Normalized defining polynomial
\( x^{22} - 11 x^{20} + 43 x^{18} - 62 x^{16} - 16 x^{14} + 116 x^{12} - 53 x^{10} - 56 x^{8} + 32 x^{6} + \cdots - 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[14, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(13844017509319141075535476685799424\) \(\medspace = 2^{22}\cdot 233^{2}\cdot 246572816873^{2}\) | 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: | \(35.63\) | 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: | \(2\), \(233\), \(246572816873\) | 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)
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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: | $17$ | 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: | $a^{21}-11a^{19}+43a^{17}-62a^{15}-16a^{13}+116a^{11}-53a^{9}-56a^{7}+32a^{5}+10a^{3}-4a$, $2a^{20}-23a^{18}+97a^{16}-167a^{14}+30a^{12}+248a^{10}-222a^{8}-58a^{6}+115a^{4}-8a^{2}-13$, $a^{21}-11a^{19}+43a^{17}-62a^{15}-16a^{13}+116a^{11}-53a^{9}-56a^{7}+32a^{5}+10a^{3}-5a$, $11a^{21}-122a^{19}+485a^{17}-737a^{15}-62a^{13}+1205a^{11}-702a^{9}-414a^{7}+330a^{5}+13a^{3}-23a$, $2a^{20}-23a^{18}+97a^{16}-167a^{14}+30a^{12}+248a^{10}-222a^{8}-58a^{6}+115a^{4}-9a^{2}-11$, $2a^{18}-21a^{16}+76a^{14}-91a^{12}-61a^{10}+187a^{8}-35a^{6}-93a^{4}+23a^{2}+12$, $7a^{20}-79a^{18}+323a^{16}-521a^{14}+22a^{12}+811a^{10}-573a^{8}-247a^{6}+272a^{4}-2a^{2}-22$, $9a^{20}-100a^{18}+399a^{16}-612a^{14}-39a^{12}+998a^{10}-608a^{8}-341a^{6}+298a^{4}+11a^{2}-25$, $13a^{20}-147a^{18}+604a^{16}-989a^{14}+84a^{12}+1501a^{10}-1144a^{8}-413a^{6}+552a^{4}-21a^{2}-50$, $4a^{20}-41a^{18}+140a^{16}-130a^{14}-208a^{12}+363a^{10}+96a^{8}-261a^{6}-47a^{4}+58a^{2}+14$, $a-1$, $6a^{20}-65a^{18}+249a^{16}-349a^{14}-91a^{12}+614a^{10}-275a^{8}-242a^{6}+137a^{4}+17a^{2}+a-8$, $5a^{21}+4a^{20}-54a^{19}-42a^{18}+206a^{17}+152a^{16}-287a^{15}-182a^{14}-75a^{13}-121a^{12}+498a^{11}+366a^{10}-222a^{9}-53a^{8}-186a^{7}-184a^{6}+105a^{5}+22a^{4}+8a^{3}+26a^{2}-6a+3$, $2a^{21}+4a^{20}-24a^{19}-42a^{18}+108a^{17}+152a^{16}-210a^{15}-182a^{14}+93a^{13}-121a^{12}+255a^{11}+366a^{10}-313a^{9}-53a^{8}-19a^{7}-184a^{6}+142a^{5}+22a^{4}-16a^{3}+26a^{2}-15a+3$, $10a^{21}+4a^{20}-109a^{19}-44a^{18}+421a^{17}+173a^{16}-598a^{15}-258a^{14}-146a^{13}-30a^{12}+1053a^{11}+427a^{10}-475a^{9}-239a^{8}-427a^{7}-153a^{6}+234a^{5}+115a^{4}+36a^{3}+9a^{2}-14a-8$, $6a^{21}-10a^{20}-61a^{19}+106a^{18}+205a^{17}-389a^{16}-178a^{15}+479a^{14}-331a^{13}+299a^{12}+534a^{11}-977a^{10}+176a^{9}+179a^{8}-403a^{7}+503a^{6}-79a^{5}-90a^{4}+92a^{3}-75a^{2}+22a-6$, $11a^{21}-a^{20}-119a^{19}+10a^{18}+454a^{17}-33a^{16}-627a^{15}+29a^{14}-191a^{13}+45a^{12}+1124a^{11}-71a^{10}-457a^{9}-18a^{8}-465a^{7}+38a^{6}+228a^{5}+6a^{4}+39a^{3}-3a^{2}-13a-1$ (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: | \( 1553582369.31 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{4}\cdot 1553582369.31 \cdot 1}{2\cdot\sqrt{13844017509319141075535476685799424}}\cr\approx \mathstrut & 0.168582554819 \end{aligned}\] (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
11.11.57451466331409.1 |
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: | 22.14.94400044432437843723463983111902775006545021062746810745120207719179974529050908880259938998245021182323947279057906110683377561476774610792433232547713908736.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.7.0.1}{7} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.9.0.1}{9} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | $16{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.11.0.1}{11} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 $22$ | $2$ | $11$ | $22$ | |||
\(233\) | Deg $4$ | $2$ | $2$ | $2$ | |||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
\(246572816873\) | Deg $4$ | $2$ | $2$ | $2$ | |||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |