Normalized defining polynomial
\( x^{22} + 4x^{20} + x^{18} - 20x^{16} - 7x^{14} + 93x^{12} + 46x^{10} - 121x^{8} - 80x^{6} - 34x^{4} - 9x^{2} - 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(716463968651672990739595264\) \(\medspace = 2^{22}\cdot 11^{4}\cdot 19^{4}\cdot 547^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.62\) | 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: | \(2\), \(11\), \(19\), \(547\) | 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)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{19}-\frac{1}{4}a^{18}-\frac{1}{4}a^{17}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}+\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{431991212}a^{20}+\frac{19757011}{215995606}a^{18}-\frac{1}{4}a^{17}-\frac{12181834}{107997803}a^{16}-\frac{1}{4}a^{15}+\frac{19879178}{107997803}a^{14}+\frac{51748585}{215995606}a^{12}-\frac{1}{4}a^{11}-\frac{81565729}{215995606}a^{10}-\frac{1}{4}a^{9}-\frac{17188587}{215995606}a^{8}+\frac{559025}{107997803}a^{6}+\frac{1}{4}a^{5}-\frac{134894503}{431991212}a^{4}-\frac{1}{4}a^{3}-\frac{7104976}{107997803}a^{2}+\frac{1}{4}a-\frac{115094833}{431991212}$, $\frac{1}{431991212}a^{21}+\frac{19757011}{215995606}a^{19}-\frac{1}{4}a^{18}-\frac{12181834}{107997803}a^{17}-\frac{1}{4}a^{16}+\frac{19879178}{107997803}a^{15}+\frac{51748585}{215995606}a^{13}-\frac{1}{4}a^{12}-\frac{81565729}{215995606}a^{11}-\frac{1}{4}a^{10}-\frac{17188587}{215995606}a^{9}+\frac{559025}{107997803}a^{7}+\frac{1}{4}a^{6}-\frac{134894503}{431991212}a^{5}-\frac{1}{4}a^{4}-\frac{7104976}{107997803}a^{3}+\frac{1}{4}a^{2}-\frac{115094833}{431991212}a$
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)
| |
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)
| |
Fundamental units: | $\frac{59479491}{215995606}a^{20}+\frac{119777647}{107997803}a^{18}+\frac{32631081}{107997803}a^{16}-\frac{594831634}{107997803}a^{14}-\frac{224868541}{107997803}a^{12}+\frac{2762127436}{107997803}a^{10}+\frac{1436154523}{107997803}a^{8}-\frac{3570183638}{107997803}a^{6}-\frac{4974261389}{215995606}a^{4}-\frac{1157994389}{107997803}a^{2}-\frac{499348735}{215995606}$, $\frac{17813216}{107997803}a^{21}-\frac{1950579}{431991212}a^{20}+\frac{139564123}{215995606}a^{19}+\frac{634871}{107997803}a^{18}+\frac{48758455}{431991212}a^{17}+\frac{13929455}{215995606}a^{16}-\frac{1422835025}{431991212}a^{15}+\frac{4074737}{215995606}a^{14}-\frac{90677920}{107997803}a^{13}-\frac{48297890}{107997803}a^{12}+\frac{6650233363}{431991212}a^{11}-\frac{21312055}{215995606}a^{10}+\frac{2630593413}{431991212}a^{9}+\frac{426887341}{215995606}a^{8}-\frac{2209352132}{107997803}a^{7}-\frac{76606387}{107997803}a^{6}-\frac{4529755251}{431991212}a^{5}-\frac{1302505903}{431991212}a^{4}-\frac{1977143779}{431991212}a^{3}+\frac{106908932}{107997803}a^{2}-\frac{1172497097}{431991212}a+\frac{267289633}{431991212}$, $\frac{132360061}{215995606}a^{20}+\frac{250891423}{107997803}a^{18}+\frac{15685202}{107997803}a^{16}-\frac{1321141584}{107997803}a^{14}-\frac{190411176}{107997803}a^{12}+\frac{6161915138}{107997803}a^{10}+\frac{1778123662}{107997803}a^{8}-\frac{8219168561}{107997803}a^{6}-\frac{7222648443}{215995606}a^{4}-\frac{1611233373}{107997803}a^{2}-\frac{701178693}{215995606}$, $a$, $\frac{634026491}{431991212}a^{21}-\frac{36441761}{431991212}a^{20}+\frac{2401960403}{431991212}a^{19}-\frac{33749157}{107997803}a^{18}+\frac{55358735}{215995606}a^{17}+\frac{11957017}{431991212}a^{16}-\frac{12759575615}{431991212}a^{15}+\frac{753254899}{431991212}a^{14}-\frac{1733420379}{431991212}a^{13}+\frac{5152740}{107997803}a^{12}+\frac{14912243825}{107997803}a^{11}-\frac{3560854187}{431991212}a^{10}+\frac{16580984871}{431991212}a^{9}-\frac{576223331}{431991212}a^{8}-\frac{81676584421}{431991212}a^{7}+\frac{2672210017}{215995606}a^{6}-\frac{33837561839}{431991212}a^{5}+\frac{594264513}{215995606}a^{4}-\frac{6127745411}{215995606}a^{3}-\frac{258114897}{431991212}a^{2}-\frac{459228158}{107997803}a+\frac{25086092}{107997803}$, $\frac{801690673}{431991212}a^{21}-\frac{28503477}{215995606}a^{20}+\frac{747069777}{107997803}a^{19}-\frac{246632021}{431991212}a^{18}-\frac{910444}{107997803}a^{17}-\frac{125360657}{431991212}a^{16}-\frac{7997806645}{215995606}a^{15}+\frac{572900283}{215995606}a^{14}-\frac{619293787}{215995606}a^{13}+\frac{777737887}{431991212}a^{12}+\frac{18678227455}{107997803}a^{11}-\frac{5299350857}{431991212}a^{10}+\frac{4107340897}{107997803}a^{9}-\frac{1103020000}{107997803}a^{8}-\frac{50319976517}{215995606}a^{7}+\frac{6600837179}{431991212}a^{6}-\frac{36132604921}{431991212}a^{5}+\frac{7151416495}{431991212}a^{4}-\frac{4616616930}{107997803}a^{3}+\frac{2578318901}{431991212}a^{2}-\frac{3120969985}{431991212}a+\frac{318554237}{215995606}$, $\frac{351040197}{431991212}a^{21}-\frac{36441761}{431991212}a^{20}+\frac{1313162497}{431991212}a^{19}-\frac{33749157}{107997803}a^{18}-\frac{252398}{107997803}a^{17}+\frac{11957017}{431991212}a^{16}-\frac{7072807835}{431991212}a^{15}+\frac{753254899}{431991212}a^{14}-\frac{642910429}{431991212}a^{13}+\frac{5152740}{107997803}a^{12}+\frac{16536812159}{215995606}a^{11}-\frac{3560854187}{431991212}a^{10}+\frac{7711858587}{431991212}a^{9}-\frac{576223331}{431991212}a^{8}-\frac{45778090687}{431991212}a^{7}+\frac{2672210017}{215995606}a^{6}-\frac{16985367579}{431991212}a^{5}+\frac{594264513}{215995606}a^{4}-\frac{1320840250}{107997803}a^{3}-\frac{258114897}{431991212}a^{2}-\frac{553347583}{215995606}a+\frac{25086092}{107997803}$, $\frac{93195361}{107997803}a^{21}-\frac{152291021}{215995606}a^{20}+\frac{1349780799}{431991212}a^{19}-\frac{1137771935}{431991212}a^{18}-\frac{155874005}{431991212}a^{17}-\frac{937443}{107997803}a^{16}-\frac{1865935893}{107997803}a^{15}+\frac{6091143021}{431991212}a^{14}+\frac{204812537}{431991212}a^{13}+\frac{519859115}{431991212}a^{12}+\frac{34917577773}{431991212}a^{11}-\frac{7112540512}{107997803}a^{10}+\frac{996105478}{107997803}a^{9}-\frac{6471854929}{431991212}a^{8}-\frac{48194637315}{431991212}a^{7}+\frac{38483969429}{431991212}a^{6}-\frac{12103586863}{431991212}a^{5}+\frac{3523951453}{107997803}a^{4}-\frac{6033547311}{431991212}a^{3}+\frac{1729650170}{107997803}a^{2}-\frac{275896905}{215995606}a+\frac{995525047}{431991212}$, $\frac{18202483}{107997803}a^{21}+\frac{234854093}{431991212}a^{20}+\frac{190978927}{215995606}a^{19}+\frac{211320155}{107997803}a^{18}+\frac{401680409}{431991212}a^{17}-\frac{59085827}{215995606}a^{16}-\frac{1496385929}{431991212}a^{15}-\frac{2351355807}{215995606}a^{14}-\frac{1168427557}{215995606}a^{13}+\frac{118228813}{215995606}a^{12}+\frac{6821923311}{431991212}a^{11}+\frac{11004044703}{215995606}a^{10}+\frac{11919926347}{431991212}a^{9}+\frac{989291329}{215995606}a^{8}-\frac{3923422209}{215995606}a^{7}-\frac{7615686486}{107997803}a^{6}-\frac{17876324265}{431991212}a^{5}-\frac{6749760449}{431991212}a^{4}-\frac{5488366909}{431991212}a^{3}-\frac{1842507285}{215995606}a^{2}-\frac{1827593389}{431991212}a-\frac{486676391}{431991212}$, $\frac{93195361}{107997803}a^{21}+\frac{152291021}{215995606}a^{20}+\frac{1349780799}{431991212}a^{19}+\frac{1137771935}{431991212}a^{18}-\frac{155874005}{431991212}a^{17}+\frac{937443}{107997803}a^{16}-\frac{1865935893}{107997803}a^{15}-\frac{6091143021}{431991212}a^{14}+\frac{204812537}{431991212}a^{13}-\frac{519859115}{431991212}a^{12}+\frac{34917577773}{431991212}a^{11}+\frac{7112540512}{107997803}a^{10}+\frac{996105478}{107997803}a^{9}+\frac{6471854929}{431991212}a^{8}-\frac{48194637315}{431991212}a^{7}-\frac{38483969429}{431991212}a^{6}-\frac{12103586863}{431991212}a^{5}-\frac{3523951453}{107997803}a^{4}-\frac{6033547311}{431991212}a^{3}-\frac{1729650170}{107997803}a^{2}-\frac{275896905}{215995606}a-\frac{995525047}{431991212}$, $\frac{354180669}{215995606}a^{21}+\frac{312134259}{215995606}a^{20}+\frac{692638797}{107997803}a^{19}+\frac{1157187139}{215995606}a^{18}+\frac{109808727}{107997803}a^{17}-\frac{29767927}{215995606}a^{16}-\frac{7145182581}{215995606}a^{15}-\frac{6242455379}{215995606}a^{14}-\frac{919167608}{107997803}a^{13}-\frac{172184946}{107997803}a^{12}+\frac{16669764833}{107997803}a^{11}+\frac{29197135717}{215995606}a^{10}+\frac{6659558090}{107997803}a^{9}+\frac{2874222336}{107997803}a^{8}-\frac{45169806941}{215995606}a^{7}-\frac{19887983336}{107997803}a^{6}-\frac{12224398998}{107997803}a^{5}-\frac{6559539339}{107997803}a^{4}-\frac{4121853850}{107997803}a^{3}-\frac{3115514036}{107997803}a^{2}-\frac{2164077231}{215995606}a-\frac{1076489687}{215995606}$ (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: | \( 38833.928533 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 38833.928533 \cdot 1}{2\cdot\sqrt{716463968651672990739595264}}\cr\approx \mathstrut & 0.27825490525 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.A_{11}$ (as 22T49):
A non-solvable group of order 20437401600 |
The 200 conjugacy class representatives for $C_2^{10}.A_{11}$ are not computed |
Character table for $C_2^{10}.A_{11}$ is not computed |
Intermediate fields
11.3.836463893056.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | R | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.11.0.1}{11} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
2.18.18.124 | $x^{18} + 6 x^{14} + 6 x^{13} + 6 x^{12} + 16 x^{10} + 24 x^{9} + 36 x^{8} + 24 x^{7} + 20 x^{6} + 32 x^{5} + 56 x^{4} + 56 x^{3} + 48 x^{2} + 24 x + 8$ | $6$ | $3$ | $18$ | 18T33 | $[4/3, 4/3]_{3}^{6}$ | |
\(11\) | 11.3.2.1 | $x^{3} + 11$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
11.3.2.1 | $x^{3} + 11$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
11.6.0.1 | $x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(19\) | 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
19.7.0.1 | $x^{7} + 6 x + 17$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
19.7.0.1 | $x^{7} + 6 x + 17$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(547\) | $\Q_{547}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{547}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |