Normalized defining polynomial
\( x^{22} - 2 x^{21} - 37 x^{20} + 146 x^{19} + 494 x^{18} - 3636 x^{17} + 2299 x^{16} + 29626 x^{15} - 64260 x^{14} - 124424 x^{13} + 736428 x^{12} - 1124536 x^{11} + 59133 x^{10} + 1514350 x^{9} + 3820816 x^{8} - 32528018 x^{7} + 109540538 x^{6} - 244699026 x^{5} + 410278465 x^{4} - 525754586 x^{3} + 533811683 x^{2} - 395494894 x + 210368707 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 11]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-28542099180404529250131657131852910655063457792=-\,2^{33}\cdot 67^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $129.30$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 67$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(536=2^{3}\cdot 67\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{536}(1,·)$, $\chi_{536}(131,·)$, $\chi_{536}(129,·)$, $\chi_{536}(9,·)$, $\chi_{536}(81,·)$, $\chi_{536}(403,·)$, $\chi_{536}(89,·)$, $\chi_{536}(225,·)$, $\chi_{536}(25,·)$, $\chi_{536}(283,·)$, $\chi_{536}(427,·)$, $\chi_{536}(107,·)$, $\chi_{536}(417,·)$, $\chi_{536}(411,·)$, $\chi_{536}(193,·)$, $\chi_{536}(491,·)$, $\chi_{536}(483,·)$, $\chi_{536}(241,·)$, $\chi_{536}(91,·)$, $\chi_{536}(531,·)$, $\chi_{536}(265,·)$, $\chi_{536}(59,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{256447} a^{20} + \frac{70887}{256447} a^{19} - \frac{13300}{256447} a^{18} + \frac{107472}{256447} a^{17} + \frac{15692}{256447} a^{16} - \frac{39048}{256447} a^{15} - \frac{95840}{256447} a^{14} - \frac{126623}{256447} a^{13} + \frac{80757}{256447} a^{12} + \frac{24720}{256447} a^{11} + \frac{56656}{256447} a^{10} + \frac{68559}{256447} a^{9} + \frac{25699}{256447} a^{8} - \frac{96037}{256447} a^{7} + \frac{118286}{256447} a^{6} + \frac{54895}{256447} a^{5} + \frac{64740}{256447} a^{4} + \frac{48502}{256447} a^{3} - \frac{15421}{256447} a^{2} - \frac{124532}{256447} a - \frac{84802}{256447}$, $\frac{1}{1054969309143111993870779323772717981182590378000390692394149387699326631} a^{21} - \frac{1830602609038389833030758423431597333434495249870275176515168879632}{1054969309143111993870779323772717981182590378000390692394149387699326631} a^{20} - \frac{326955823583929362573550668005884946617705912456734446720489400258715195}{1054969309143111993870779323772717981182590378000390692394149387699326631} a^{19} - \frac{94282987979178785260144058863468768423537358513086795302532914166971101}{1054969309143111993870779323772717981182590378000390692394149387699326631} a^{18} - \frac{300533019947828482337431414130607547665475467506207969225913821174868379}{1054969309143111993870779323772717981182590378000390692394149387699326631} a^{17} - \frac{4623650781530798324304553692653056337555916571875784409022938217708768}{1054969309143111993870779323772717981182590378000390692394149387699326631} a^{16} + \frac{525905607516888678286252065706909524388165113132951060318845158103516146}{1054969309143111993870779323772717981182590378000390692394149387699326631} a^{15} - \frac{8342091726722125414103435815463791877138011817243657993421631599471426}{36378252039417654961061355992162689006296219931047954910143082334459539} a^{14} + \frac{4889130849650901794006457545275012470070631323129470529802590787337256}{1054969309143111993870779323772717981182590378000390692394149387699326631} a^{13} - \frac{180264738558033712506844459511763305398999251789801084807332330407497326}{1054969309143111993870779323772717981182590378000390692394149387699326631} a^{12} + \frac{10217446144079354917558157622635192329683835655724273283033943109344719}{36378252039417654961061355992162689006296219931047954910143082334459539} a^{11} + \frac{259809157538917667437485013327729588275506933687578525523428075503167337}{1054969309143111993870779323772717981182590378000390692394149387699326631} a^{10} - \frac{366816427302712375920985988107763934713380777714127758897216519187687668}{1054969309143111993870779323772717981182590378000390692394149387699326631} a^{9} + \frac{76422273756686806736272030509509104954517960727944513807779013651614061}{1054969309143111993870779323772717981182590378000390692394149387699326631} a^{8} - \frac{456097503260145811574859839833644038505856772611619695656557291199821530}{1054969309143111993870779323772717981182590378000390692394149387699326631} a^{7} - \frac{66282925800718490140515551219517120674577676879102946279553086011578987}{1054969309143111993870779323772717981182590378000390692394149387699326631} a^{6} + \frac{371142805298792398435220490225630682444701936207783451975902563354363661}{1054969309143111993870779323772717981182590378000390692394149387699326631} a^{5} + \frac{81511490471741065675778164339446424936261851374879283653465548532804863}{1054969309143111993870779323772717981182590378000390692394149387699326631} a^{4} + \frac{459800256539537989260960174997130615815467206588426817605368174130090050}{1054969309143111993870779323772717981182590378000390692394149387699326631} a^{3} - \frac{325725036508117036964626145186020507893935881313583958755944396502463518}{1054969309143111993870779323772717981182590378000390692394149387699326631} a^{2} + \frac{24185233808946706454497353428058247926722475282360576335018124026116886}{1054969309143111993870779323772717981182590378000390692394149387699326631} a + \frac{257672427583647223847363367051622286700829177804860668766353280833127545}{1054969309143111993870779323772717981182590378000390692394149387699326631}$
Class group and class number
$C_{232331}$, which has order $232331$ (assuming GRH)
Unit group
| Rank: | $10$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 338444542.042557 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 22 |
| The 22 conjugacy class representatives for $C_{22}$ |
| Character table for $C_{22}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-2}) \), 11.11.1822837804551761449.1 |
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 | ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ | $22$ | $22$ | ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{11}$ | $22$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$ | ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ | $22$ | $22$ | ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$ |
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 | Data not computed | ||||||
| $67$ | 67.11.10.1 | $x^{11} - 67$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |
| 67.11.10.1 | $x^{11} - 67$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ | |