Normalized defining polynomial
\( x^{22} + 12x^{20} + 58x^{18} + 144x^{16} + 193x^{14} + 130x^{12} + 21x^{10} - 40x^{8} - 45x^{6} - 18x^{4} + 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-315595142827230969392399757869056\) \(\medspace = -\,2^{22}\cdot 8674315276967^{2}\) | 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: | \(30.01\) | 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: | \(2\), \(8674315276967\) | 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: | $12$ | 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: | $a^{21}+12a^{19}+58a^{17}+144a^{15}+193a^{13}+130a^{11}+21a^{9}-40a^{7}-45a^{5}-18a^{3}$, $a^{2}+1$, $a^{20}+11a^{18}+47a^{16}+97a^{14}+96a^{12}+34a^{10}-13a^{8}-27a^{6}-18a^{4}+a^{2}+1$, $a^{20}+12a^{18}+57a^{16}+134a^{14}+156a^{12}+69a^{10}-20a^{8}-44a^{6}-34a^{4}-2a^{2}+4$, $a^{21}+12a^{19}+57a^{17}+134a^{15}+156a^{13}+69a^{11}-20a^{9}-44a^{7}-34a^{5}-2a^{3}+5a$, $2a^{20}+23a^{18}+105a^{16}+240a^{14}+281a^{12}+142a^{10}-15a^{8}-72a^{6}-58a^{4}-10a^{2}+4$, $a^{20}+11a^{18}+47a^{16}+97a^{14}+96a^{12}+34a^{10}-13a^{8}-27a^{6}-18a^{4}+a^{2}+2$, $4a^{21}+46a^{19}+209a^{17}+472a^{15}+540a^{13}+261a^{11}-35a^{9}-140a^{7}-113a^{5}-21a^{3}+7a$, $5a^{21}+57a^{19}+256a^{17}+568a^{15}+628a^{13}+273a^{11}-71a^{9}-173a^{7}-129a^{5}-14a^{3}+11a$, $2a^{21}+22a^{19}+93a^{17}+183a^{15}+145a^{13}-a^{12}-29a^{11}-8a^{10}-123a^{9}-23a^{8}-95a^{7}-29a^{6}-40a^{5}-17a^{4}+9a^{3}-6a^{2}+8a-1$, $2a^{21}-3a^{20}+27a^{19}-36a^{18}+150a^{17}-174a^{16}+441a^{15}-433a^{14}+733a^{13}-587a^{12}+674a^{11}-413a^{10}+279a^{9}-91a^{8}-67a^{7}+109a^{6}-187a^{5}+141a^{4}-131a^{3}+66a^{2}-33a+10$, $3a^{21}-2a^{20}+37a^{19}-25a^{18}+186a^{17}-126a^{16}+488a^{15}-323a^{14}+702a^{13}-425a^{12}+498a^{11}-215a^{10}+32a^{9}+91a^{8}-222a^{7}+143a^{6}-151a^{5}+41a^{4}-19a^{3}-4a^{2}+6a-3$ (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: | \( 22686333.3679 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{9}\cdot 22686333.3679 \cdot 1}{2\cdot\sqrt{315595142827230969392399757869056}}\cr\approx \mathstrut & 0.155922200954 \end{aligned}\] (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 $C_2^{10}.(C_2\times S_{11})$ |
Character table for $C_2^{10}.(C_2\times S_{11})$ |
Intermediate fields
11.9.8674315276967.1 |
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 | $22$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.9.0.1}{9} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }$ | $22$ | $16{,}\,{\href{/padicField/13.6.0.1}{6} }$ | $18{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | $22$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 $22$ | $2$ | $11$ | $22$ | |||
\(8674315276967\) | Deg $4$ | $2$ | $2$ | $2$ | |||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ |