↑

Properties

Label 22.22.1551120513...4833.1
Degree $22$
Signature $[22, 0]$
Discriminant $11^{40}\cdot 17^{11}$
Root discriminant $322.60$
Ramified primes $11, 17$
Class number $23$ (GRH)
Class group $[23]$ (GRH)
Galois group $C_{22}$ (as 22T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![110603, -5011721, 57376176, -271848962, 524533735, -50899365, -974902588, 585528031, 569624253, -422212637, -147077084, 121098216, 17916371, -17555384, -844338, 1390774, -16577, -60159, 3047, 1309, -99, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 - 99*x^20 + 1309*x^19 + 3047*x^18 - 60159*x^17 - 16577*x^16 + 1390774*x^15 - 844338*x^14 - 17555384*x^13 + 17916371*x^12 + 121098216*x^11 - 147077084*x^10 - 422212637*x^9 + 569624253*x^8 + 585528031*x^7 - 974902588*x^6 - 50899365*x^5 + 524533735*x^4 - 271848962*x^3 + 57376176*x^2 - 5011721*x + 110603)
 
gp: K = bnfinit(x^22 - 11*x^21 - 99*x^20 + 1309*x^19 + 3047*x^18 - 60159*x^17 - 16577*x^16 + 1390774*x^15 - 844338*x^14 - 17555384*x^13 + 17916371*x^12 + 121098216*x^11 - 147077084*x^10 - 422212637*x^9 + 569624253*x^8 + 585528031*x^7 - 974902588*x^6 - 50899365*x^5 + 524533735*x^4 - 271848962*x^3 + 57376176*x^2 - 5011721*x + 110603, 1)
 

Normalized defining polynomial

\( x^{22} - 11 x^{21} - 99 x^{20} + 1309 x^{19} + 3047 x^{18} - 60159 x^{17} - 16577 x^{16} + 1390774 x^{15} - 844338 x^{14} - 17555384 x^{13} + 17916371 x^{12} + 121098216 x^{11} - 147077084 x^{10} - 422212637 x^{9} + 569624253 x^{8} + 585528031 x^{7} - 974902588 x^{6} - 50899365 x^{5} + 524533735 x^{4} - 271848962 x^{3} + 57376176 x^{2} - 5011721 x + 110603 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15511205137931876981971247794361198940089877934973874833=11^{40}\cdot 17^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $322.60$
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:  $11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2057=11^{2}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{2057}(1,·)$, $\chi_{2057}(67,·)$, $\chi_{2057}(1871,·)$, $\chi_{2057}(1937,·)$, $\chi_{2057}(1684,·)$, $\chi_{2057}(1750,·)$, $\chi_{2057}(1497,·)$, $\chi_{2057}(1563,·)$, $\chi_{2057}(1310,·)$, $\chi_{2057}(1376,·)$, $\chi_{2057}(1123,·)$, $\chi_{2057}(1189,·)$, $\chi_{2057}(936,·)$, $\chi_{2057}(1002,·)$, $\chi_{2057}(749,·)$, $\chi_{2057}(815,·)$, $\chi_{2057}(562,·)$, $\chi_{2057}(628,·)$, $\chi_{2057}(375,·)$, $\chi_{2057}(441,·)$, $\chi_{2057}(188,·)$, $\chi_{2057}(254,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{1}{9} a^{6} + \frac{2}{9} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} + \frac{1}{9} a^{7} + \frac{2}{9} a^{3} + \frac{1}{9} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{10} + \frac{1}{9} a^{8} + \frac{1}{9} a^{6} + \frac{2}{9} a^{4} + \frac{1}{3} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{11} + \frac{1}{9} a^{9} + \frac{1}{9} a^{7} + \frac{2}{9} a^{5} + \frac{1}{3} a^{3} + \frac{1}{9} a$, $\frac{1}{81} a^{16} + \frac{4}{81} a^{15} - \frac{1}{81} a^{14} - \frac{1}{27} a^{13} - \frac{4}{81} a^{12} - \frac{1}{81} a^{11} + \frac{5}{81} a^{10} + \frac{4}{81} a^{9} + \frac{1}{81} a^{7} - \frac{2}{81} a^{6} + \frac{8}{81} a^{5} - \frac{35}{81} a^{4} + \frac{8}{27} a^{3} - \frac{8}{81} a^{2} - \frac{35}{81} a - \frac{22}{81}$, $\frac{1}{81} a^{17} + \frac{1}{81} a^{15} + \frac{1}{81} a^{14} - \frac{1}{81} a^{13} - \frac{1}{27} a^{12} + \frac{2}{81} a^{10} + \frac{2}{81} a^{9} + \frac{1}{81} a^{8} + \frac{1}{27} a^{7} - \frac{2}{81} a^{6} - \frac{31}{81} a^{5} + \frac{2}{81} a^{4} + \frac{13}{81} a^{3} - \frac{13}{27} a^{2} - \frac{35}{81} a - \frac{11}{81}$, $\frac{1}{243} a^{18} - \frac{1}{243} a^{17} - \frac{1}{243} a^{16} - \frac{8}{243} a^{15} + \frac{1}{27} a^{14} + \frac{4}{243} a^{13} - \frac{7}{243} a^{12} + \frac{31}{243} a^{11} + \frac{26}{243} a^{10} - \frac{1}{27} a^{9} + \frac{11}{243} a^{8} - \frac{34}{243} a^{7} - \frac{7}{243} a^{6} - \frac{10}{243} a^{5} - \frac{13}{27} a^{4} - \frac{73}{243} a^{3} + \frac{92}{243} a^{2} - \frac{14}{243} a + \frac{73}{243}$, $\frac{1}{243} a^{19} + \frac{1}{243} a^{17} + \frac{13}{243} a^{15} + \frac{7}{243} a^{14} - \frac{2}{81} a^{13} + \frac{2}{81} a^{12} - \frac{11}{81} a^{11} - \frac{40}{243} a^{10} + \frac{17}{243} a^{9} - \frac{20}{243} a^{8} - \frac{23}{243} a^{7} - \frac{14}{243} a^{6} + \frac{41}{243} a^{5} - \frac{13}{243} a^{4} + \frac{85}{243} a^{3} + \frac{8}{81} a^{2} - \frac{118}{243} a + \frac{112}{243}$, $\frac{1}{333153} a^{20} - \frac{661}{333153} a^{19} - \frac{218}{111051} a^{18} - \frac{16}{12339} a^{17} + \frac{1613}{333153} a^{16} - \frac{17380}{333153} a^{15} - \frac{11116}{333153} a^{14} - \frac{1408}{333153} a^{13} - \frac{701}{333153} a^{12} - \frac{14057}{333153} a^{11} - \frac{19397}{333153} a^{10} + \frac{17114}{333153} a^{9} + \frac{30325}{333153} a^{8} - \frac{5966}{333153} a^{7} + \frac{42137}{333153} a^{6} + \frac{131557}{333153} a^{5} - \frac{152788}{333153} a^{4} - \frac{39017}{111051} a^{3} + \frac{14972}{111051} a^{2} - \frac{67046}{333153} a - \frac{64592}{333153}$, $\frac{1}{91708787646477432213543495972829748543519237896135353505941} a^{21} + \frac{82536824671088886066319459995277100618906728604463764}{91708787646477432213543495972829748543519237896135353505941} a^{20} + \frac{27831659872704501974375696245283270235031104902506663070}{30569595882159144071181165324276582847839745965378451168647} a^{19} + \frac{31147811480895577571268854620574105824020504807013258583}{30569595882159144071181165324276582847839745965378451168647} a^{18} - \frac{471809294362121600750708867069589078999500293896719198170}{91708787646477432213543495972829748543519237896135353505941} a^{17} - \frac{314010261023808796267759718617743284656345038612207839263}{91708787646477432213543495972829748543519237896135353505941} a^{16} + \frac{4603411715252833160642269790540540783273399230948017815567}{91708787646477432213543495972829748543519237896135353505941} a^{15} + \frac{1275359089488027784183189160311755080181553808568663661406}{91708787646477432213543495972829748543519237896135353505941} a^{14} - \frac{3217055019475225856664410265905289294226179167828468982326}{91708787646477432213543495972829748543519237896135353505941} a^{13} + \frac{4632039890874989931999935569541835909961593406573091649367}{91708787646477432213543495972829748543519237896135353505941} a^{12} + \frac{11793031489866629390737701569843312084454065194479133873435}{91708787646477432213543495972829748543519237896135353505941} a^{11} + \frac{320297903787179100633515118769963851605197981839840664865}{91708787646477432213543495972829748543519237896135353505941} a^{10} - \frac{5226732676924746358536490846270114498086809295527071881995}{91708787646477432213543495972829748543519237896135353505941} a^{9} - \frac{1511897492630540165148793974803685624366274788668707739666}{91708787646477432213543495972829748543519237896135353505941} a^{8} - \frac{13996814618096021646150494367228275206043508163981274410246}{91708787646477432213543495972829748543519237896135353505941} a^{7} - \frac{7123482124458194054619806519514246468360233557348546856855}{91708787646477432213543495972829748543519237896135353505941} a^{6} + \frac{33779915048570628938494885596609579507060925720683257667101}{91708787646477432213543495972829748543519237896135353505941} a^{5} + \frac{1967010131911508470476209550030499751562062022568004115871}{30569595882159144071181165324276582847839745965378451168647} a^{4} + \frac{14736743926734849980227308140821689258382578426503103667448}{30569595882159144071181165324276582847839745965378451168647} a^{3} - \frac{8542243919726653726720035549963597409170856245138488718530}{91708787646477432213543495972829748543519237896135353505941} a^{2} + \frac{794295900651815305542248967833782282922107574218075802251}{91708787646477432213543495972829748543519237896135353505941} a + \frac{10807122527603123780177566808016441971413339461496290942506}{30569595882159144071181165324276582847839745965378451168647}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{23}$, which has order $23$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $21$
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:  \( 33318410992799138000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{22}$ (as 22T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 22
The 22 conjugacy class representatives for $C_{22}$
Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 11.11.672749994932560009201.1

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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{11}$ $22$ $22$ R ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ $22$ $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
11Data not computed
17Data not computed