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Properties

Label 44.44.7829660228...9952.1
Degree $44$
Signature $[44, 0]$
Discriminant $2^{121}\cdot 23^{40}$
Root discriminant $116.35$
Ramified primes $2, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{44}$ (as 44T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2048, 0, -270336, 0, 8853504, 0, -127395840, 0, 982631936, 0, -4610997248, 0, 14289749760, 0, -30880705024, 0, 48285946240, 0, -56035526656, 0, 49105446400, 0, -32860916224, 0, 16898234240, 0, -6691379968, 0, 2036712064, 0, -473860064, 0, 83449416, 0, -10957792, 0, 1048936, 0, -70680, 0, 3162, 0, -84, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^44 - 84*x^42 + 3162*x^40 - 70680*x^38 + 1048936*x^36 - 10957792*x^34 + 83449416*x^32 - 473860064*x^30 + 2036712064*x^28 - 6691379968*x^26 + 16898234240*x^24 - 32860916224*x^22 + 49105446400*x^20 - 56035526656*x^18 + 48285946240*x^16 - 30880705024*x^14 + 14289749760*x^12 - 4610997248*x^10 + 982631936*x^8 - 127395840*x^6 + 8853504*x^4 - 270336*x^2 + 2048)
 
gp: K = bnfinit(x^44 - 84*x^42 + 3162*x^40 - 70680*x^38 + 1048936*x^36 - 10957792*x^34 + 83449416*x^32 - 473860064*x^30 + 2036712064*x^28 - 6691379968*x^26 + 16898234240*x^24 - 32860916224*x^22 + 49105446400*x^20 - 56035526656*x^18 + 48285946240*x^16 - 30880705024*x^14 + 14289749760*x^12 - 4610997248*x^10 + 982631936*x^8 - 127395840*x^6 + 8853504*x^4 - 270336*x^2 + 2048, 1)
 

Normalized defining polynomial

\( x^{44} - 84 x^{42} + 3162 x^{40} - 70680 x^{38} + 1048936 x^{36} - 10957792 x^{34} + 83449416 x^{32} - 473860064 x^{30} + 2036712064 x^{28} - 6691379968 x^{26} + 16898234240 x^{24} - 32860916224 x^{22} + 49105446400 x^{20} - 56035526656 x^{18} + 48285946240 x^{16} - 30880705024 x^{14} + 14289749760 x^{12} - 4610997248 x^{10} + 982631936 x^{8} - 127395840 x^{6} + 8853504 x^{4} - 270336 x^{2} + 2048 \)

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

Invariants

Degree:  $44$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[44, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7829660228065619245582194641412012312544945884150589900838471630076269829766255604192509952=2^{121}\cdot 23^{40}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $116.35$
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:  $2, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(368=2^{4}\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{368}(1,·)$, $\chi_{368}(261,·)$, $\chi_{368}(257,·)$, $\chi_{368}(9,·)$, $\chi_{368}(13,·)$, $\chi_{368}(301,·)$, $\chi_{368}(269,·)$, $\chi_{368}(277,·)$, $\chi_{368}(25,·)$, $\chi_{368}(29,·)$, $\chi_{368}(133,·)$, $\chi_{368}(289,·)$, $\chi_{368}(165,·)$, $\chi_{368}(325,·)$, $\chi_{368}(305,·)$, $\chi_{368}(41,·)$, $\chi_{368}(173,·)$, $\chi_{368}(93,·)$, $\chi_{368}(49,·)$, $\chi_{368}(265,·)$, $\chi_{368}(185,·)$, $\chi_{368}(317,·)$, $\chi_{368}(193,·)$, $\chi_{368}(197,·)$, $\chi_{368}(353,·)$, $\chi_{368}(73,·)$, $\chi_{368}(77,·)$, $\chi_{368}(141,·)$, $\chi_{368}(81,·)$, $\chi_{368}(85,·)$, $\chi_{368}(105,·)$, $\chi_{368}(361,·)$, $\chi_{368}(349,·)$, $\chi_{368}(357,·)$, $\chi_{368}(225,·)$, $\chi_{368}(101,·)$, $\chi_{368}(209,·)$, $\chi_{368}(233,·)$, $\chi_{368}(177,·)$, $\chi_{368}(285,·)$, $\chi_{368}(117,·)$, $\chi_{368}(169,·)$, $\chi_{368}(121,·)$, $\chi_{368}(213,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{8} a^{13}$, $\frac{1}{8} a^{14}$, $\frac{1}{8} a^{15}$, $\frac{1}{16} a^{16}$, $\frac{1}{16} a^{17}$, $\frac{1}{16} a^{18}$, $\frac{1}{16} a^{19}$, $\frac{1}{32} a^{20}$, $\frac{1}{32} a^{21}$, $\frac{1}{32} a^{22}$, $\frac{1}{32} a^{23}$, $\frac{1}{64} a^{24}$, $\frac{1}{64} a^{25}$, $\frac{1}{64} a^{26}$, $\frac{1}{64} a^{27}$, $\frac{1}{128} a^{28}$, $\frac{1}{128} a^{29}$, $\frac{1}{128} a^{30}$, $\frac{1}{128} a^{31}$, $\frac{1}{256} a^{32}$, $\frac{1}{256} a^{33}$, $\frac{1}{12032} a^{34} + \frac{9}{6016} a^{32} - \frac{5}{3008} a^{30} + \frac{11}{3008} a^{28} + \frac{1}{752} a^{26} + \frac{5}{752} a^{24} - \frac{15}{1504} a^{22} - \frac{23}{1504} a^{20} + \frac{21}{752} a^{18} - \frac{3}{188} a^{16} - \frac{13}{376} a^{14} - \frac{3}{94} a^{12} + \frac{7}{188} a^{10} + \frac{3}{188} a^{8} - \frac{15}{94} a^{6} + \frac{11}{47} a^{4} - \frac{15}{47} a^{2} + \frac{19}{47}$, $\frac{1}{12032} a^{35} + \frac{9}{6016} a^{33} - \frac{5}{3008} a^{31} + \frac{11}{3008} a^{29} + \frac{1}{752} a^{27} + \frac{5}{752} a^{25} - \frac{15}{1504} a^{23} - \frac{23}{1504} a^{21} + \frac{21}{752} a^{19} - \frac{3}{188} a^{17} - \frac{13}{376} a^{15} - \frac{3}{94} a^{13} + \frac{7}{188} a^{11} + \frac{3}{188} a^{9} - \frac{15}{94} a^{7} + \frac{11}{47} a^{5} - \frac{15}{47} a^{3} + \frac{19}{47} a$, $\frac{1}{24064} a^{36} + \frac{1}{752} a^{32} + \frac{7}{6016} a^{30} - \frac{3}{3008} a^{28} + \frac{21}{3008} a^{26} - \frac{7}{3008} a^{24} + \frac{3}{752} a^{22} - \frac{7}{1504} a^{20} - \frac{7}{752} a^{18} + \frac{1}{752} a^{16} + \frac{17}{376} a^{14} + \frac{21}{376} a^{12} + \frac{9}{188} a^{10} + \frac{5}{188} a^{8} + \frac{5}{94} a^{6} + \frac{11}{47} a^{4} - \frac{20}{47} a^{2} + \frac{17}{47}$, $\frac{1}{24064} a^{37} + \frac{1}{752} a^{33} + \frac{7}{6016} a^{31} - \frac{3}{3008} a^{29} + \frac{21}{3008} a^{27} - \frac{7}{3008} a^{25} + \frac{3}{752} a^{23} - \frac{7}{1504} a^{21} - \frac{7}{752} a^{19} + \frac{1}{752} a^{17} + \frac{17}{376} a^{15} + \frac{21}{376} a^{13} + \frac{9}{188} a^{11} + \frac{5}{188} a^{9} + \frac{5}{94} a^{7} + \frac{11}{47} a^{5} - \frac{20}{47} a^{3} + \frac{17}{47} a$, $\frac{1}{24064} a^{38} + \frac{1}{1504} a^{32} + \frac{13}{6016} a^{30} + \frac{19}{6016} a^{28} + \frac{23}{3008} a^{26} + \frac{21}{3008} a^{24} - \frac{1}{752} a^{22} - \frac{11}{752} a^{20} - \frac{3}{376} a^{18} - \frac{9}{752} a^{16} - \frac{3}{188} a^{14} + \frac{11}{188} a^{12} - \frac{13}{188} a^{10} + \frac{9}{188} a^{8} - \frac{10}{47} a^{6} - \frac{8}{47} a^{4} + \frac{22}{47} a^{2} - \frac{22}{47}$, $\frac{1}{24064} a^{39} + \frac{1}{1504} a^{33} + \frac{13}{6016} a^{31} + \frac{19}{6016} a^{29} + \frac{23}{3008} a^{27} + \frac{21}{3008} a^{25} - \frac{1}{752} a^{23} - \frac{11}{752} a^{21} - \frac{3}{376} a^{19} - \frac{9}{752} a^{17} - \frac{3}{188} a^{15} + \frac{11}{188} a^{13} - \frac{13}{188} a^{11} + \frac{9}{188} a^{9} - \frac{10}{47} a^{7} - \frac{8}{47} a^{5} + \frac{22}{47} a^{3} - \frac{22}{47} a$, $\frac{1}{6593536} a^{40} + \frac{61}{3296768} a^{38} + \frac{21}{3296768} a^{36} + \frac{9}{824192} a^{34} - \frac{419}{824192} a^{32} - \frac{547}{412096} a^{30} - \frac{2913}{824192} a^{28} - \frac{181}{412096} a^{26} - \frac{91}{12878} a^{24} + \frac{1381}{206048} a^{22} + \frac{24}{6439} a^{20} + \frac{2291}{103024} a^{18} - \frac{387}{25756} a^{16} + \frac{1747}{51512} a^{14} - \frac{701}{51512} a^{12} - \frac{333}{12878} a^{10} - \frac{489}{6439} a^{8} - \frac{111}{12878} a^{6} + \frac{1649}{12878} a^{4} + \frac{2440}{6439} a^{2} - \frac{484}{6439}$, $\frac{1}{6593536} a^{41} + \frac{61}{3296768} a^{39} + \frac{21}{3296768} a^{37} + \frac{9}{824192} a^{35} - \frac{419}{824192} a^{33} - \frac{547}{412096} a^{31} - \frac{2913}{824192} a^{29} - \frac{181}{412096} a^{27} - \frac{91}{12878} a^{25} + \frac{1381}{206048} a^{23} + \frac{24}{6439} a^{21} + \frac{2291}{103024} a^{19} - \frac{387}{25756} a^{17} + \frac{1747}{51512} a^{15} - \frac{701}{51512} a^{13} - \frac{333}{12878} a^{11} - \frac{489}{6439} a^{9} - \frac{111}{12878} a^{7} + \frac{1649}{12878} a^{5} + \frac{2440}{6439} a^{3} - \frac{484}{6439} a$, $\frac{1}{188725085694020834775013884886713344} a^{42} + \frac{9010902099522122258973228637}{188725085694020834775013884886713344} a^{40} - \frac{682461338330456828937564857705}{47181271423505208693753471221678336} a^{38} - \frac{1476393579344526752306541404885}{94362542847010417387506942443356672} a^{36} - \frac{153286335230245718827606135653}{11795317855876302173438367805419584} a^{34} - \frac{19381390834622955619133630473921}{47181271423505208693753471221678336} a^{32} + \frac{64380917372913753634450047477113}{23590635711752604346876735610839168} a^{30} - \frac{51851247829906776057697830806779}{23590635711752604346876735610839168} a^{28} - \frac{60895284846110634405086991251197}{11795317855876302173438367805419584} a^{26} + \frac{30676629989127855177825650190699}{5897658927938151086719183902709792} a^{24} + \frac{77962790774085599562716697064117}{5897658927938151086719183902709792} a^{22} + \frac{26107984131268431756871062294453}{5897658927938151086719183902709792} a^{20} + \frac{17408760370534654124950526757271}{1474414731984537771679795975677448} a^{18} - \frac{11469683266188456153348111139933}{368603682996134442919948993919362} a^{16} - \frac{48527130204531913903338090697325}{1474414731984537771679795975677448} a^{14} + \frac{21749098623519347560231750471615}{368603682996134442919948993919362} a^{12} + \frac{21124523397296623394237570667623}{737207365992268885839897987838724} a^{10} + \frac{31200496312796428093218961015827}{737207365992268885839897987838724} a^{8} + \frac{24670767956895701185089688661871}{184301841498067221459974496959681} a^{6} + \frac{41636672806877960571986394481899}{368603682996134442919948993919362} a^{4} - \frac{16605634311707652674731203981587}{184301841498067221459974496959681} a^{2} + \frac{91356427128471149976212419850127}{184301841498067221459974496959681}$, $\frac{1}{188725085694020834775013884886713344} a^{43} + \frac{9010902099522122258973228637}{188725085694020834775013884886713344} a^{41} - \frac{682461338330456828937564857705}{47181271423505208693753471221678336} a^{39} - \frac{1476393579344526752306541404885}{94362542847010417387506942443356672} a^{37} - \frac{153286335230245718827606135653}{11795317855876302173438367805419584} a^{35} - \frac{19381390834622955619133630473921}{47181271423505208693753471221678336} a^{33} + \frac{64380917372913753634450047477113}{23590635711752604346876735610839168} a^{31} - \frac{51851247829906776057697830806779}{23590635711752604346876735610839168} a^{29} - \frac{60895284846110634405086991251197}{11795317855876302173438367805419584} a^{27} + \frac{30676629989127855177825650190699}{5897658927938151086719183902709792} a^{25} + \frac{77962790774085599562716697064117}{5897658927938151086719183902709792} a^{23} + \frac{26107984131268431756871062294453}{5897658927938151086719183902709792} a^{21} + \frac{17408760370534654124950526757271}{1474414731984537771679795975677448} a^{19} - \frac{11469683266188456153348111139933}{368603682996134442919948993919362} a^{17} - \frac{48527130204531913903338090697325}{1474414731984537771679795975677448} a^{15} + \frac{21749098623519347560231750471615}{368603682996134442919948993919362} a^{13} + \frac{21124523397296623394237570667623}{737207365992268885839897987838724} a^{11} + \frac{31200496312796428093218961015827}{737207365992268885839897987838724} a^{9} + \frac{24670767956895701185089688661871}{184301841498067221459974496959681} a^{7} + \frac{41636672806877960571986394481899}{368603682996134442919948993919362} a^{5} - \frac{16605634311707652674731203981587}{184301841498067221459974496959681} a^{3} + \frac{91356427128471149976212419850127}{184301841498067221459974496959681} a$

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:  $43$
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:  \( 78348006818091075000000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{44}$ (as 44T1):

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

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{23})^+\), 22.22.14741666340843480753092741810452692992.1

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 $44$ $44$ $22^{2}$ $44$ $44$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{4}$ $44$ R $44$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{4}$ $44$ $22^{2}$ $44$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{44}$ $44$ $44$

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
2Data not computed
23Data not computed