Normalized defining polynomial
\( x^{22} - 11 x^{21} + 43 x^{20} - 45 x^{19} - 152 x^{18} + 456 x^{17} - 114 x^{16} - 1026 x^{15} + 1114 x^{14} + 790 x^{13} - 1814 x^{12} + 120 x^{11} + 1304 x^{10} - 464 x^{9} - 493 x^{8} + \cdots - 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[16, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-289690752824370116310308034941759\) \(\medspace = -\,809\cdot 4759\cdot 8674315276967^{2}\) | 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: | \(29.89\) | 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: | $809^{1/2}4759^{1/2}8674315276967^{1/2}\approx 5778960349.413772$ | ||
Ramified primes: | \(809\), \(4759\), \(8674315276967\) | 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: | $\Q(\sqrt{-3850031}$) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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: | $18$ | 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: | $a^{20}-10a^{19}+33a^{18}-12a^{17}-165a^{16}+300a^{15}+160a^{14}-862a^{13}+369a^{12}+984a^{11}-930a^{10}-405a^{9}+749a^{8}-32a^{7}-265a^{6}+17a^{5}+77a^{4}+26a^{3}-30a^{2}-5a+3$, $a^{2}-a-1$, $a$, $a^{21}-10a^{20}+33a^{19}-12a^{18}-164a^{17}+292a^{16}+178a^{15}-848a^{14}+266a^{13}+1056a^{12}-758a^{11}-638a^{10}+666a^{9}+202a^{8}-291a^{7}-74a^{6}+96a^{5}+50a^{4}-27a^{3}-18a^{2}$, $a^{20}-10a^{19}+34a^{18}-21a^{17}-139a^{16}+296a^{15}+43a^{14}-687a^{13}+470a^{12}+573a^{11}-771a^{10}-78a^{9}+455a^{8}-87a^{7}-125a^{6}+5a^{5}+50a^{4}+9a^{3}-19a^{2}+a+1$, $2a^{20}-20a^{19}+67a^{18}-33a^{17}-303a^{16}+588a^{15}+222a^{14}-1542a^{13}+749a^{12}+1642a^{11}-1592a^{10}-687a^{9}+1205a^{8}+34a^{7}-446a^{6}-17a^{5}+142a^{4}+51a^{3}-48a^{2}-14a+3$, $2a^{20}-20a^{19}+68a^{18}-42a^{17}-277a^{16}+584a^{15}+105a^{14}-1367a^{13}+850a^{12}+1231a^{11}-1433a^{10}-360a^{9}+912a^{8}-25a^{7}-304a^{6}-21a^{5}+107a^{4}+32a^{3}-33a^{2}-9a+1$, $4a^{20}-40a^{19}+135a^{18}-75a^{17}-581a^{16}+1180a^{15}+308a^{14}-2916a^{13}+1688a^{12}+2794a^{11}-3143a^{10}-853a^{9}+2150a^{8}-154a^{7}-725a^{6}+21a^{5}+241a^{4}+61a^{3}-84a^{2}-11a+8$, $a^{20}-10a^{19}+33a^{18}-12a^{17}-165a^{16}+300a^{15}+160a^{14}-862a^{13}+369a^{12}+984a^{11}-930a^{10}-405a^{9}+749a^{8}-32a^{7}-265a^{6}+17a^{5}+78a^{4}+24a^{3}-32a^{2}-2a+5$, $3a^{20}-30a^{19}+101a^{18}-54a^{17}-442a^{16}+884a^{15}+265a^{14}-2229a^{13}+1219a^{12}+2215a^{11}-2363a^{10}-765a^{9}+1661a^{8}-57a^{7}-569a^{6}-4a^{5}+185a^{4}+56a^{3}-65a^{2}-11a+5$, $4a^{21}-45a^{20}+183a^{19}-224a^{18}-554a^{17}+1951a^{16}-917a^{15}-3848a^{14}+5310a^{13}+1815a^{12}-7473a^{11}+2298a^{10}+4335a^{9}-2800a^{8}-1055a^{7}+991a^{6}+358a^{5}-213a^{4}-231a^{3}+82a^{2}+35a-7$, $a^{21}-11a^{20}+43a^{19}-45a^{18}-152a^{17}+456a^{16}-114a^{15}-1026a^{14}+1114a^{13}+790a^{12}-1814a^{11}+120a^{10}+1304a^{9}-464a^{8}-493a^{7}+217a^{6}+170a^{5}-46a^{4}-77a^{3}+9a^{2}+17a$, $2a^{21}-21a^{20}+77a^{19}-67a^{18}-283a^{17}+735a^{16}-93a^{15}-1592a^{14}+1525a^{13}+1094a^{12}-2289a^{11}+287a^{10}+1326a^{9}-618a^{8}-324a^{7}+198a^{6}+110a^{5}-30a^{4}-55a^{3}+20a^{2}+5a-3$, $a^{21}-11a^{20}+43a^{19}-45a^{18}-153a^{17}+465a^{16}-140a^{15}-1022a^{14}+1231a^{13}+615a^{12}-1914a^{11}+525a^{10}+1154a^{9}-781a^{8}-233a^{7}+282a^{6}+61a^{5}-54a^{4}-56a^{3}+30a^{2}+6a-3$, $a^{21}-7a^{20}+3a^{19}+89a^{18}-218a^{17}-151a^{16}+1070a^{15}-602a^{14}-1970a^{13}+2364a^{12}+1378a^{11}-3119a^{10}+95a^{9}+1896a^{8}-511a^{7}-618a^{6}+151a^{5}+226a^{4}+9a^{3}-83a^{2}-3a+7$, $4a^{20}-40a^{19}+135a^{18}-75a^{17}-581a^{16}+1180a^{15}+308a^{14}-2916a^{13}+1689a^{12}+2788a^{11}-3134a^{10}-843a^{9}+2116a^{8}-144a^{7}-694a^{6}+a^{5}+234a^{4}+67a^{3}-82a^{2}-12a+5$, $3a^{21}-34a^{20}+140a^{19}-179a^{18}-401a^{17}+1487a^{16}-785a^{15}-2807a^{14}+4086a^{13}+1111a^{12}-5480a^{11}+1891a^{10}+2987a^{9}-2052a^{8}-659a^{7}+684a^{6}+238a^{5}-152a^{4}-151a^{3}+55a^{2}+16a-5$, $2a^{21}-23a^{20}+97a^{19}-133a^{18}-259a^{17}+1064a^{16}-685a^{15}-1930a^{14}+3235a^{13}+459a^{12}-4329a^{11}+1975a^{10}+2370a^{9}-2038a^{8}-488a^{7}+760a^{6}+150a^{5}-196a^{4}-122a^{3}+73a^{2}+26a-8$ (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: | \( 334750790.993 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{3}\cdot 334750790.993 \cdot 1}{2\cdot\sqrt{289690752824370116310308034941759}}\cr\approx \mathstrut & 0.159861482349 \end{aligned}\] (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
11.9.8674315276967.1 |
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 | ${\href{/padicField/2.11.0.1}{11} }^{2}$ | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.9.0.1}{9} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | $22$ | $16{,}\,{\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.9.0.1}{9} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
---|---|---|---|---|---|---|---|
\(809\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(4759\) | $\Q_{4759}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{4759}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{4759}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{4759}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
\(8674315276967\) | Deg $4$ | $2$ | $2$ | $2$ | |||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |