Properties

Label 44.0.681...976.2
Degree $44$
Signature $[0, 22]$
Discriminant $6.820\times 10^{84}$
Root discriminant $84.73$
Ramified primes $2, 3, 23$
Class number not computed
Class group not computed
Galois group $C_2\times C_{22}$ (as 44T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 + 42*x^42 + 1004*x^40 + 16416*x^38 + 203056*x^36 + 1976896*x^34 + 15592256*x^32 + 100902784*x^30 + 540972800*x^28 + 2406430208*x^26 + 8885665792*x^24 + 27062865920*x^22 + 67552104448*x^20 + 136133230592*x^18 + 218558169088*x^16 + 271461023744*x^14 + 255644925952*x^12 + 171145035776*x^10 + 80371253248*x^8 + 21592276992*x^6 + 3817865216*x^4 + 138412032*x^2 + 4194304)
 
gp: K = bnfinit(x^44 + 42*x^42 + 1004*x^40 + 16416*x^38 + 203056*x^36 + 1976896*x^34 + 15592256*x^32 + 100902784*x^30 + 540972800*x^28 + 2406430208*x^26 + 8885665792*x^24 + 27062865920*x^22 + 67552104448*x^20 + 136133230592*x^18 + 218558169088*x^16 + 271461023744*x^14 + 255644925952*x^12 + 171145035776*x^10 + 80371253248*x^8 + 21592276992*x^6 + 3817865216*x^4 + 138412032*x^2 + 4194304, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4194304, 0, 138412032, 0, 3817865216, 0, 21592276992, 0, 80371253248, 0, 171145035776, 0, 255644925952, 0, 271461023744, 0, 218558169088, 0, 136133230592, 0, 67552104448, 0, 27062865920, 0, 8885665792, 0, 2406430208, 0, 540972800, 0, 100902784, 0, 15592256, 0, 1976896, 0, 203056, 0, 16416, 0, 1004, 0, 42, 0, 1]);
 

\(x^{44} + 42 x^{42} + 1004 x^{40} + 16416 x^{38} + 203056 x^{36} + 1976896 x^{34} + 15592256 x^{32} + 100902784 x^{30} + 540972800 x^{28} + 2406430208 x^{26} + 8885665792 x^{24} + 27062865920 x^{22} + 67552104448 x^{20} + 136133230592 x^{18} + 218558169088 x^{16} + 271461023744 x^{14} + 255644925952 x^{12} + 171145035776 x^{10} + 80371253248 x^{8} + 21592276992 x^{6} + 3817865216 x^{4} + 138412032 x^{2} + 4194304\)  Toggle raw display

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

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 22]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(681\!\cdots\!976\)\(\medspace = 2^{66}\cdot 3^{22}\cdot 23^{40}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $84.73$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 23$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $44$
This field is Galois and abelian over $\Q$.
Conductor:  \(552=2^{3}\cdot 3\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{552}(1,·)$, $\chi_{552}(277,·)$, $\chi_{552}(133,·)$, $\chi_{552}(257,·)$, $\chi_{552}(265,·)$, $\chi_{552}(13,·)$, $\chi_{552}(77,·)$, $\chi_{552}(301,·)$, $\chi_{552}(269,·)$, $\chi_{552}(533,·)$, $\chi_{552}(25,·)$, $\chi_{552}(409,·)$, $\chi_{552}(29,·)$, $\chi_{552}(325,·)$, $\chi_{552}(545,·)$, $\chi_{552}(49,·)$, $\chi_{552}(41,·)$, $\chi_{552}(173,·)$, $\chi_{552}(541,·)$, $\chi_{552}(305,·)$, $\chi_{552}(449,·)$, $\chi_{552}(185,·)$, $\chi_{552}(317,·)$, $\chi_{552}(169,·)$, $\chi_{552}(193,·)$, $\chi_{552}(197,·)$, $\chi_{552}(289,·)$, $\chi_{552}(73,·)$, $\chi_{552}(461,·)$, $\chi_{552}(397,·)$, $\chi_{552}(209,·)$, $\chi_{552}(469,·)$, $\chi_{552}(121,·)$, $\chi_{552}(473,·)$, $\chi_{552}(349,·)$, $\chi_{552}(101,·)$, $\chi_{552}(353,·)$, $\chi_{552}(485,·)$, $\chi_{552}(233,·)$, $\chi_{552}(445,·)$, $\chi_{552}(361,·)$, $\chi_{552}(377,·)$, $\chi_{552}(509,·)$, $\chi_{552}(85,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{4096} a^{24}$, $\frac{1}{4096} a^{25}$, $\frac{1}{8192} a^{26}$, $\frac{1}{8192} a^{27}$, $\frac{1}{16384} a^{28}$, $\frac{1}{16384} a^{29}$, $\frac{1}{32768} a^{30}$, $\frac{1}{32768} a^{31}$, $\frac{1}{65536} a^{32}$, $\frac{1}{65536} a^{33}$, $\frac{1}{131072} a^{34}$, $\frac{1}{131072} a^{35}$, $\frac{1}{262144} a^{36}$, $\frac{1}{262144} a^{37}$, $\frac{1}{524288} a^{38}$, $\frac{1}{524288} a^{39}$, $\frac{1}{40373321728} a^{40} + \frac{7487}{10093330432} a^{38} - \frac{7545}{10093330432} a^{36} + \frac{11609}{5046665216} a^{34} + \frac{12115}{2523332608} a^{32} - \frac{247}{78854144} a^{30} + \frac{473}{39427072} a^{28} + \frac{2463}{78854144} a^{26} + \frac{3271}{157708288} a^{24} + \frac{1295}{39427072} a^{22} - \frac{2219}{19713536} a^{20} - \frac{12705}{19713536} a^{18} - \frac{13257}{9856768} a^{16} - \frac{10861}{4928384} a^{14} - \frac{1133}{154012} a^{12} + \frac{5037}{1232096} a^{10} + \frac{8413}{308024} a^{8} + \frac{11491}{308024} a^{6} - \frac{15191}{154012} a^{4} + \frac{10591}{77006} a^{2} + \frac{10893}{38503}$, $\frac{1}{40373321728} a^{41} + \frac{7487}{10093330432} a^{39} - \frac{7545}{10093330432} a^{37} + \frac{11609}{5046665216} a^{35} + \frac{12115}{2523332608} a^{33} - \frac{247}{78854144} a^{31} + \frac{473}{39427072} a^{29} + \frac{2463}{78854144} a^{27} + \frac{3271}{157708288} a^{25} + \frac{1295}{39427072} a^{23} - \frac{2219}{19713536} a^{21} - \frac{12705}{19713536} a^{19} - \frac{13257}{9856768} a^{17} - \frac{10861}{4928384} a^{15} - \frac{1133}{154012} a^{13} + \frac{5037}{1232096} a^{11} + \frac{8413}{308024} a^{9} + \frac{11491}{308024} a^{7} - \frac{15191}{154012} a^{5} + \frac{10591}{77006} a^{3} + \frac{10893}{38503} a$, $\frac{1}{82942646971370450752680255882097651647191413849849856} a^{42} + \frac{190427017626724063368374735989346701359275}{41471323485685225376340127941048825823595706924924928} a^{40} - \frac{786263041018262723491903013753758681097995925}{10367830871421306344085031985262206455898926731231232} a^{38} + \frac{3021604855640066456005666421892701219533034743}{5183915435710653172042515992631103227949463365615616} a^{36} + \frac{4839410230371102624831550392114737591100166649}{2591957717855326586021257996315551613974731682807808} a^{34} + \frac{4201109245464735669564270054376497698354553051}{2591957717855326586021257996315551613974731682807808} a^{32} + \frac{6509596457002940563422887376183362827233688333}{1295978858927663293010628998157775806987365841403904} a^{30} + \frac{14115369888379949737298339844354461890443385011}{647989429463831646505314499078887903493682920701952} a^{28} + \frac{6362405713060001224166813557596832651604022103}{161997357365957911626328624769721975873420730175488} a^{26} - \frac{2282591661158467103086857524797610055995998357}{80998678682978955813164312384860987936710365087744} a^{24} + \frac{1011866770333132709080117347153082936581678729}{80998678682978955813164312384860987936710365087744} a^{22} + \frac{12931447348997202512424621198798701576288144755}{40499339341489477906582156192430493968355182543872} a^{20} + \frac{3176086187703292098891266666860749869761575059}{5062417417686184738322769524053811746044397817984} a^{18} + \frac{13195457438765117428959233296641609764416909133}{10124834835372369476645539048107623492088795635968} a^{16} - \frac{354851102227217888417715375986089734028109547}{316401088605386546145173095253363234127774863624} a^{14} + \frac{6919763439996626467828935579443659919679065619}{2531208708843092369161384762026905873022198908992} a^{12} - \frac{423412761248501722179481862957157107241987087}{39550136075673318268146636906670404265971857953} a^{10} - \frac{16073345197087539024686008518179841593817771065}{632802177210773092290346190506726468255549727248} a^{8} - \frac{8586853527812866040595966388217727893865367217}{158200544302693273072586547626681617063887431812} a^{6} + \frac{14354073175149464966464365281147237239181017305}{158200544302693273072586547626681617063887431812} a^{4} - \frac{8593052920207576287145858416148939981865913619}{39550136075673318268146636906670404265971857953} a^{2} - \frac{18088117971194247705875516453344668155799592221}{39550136075673318268146636906670404265971857953}$, $\frac{1}{82942646971370450752680255882097651647191413849849856} a^{43} + \frac{190427017626724063368374735989346701359275}{41471323485685225376340127941048825823595706924924928} a^{41} - \frac{786263041018262723491903013753758681097995925}{10367830871421306344085031985262206455898926731231232} a^{39} + \frac{3021604855640066456005666421892701219533034743}{5183915435710653172042515992631103227949463365615616} a^{37} + \frac{4839410230371102624831550392114737591100166649}{2591957717855326586021257996315551613974731682807808} a^{35} + \frac{4201109245464735669564270054376497698354553051}{2591957717855326586021257996315551613974731682807808} a^{33} + \frac{6509596457002940563422887376183362827233688333}{1295978858927663293010628998157775806987365841403904} a^{31} + \frac{14115369888379949737298339844354461890443385011}{647989429463831646505314499078887903493682920701952} a^{29} + \frac{6362405713060001224166813557596832651604022103}{161997357365957911626328624769721975873420730175488} a^{27} - \frac{2282591661158467103086857524797610055995998357}{80998678682978955813164312384860987936710365087744} a^{25} + \frac{1011866770333132709080117347153082936581678729}{80998678682978955813164312384860987936710365087744} a^{23} + \frac{12931447348997202512424621198798701576288144755}{40499339341489477906582156192430493968355182543872} a^{21} + \frac{3176086187703292098891266666860749869761575059}{5062417417686184738322769524053811746044397817984} a^{19} + \frac{13195457438765117428959233296641609764416909133}{10124834835372369476645539048107623492088795635968} a^{17} - \frac{354851102227217888417715375986089734028109547}{316401088605386546145173095253363234127774863624} a^{15} + \frac{6919763439996626467828935579443659919679065619}{2531208708843092369161384762026905873022198908992} a^{13} - \frac{423412761248501722179481862957157107241987087}{39550136075673318268146636906670404265971857953} a^{11} - \frac{16073345197087539024686008518179841593817771065}{632802177210773092290346190506726468255549727248} a^{9} - \frac{8586853527812866040595966388217727893865367217}{158200544302693273072586547626681617063887431812} a^{7} + \frac{14354073175149464966464365281147237239181017305}{158200544302693273072586547626681617063887431812} a^{5} - \frac{8593052920207576287145858416148939981865913619}{39550136075673318268146636906670404265971857953} a^{3} - \frac{18088117971194247705875516453344668155799592221}{39550136075673318268146636906670404265971857953} a$  Toggle raw display

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

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $21$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{414556177818937525350558483019015320605052303}{41471323485685225376340127941048825823595706924924928} a^{42} - \frac{17385663921699872161341757065009211580446431211}{41471323485685225376340127941048825823595706924924928} a^{40} - \frac{207571435231570322670936516059315369508076226499}{20735661742842612688170063970524412911797853462462464} a^{38} - \frac{423742173509196008995238865786615125793968494271}{2591957717855326586021257996315551613974731682807808} a^{36} - \frac{10470482522236172424103405482756392899998492353119}{5183915435710653172042515992631103227949463365615616} a^{34} - \frac{6362807107977430776479524171323280007627947299131}{323994714731915823252657249539443951746841460350976} a^{32} - \frac{100227662715035713281310453796142886276182769225661}{647989429463831646505314499078887903493682920701952} a^{30} - \frac{161889454796178142746571197659340991714810819002625}{161997357365957911626328624769721975873420730175488} a^{28} - \frac{1732686555624062704122396534404865341348735205808011}{323994714731915823252657249539443951746841460350976} a^{26} - \frac{1922700536593001496491465570338732439893212888516475}{80998678682978955813164312384860987936710365087744} a^{24} - \frac{7081243878823882957966445691871830577051513278050789}{80998678682978955813164312384860987936710365087744} a^{22} - \frac{10749514882813051417553785399715272103359330343705585}{40499339341489477906582156192430493968355182543872} a^{20} - \frac{13364002198568950540869434595421413010164419072808287}{20249669670744738953291078096215246984177591271936} a^{18} - \frac{13398445484858974345060546099950451427258482198520615}{10124834835372369476645539048107623492088795635968} a^{16} - \frac{5343417171040549871852619428971842726985846840924477}{2531208708843092369161384762026905873022198908992} a^{14} - \frac{6578200791580392859217287085261064583363924327001495}{2531208708843092369161384762026905873022198908992} a^{12} - \frac{3061668823017677054286700470270645279848404039106171}{1265604354421546184580692381013452936511099454496} a^{10} - \frac{62876845523727485055108361330912641191341407209893}{39550136075673318268146636906670404265971857953} a^{8} - \frac{28883149982394774867804473309861945316069853721436}{39550136075673318268146636906670404265971857953} a^{6} - \frac{106108068317830159407054267426559430134303497265}{571121098565679686182622915619789231277571956} a^{4} - \frac{2664524270969116664641531806015314757346935267351}{79100272151346636536293273813340808531943715906} a^{2} - \frac{8712160285006901554448814973479920152297532642}{39550136075673318268146636906670404265971857953} \) (order $6$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed  Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

$C_2\times C_{22}$ (as 44T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
An abelian group of order 44
The 44 conjugacy class representatives for $C_2\times C_{22}$
Character table for $C_2\times C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\zeta_{23})^+\), 22.22.14741666340843480753092741810452692992.1, 22.0.2611441967281400084968119933496263205453824.1, 22.0.304011857053427966889939263171547.1

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 R $22^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ R $22^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$ $22^{2}$ $22^{2}$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
3Data not computed
23Data not computed