{14}+21a^{13}+16a^{12}+10a^{11}-65a^{10}-45a^{9}-17a^{8}-7a^{7}+71a^{6}+33a^{5}+11a^{4}+5a^{3}-25a^{2}-4a-3$, $a^{21}+4a^{20}-2a^{19}-9a^{18}-8a^{17}-33a^{16}+14a^{15}+71a^{14}+23a^{13}+96a^{12}-30a^{11}-186a^{10}-31a^{9}-122a^{8}+22a^{7}+192a^{6}+18a^{5}+60a^{4}-2a^{3}-65a^{2}-a-4$, $3a^{20}-8a^{18}+a^{17}-21a^{16}-2a^{15}+60a^{14}-8a^{13}+48a^{12}+13a^{11}-147a^{10}+22a^{9}-49a^{8}-23a^{7}+141a^{6}-24a^{5}+25a^{4}+10a^{3}-42a^{2}+7a-3$, $2a^{21}-a^{20}-5a^{19}+2a^{18}-15a^{17}+9a^{16}+38a^{15}-17a^{14}+38a^{13}-29a^{12}-94a^{11}+49a^{10}-41a^{9}+41a^{8}+89a^{7}-55a^{6}+15a^{5}-23a^{4}-26a^{3}+19a^{2}+2a+2$ (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: | \( 879851069.403 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{5}\cdot 879851069.403 \cdot 1}{2\cdot\sqrt{12701184200335173359101913935642624}}\cr\approx \mathstrut & 0.156572812593 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.(C_2\times S_{11})$ (as 22T53):
A non-solvable group of order 81749606400 |
The 752 conjugacy class representatives for $C_2^{10}.(C_2\times S_{11})$ are not computed |
Character table for $C_2^{10}.(C_2\times S_{11})$ is not computed |
Intermediate fields
11.11.55029067682009.1 |
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 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | $22$ | $22$ | ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | $22$ | ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $22$ | $20{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 $22$ | $2$ | $11$ | $22$ | |||
\(55029067682009\) | $\Q_{55029067682009}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{55029067682009}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |