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Properties

Label 22.0.11019960576...0000.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{44}\cdot 3^{16}\cdot 5^{36}$
Root discriminant $123.83$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T27

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8100, 0, 9839025, 0, 148121250, 0, -480575, 0, 4556900, 0, -549974, 0, 69824, 0, 4390, 0, 680, 0, -35, 0, 14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 14*x^20 - 35*x^18 + 680*x^16 + 4390*x^14 + 69824*x^12 - 549974*x^10 + 4556900*x^8 - 480575*x^6 + 148121250*x^4 + 9839025*x^2 + 8100)
 
gp: K = bnfinit(x^22 + 14*x^20 - 35*x^18 + 680*x^16 + 4390*x^14 + 69824*x^12 - 549974*x^10 + 4556900*x^8 - 480575*x^6 + 148121250*x^4 + 9839025*x^2 + 8100, 1)
 

Normalized defining polynomial

\( x^{22} + 14 x^{20} - 35 x^{18} + 680 x^{16} + 4390 x^{14} + 69824 x^{12} - 549974 x^{10} + 4556900 x^{8} - 480575 x^{6} + 148121250 x^{4} + 9839025 x^{2} + 8100 \)

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:  \(-11019960576000000000000000000000000000000000000=-\,2^{44}\cdot 3^{16}\cdot 5^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $123.83$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{3}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} - \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{10} - \frac{1}{4} a^{8} + \frac{1}{8} a^{6} + \frac{1}{4} a^{4} + \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{40} a^{15} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{7} + \frac{1}{4} a^{6} - \frac{2}{5} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{80} a^{16} - \frac{1}{16} a^{14} - \frac{1}{16} a^{12} - \frac{1}{16} a^{10} - \frac{1}{16} a^{8} + \frac{29}{80} a^{6} - \frac{1}{2} a^{5} + \frac{1}{16} a^{4} - \frac{1}{2} a^{3} - \frac{7}{16} a^{2} + \frac{1}{4}$, $\frac{1}{80} a^{17} - \frac{1}{80} a^{15} - \frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{1}{16} a^{9} + \frac{9}{80} a^{7} - \frac{1}{2} a^{6} + \frac{21}{80} a^{5} - \frac{1}{2} a^{4} + \frac{1}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{480} a^{18} + \frac{1}{240} a^{16} + \frac{1}{48} a^{14} - \frac{1}{48} a^{12} + \frac{1}{12} a^{10} - \frac{23}{240} a^{8} - \frac{31}{240} a^{6} - \frac{1}{2} a^{5} - \frac{19}{48} a^{4} - \frac{1}{2} a^{3} - \frac{25}{96} a^{2} - \frac{3}{8}$, $\frac{1}{2880} a^{19} - \frac{1}{960} a^{18} - \frac{1}{180} a^{17} + \frac{1}{240} a^{16} - \frac{1}{90} a^{15} + \frac{1}{48} a^{14} - \frac{1}{18} a^{13} - \frac{1}{48} a^{12} - \frac{17}{288} a^{11} + \frac{11}{96} a^{10} + \frac{101}{720} a^{9} - \frac{13}{120} a^{8} - \frac{11}{720} a^{7} + \frac{37}{120} a^{6} - \frac{13}{180} a^{5} - \frac{7}{48} a^{4} + \frac{281}{576} a^{3} - \frac{29}{192} a^{2} + \frac{11}{48} a + \frac{1}{16}$, $\frac{1}{784873125075634732415295009922734720} a^{20} + \frac{392239408933558677239075601106661}{784873125075634732415295009922734720} a^{18} - \frac{30061088713206533313916483458091}{98109140634454341551911876240341840} a^{16} + \frac{105964639294718819229390158963291}{4905457031722717077595593812017092} a^{14} + \frac{1938800886683345502086763451452799}{78487312507563473241529500992273472} a^{12} + \frac{45358808213541116914565967270472477}{392436562537817366207647504961367360} a^{10} - \frac{18727423820446203892263902903057}{91179498730905521888393937026340} a^{8} + \frac{95654902619460242283100928924811257}{196218281268908683103823752480683680} a^{6} - \frac{1}{2} a^{5} + \frac{21233270283537209587020298296398441}{156974625015126946483059001984546944} a^{4} - \frac{13470569239999297490821328726520733}{52324875005042315494353000661515648} a^{2} - \frac{1}{2} a + \frac{448249577126645559072203458525935}{1453468750140064319287583351708768}$, $\frac{1}{4709238750453808394491770059536408320} a^{21} - \frac{1}{1569746250151269464830590019845469440} a^{20} + \frac{392239408933558677239075601106661}{4709238750453808394491770059536408320} a^{19} + \frac{1242912934974013681959455669565703}{1569746250151269464830590019845469440} a^{18} + \frac{484533485429630401096776084510091}{117730968761345209862294251488410208} a^{17} - \frac{787515083240579646285349151878091}{196218281268908683103823752480683680} a^{16} - \frac{166717864983491077104996863371363}{294327421903363024655735628721025520} a^{15} - \frac{620717364566330728258413135594673}{19621828126890868310382375248068368} a^{14} - \frac{17683027240207522808295611796615569}{470923875045380839449177005953640832} a^{13} - \frac{8479410262313634938880888534142255}{156974625015126946483059001984546944} a^{12} - \frac{52750332420913224637345908969869363}{2354619375226904197245885029768204160} a^{11} - \frac{86237616811230425894529249037281577}{784873125075634732415295009922734720} a^{10} + \frac{167699024503645016464358552503151}{1094153984770866262660727244316080} a^{9} - \frac{29901642169370074448212863510991}{182358997461811043776787874052680} a^{8} + \frac{27960803180992939196292254646593017}{235461937522690419724588502976820416} a^{7} - \frac{118547035434166255311880366714224353}{392436562537817366207647504961367360} a^{6} - \frac{1934503773778964256344665534317118067}{4709238750453808394491770059536408320} a^{5} + \frac{83416479726547421401685703026632855}{313949250030253892966118003969093888} a^{4} + \frac{130422837023867070118649423092647299}{313949250030253892966118003969093888} a^{3} + \frac{48898870024663365273456172924421953}{104649750010084630988706001323031296} a^{2} + \frac{3355187077406774197647370161943471}{8720812500840385915725500110252608} a - \frac{1356667545964185758626943053343915}{2906937500280128638575166703417536}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( -\frac{3745558129951553}{932376366653716381140} a^{21} - \frac{83852878676281567}{1491802186645946209824} a^{19} + \frac{26300154785733491}{186475273330743276228} a^{17} - \frac{5096411491171678553}{1864752733307432762280} a^{15} - \frac{3284551904945146129}{186475273330743276228} a^{13} - \frac{1045618625095601418583}{3729505466614865524560} a^{11} + \frac{3066068302737182345}{1386433258964634024} a^{9} - \frac{1708518111585445742503}{93237636665371638114} a^{7} + \frac{1937930255034028395529}{932376366653716381140} a^{5} - \frac{295926531683729374086245}{497267395548648736608} a^{3} - \frac{161746381671304970515}{4604327736561562376} a \) (order $4$)
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:  \( 4785285337040000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T27:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15840
The 20 conjugacy class representatives for t22n27
Character table for t22n27

Intermediate fields

\(\Q(\sqrt{-1}) \), 11.3.6561000000000000000000.1

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 24 sibling: data not computed
Degree 44 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R $22$ $22$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$2$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.8.18.66$x^{8} + 6 x^{4} + 4 x^{3} + 2$$8$$1$$18$$S_4\times C_2$$[2, 8/3, 8/3]_{3}^{2}$
2.12.24.419$x^{12} + 4 x^{8} + 4 x^{7} - 2 x^{6} + 4 x^{5} + 4 x + 2$$12$$1$$24$$C_2 \times S_4$$[2, 8/3, 8/3]_{3}^{2}$
3Data not computed
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.5.9.4$x^{5} + 30$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.9.4$x^{5} + 30$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.9.4$x^{5} + 30$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.9.4$x^{5} + 30$$5$$1$$9$$F_5$$[9/4]_{4}$