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Properties

Label 22.0.45838394542...1875.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,3^{21}\cdot 5^{20}\cdot 11^{16}$
Root discriminant $70.51$
Ramified primes $3, 5, 11$
Class number $93$ (GRH)
Class group $[93]$ (GRH)
Galois group $C_2\times C_{11}:C_5$ (as 22T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 44, 441, -760, 5605, -4332, 18462, -15537, 42285, -29240, 50648, -36183, 42773, -22580, 15510, -4827, 2667, -597, 325, -40, 21, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 21*x^20 - 40*x^19 + 325*x^18 - 597*x^17 + 2667*x^16 - 4827*x^15 + 15510*x^14 - 22580*x^13 + 42773*x^12 - 36183*x^11 + 50648*x^10 - 29240*x^9 + 42285*x^8 - 15537*x^7 + 18462*x^6 - 4332*x^5 + 5605*x^4 - 760*x^3 + 441*x^2 + 44*x + 16)
 
gp: K = bnfinit(x^22 - x^21 + 21*x^20 - 40*x^19 + 325*x^18 - 597*x^17 + 2667*x^16 - 4827*x^15 + 15510*x^14 - 22580*x^13 + 42773*x^12 - 36183*x^11 + 50648*x^10 - 29240*x^9 + 42285*x^8 - 15537*x^7 + 18462*x^6 - 4332*x^5 + 5605*x^4 - 760*x^3 + 441*x^2 + 44*x + 16, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 21 x^{20} - 40 x^{19} + 325 x^{18} - 597 x^{17} + 2667 x^{16} - 4827 x^{15} + 15510 x^{14} - 22580 x^{13} + 42773 x^{12} - 36183 x^{11} + 50648 x^{10} - 29240 x^{9} + 42285 x^{8} - 15537 x^{7} + 18462 x^{6} - 4332 x^{5} + 5605 x^{4} - 760 x^{3} + 441 x^{2} + 44 x + 16 \)

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

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:  \(-45838394542255573972509544658660888671875=-\,3^{21}\cdot 5^{20}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.51$
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:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{5} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{18} + \frac{1}{3} a^{10} + \frac{1}{9} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{2}{9}$, $\frac{1}{9} a^{19} - \frac{2}{9} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{85149} a^{20} - \frac{1132}{85149} a^{19} + \frac{128}{9461} a^{18} - \frac{4018}{28383} a^{17} + \frac{4714}{28383} a^{16} + \frac{3841}{28383} a^{15} + \frac{2713}{28383} a^{14} - \frac{1304}{9461} a^{13} - \frac{872}{28383} a^{12} - \frac{5048}{85149} a^{11} - \frac{23446}{85149} a^{10} - \frac{4339}{28383} a^{9} + \frac{433}{9461} a^{8} + \frac{12466}{28383} a^{7} - \frac{11153}{28383} a^{6} - \frac{9608}{28383} a^{5} - \frac{12601}{28383} a^{4} + \frac{10774}{28383} a^{3} + \frac{12910}{85149} a^{2} - \frac{21361}{85149} a + \frac{3670}{9461}$, $\frac{1}{350232408000301995010299381256198908} a^{21} + \frac{518243709899895822445913730515}{350232408000301995010299381256198908} a^{20} - \frac{740661317235952272926148913995635}{38914712000033555001144375695133212} a^{19} - \frac{592184640954334491153745108147855}{87558102000075498752574845314049727} a^{18} + \frac{13330019996902567650254788129343767}{116744136000100665003433127085399636} a^{17} - \frac{9247663846139829533952228330805903}{116744136000100665003433127085399636} a^{16} + \frac{9452609158538588228214107297871469}{116744136000100665003433127085399636} a^{15} - \frac{14653469105147627148735288942639545}{116744136000100665003433127085399636} a^{14} + \frac{4016804536024267602701301552362977}{58372068000050332501716563542699818} a^{13} - \frac{3862811428715500672235626974280685}{87558102000075498752574845314049727} a^{12} + \frac{34074332291926711933679622108648401}{350232408000301995010299381256198908} a^{11} - \frac{12982848149076237661236407997304419}{38914712000033555001144375695133212} a^{10} - \frac{13831553435293011350866194806862295}{87558102000075498752574845314049727} a^{9} - \frac{2411118334238758057662810161387959}{29186034000025166250858281771349909} a^{8} + \frac{27263532715266979903785463711727267}{116744136000100665003433127085399636} a^{7} + \frac{48759508823731123766281179664183993}{116744136000100665003433127085399636} a^{6} + \frac{16269282934032645451398631051659955}{58372068000050332501716563542699818} a^{5} + \frac{2448245085676215609002396778431636}{29186034000025166250858281771349909} a^{4} - \frac{59272884145287715532034466754523227}{350232408000301995010299381256198908} a^{3} + \frac{25863765080426175107553426427866599}{87558102000075498752574845314049727} a^{2} - \frac{45000190385953274310409491120392561}{116744136000100665003433127085399636} a + \frac{28596379252675640123940268203160388}{87558102000075498752574845314049727}$

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

Class group and class number

$C_{93}$, which has order $93$ (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:  $10$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{104522909429524397417462995231}{12339513370690272170323763564676} a^{21} - \frac{135615540564984640144146330223}{12339513370690272170323763564676} a^{20} + \frac{739580511054785356172269075945}{4113171123563424056774587854892} a^{19} - \frac{402002291561820303504300494085}{1028292780890856014193646963723} a^{18} + \frac{35058102879591172074202065471815}{12339513370690272170323763564676} a^{17} - \frac{72161729064127908067782060231227}{12339513370690272170323763564676} a^{16} + \frac{294862467311817794895570220987565}{12339513370690272170323763564676} a^{15} - \frac{582330242312467355615176475069345}{12339513370690272170323763564676} a^{14} + \frac{291755928291358940500190439434635}{2056585561781712028387293927446} a^{13} - \frac{700319487607551004140178138418420}{3084878342672568042580940891169} a^{12} + \frac{5050201182831686540261803195258463}{12339513370690272170323763564676} a^{11} - \frac{4915234392605293094308966267421545}{12339513370690272170323763564676} a^{10} + \frac{1517154493768489799418971905275610}{3084878342672568042580940891169} a^{9} - \frac{1074441319612265449921678840868285}{3084878342672568042580940891169} a^{8} + \frac{4930183336038027177249718603848095}{12339513370690272170323763564676} a^{7} - \frac{2674287955956884106600587277755207}{12339513370690272170323763564676} a^{6} + \frac{1050844151958840476128175392408489}{6169756685345136085161881782338} a^{5} - \frac{72801386561686339033292963230785}{1028292780890856014193646963723} a^{4} + \frac{599846256212853706900949428000975}{12339513370690272170323763564676} a^{3} - \frac{57477816529598070181634585737870}{3084878342672568042580940891169} a^{2} + \frac{37821646282860949356550046746651}{12339513370690272170323763564676} a + \frac{687313319118049557177777198065}{3084878342672568042580940891169} \) (order $6$)
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:  \( 12475736338.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{11}:C_5$ (as 22T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 110
The 14 conjugacy class representatives for $C_2\times C_{11}:C_5$
Character table for $C_2\times C_{11}:C_5$

Intermediate fields

\(\Q(\sqrt{-3}) \), 11.11.123610132462587890625.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R R ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
3Data not computed
5Data not computed
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$