Properties

Label 44.0.123...000.1
Degree $44$
Signature $[0, 22]$
Discriminant $1.235\times 10^{83}$
Root discriminant $77.35$
Ramified primes $2, 5, 23$
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 + 63*x^42 + 1771*x^40 + 29412*x^38 + 322363*x^36 + 2470134*x^34 + 13694893*x^32 + 56167749*x^30 + 172989335*x^28 + 404160705*x^26 + 720751936*x^24 + 983588334*x^22 + 1025844301*x^20 + 813203334*x^18 + 484862836*x^16 + 213849639*x^14 + 68058536*x^12 + 15071199*x^10 + 2201056*x^8 + 195624*x^6 + 9361*x^4 + 198*x^2 + 1)
 
gp: K = bnfinit(x^44 + 63*x^42 + 1771*x^40 + 29412*x^38 + 322363*x^36 + 2470134*x^34 + 13694893*x^32 + 56167749*x^30 + 172989335*x^28 + 404160705*x^26 + 720751936*x^24 + 983588334*x^22 + 1025844301*x^20 + 813203334*x^18 + 484862836*x^16 + 213849639*x^14 + 68058536*x^12 + 15071199*x^10 + 2201056*x^8 + 195624*x^6 + 9361*x^4 + 198*x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 198, 0, 9361, 0, 195624, 0, 2201056, 0, 15071199, 0, 68058536, 0, 213849639, 0, 484862836, 0, 813203334, 0, 1025844301, 0, 983588334, 0, 720751936, 0, 404160705, 0, 172989335, 0, 56167749, 0, 13694893, 0, 2470134, 0, 322363, 0, 29412, 0, 1771, 0, 63, 0, 1]);
 

\( x^{44} + 63 x^{42} + 1771 x^{40} + 29412 x^{38} + 322363 x^{36} + 2470134 x^{34} + 13694893 x^{32} + 56167749 x^{30} + 172989335 x^{28} + 404160705 x^{26} + 720751936 x^{24} + 983588334 x^{22} + 1025844301 x^{20} + 813203334 x^{18} + 484862836 x^{16} + 213849639 x^{14} + 68058536 x^{12} + 15071199 x^{10} + 2201056 x^{8} + 195624 x^{6} + 9361 x^{4} + 198 x^{2} + 1 \)

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:  \(123\!\cdots\!000\)\(\medspace = 2^{44}\cdot 5^{22}\cdot 23^{40}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $77.35$
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:  $2, 5, 23$
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:  \(460=2^{2}\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{460}(1,·)$, $\chi_{460}(259,·)$, $\chi_{460}(261,·)$, $\chi_{460}(9,·)$, $\chi_{460}(139,·)$, $\chi_{460}(269,·)$, $\chi_{460}(271,·)$, $\chi_{460}(131,·)$, $\chi_{460}(279,·)$, $\chi_{460}(409,·)$, $\chi_{460}(29,·)$, $\chi_{460}(31,·)$, $\chi_{460}(289,·)$, $\chi_{460}(39,·)$, $\chi_{460}(41,·)$, $\chi_{460}(301,·)$, $\chi_{460}(49,·)$, $\chi_{460}(179,·)$, $\chi_{460}(439,·)$, $\chi_{460}(441,·)$, $\chi_{460}(59,·)$, $\chi_{460}(151,·)$, $\chi_{460}(449,·)$, $\chi_{460}(331,·)$, $\chi_{460}(71,·)$, $\chi_{460}(311,·)$, $\chi_{460}(141,·)$, $\chi_{460}(81,·)$, $\chi_{460}(211,·)$, $\chi_{460}(399,·)$, $\chi_{460}(219,·)$, $\chi_{460}(349,·)$, $\chi_{460}(351,·)$, $\chi_{460}(101,·)$, $\chi_{460}(209,·)$, $\chi_{460}(361,·)$, $\chi_{460}(231,·)$, $\chi_{460}(239,·)$, $\chi_{460}(369,·)$, $\chi_{460}(371,·)$, $\chi_{460}(169,·)$, $\chi_{460}(121,·)$, $\chi_{460}(119,·)$, $\chi_{460}(381,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $\frac{1}{139} a^{40} + \frac{45}{139} a^{38} + \frac{28}{139} a^{36} - \frac{11}{139} a^{34} - \frac{50}{139} a^{32} + \frac{8}{139} a^{30} - \frac{2}{139} a^{28} + \frac{12}{139} a^{26} + \frac{64}{139} a^{24} + \frac{19}{139} a^{22} - \frac{44}{139} a^{20} + \frac{32}{139} a^{18} + \frac{3}{139} a^{16} - \frac{67}{139} a^{14} + \frac{53}{139} a^{12} + \frac{65}{139} a^{10} - \frac{57}{139} a^{8} - \frac{3}{139} a^{6} - \frac{11}{139} a^{4} - \frac{10}{139} a^{2} + \frac{66}{139}$, $\frac{1}{139} a^{41} + \frac{45}{139} a^{39} + \frac{28}{139} a^{37} - \frac{11}{139} a^{35} - \frac{50}{139} a^{33} + \frac{8}{139} a^{31} - \frac{2}{139} a^{29} + \frac{12}{139} a^{27} + \frac{64}{139} a^{25} + \frac{19}{139} a^{23} - \frac{44}{139} a^{21} + \frac{32}{139} a^{19} + \frac{3}{139} a^{17} - \frac{67}{139} a^{15} + \frac{53}{139} a^{13} + \frac{65}{139} a^{11} - \frac{57}{139} a^{9} - \frac{3}{139} a^{7} - \frac{11}{139} a^{5} - \frac{10}{139} a^{3} + \frac{66}{139} a$, $\frac{1}{903797337968266405995539176965434889261299} a^{42} + \frac{130177666064561621313887748338781075730}{903797337968266405995539176965434889261299} a^{40} + \frac{99058032285335090257918538339087825780952}{903797337968266405995539176965434889261299} a^{38} + \frac{338186811088381392638801068780181294361127}{903797337968266405995539176965434889261299} a^{36} - \frac{7385849353393373698483025322163291875960}{903797337968266405995539176965434889261299} a^{34} + \frac{32513950234371015415844648090714104490715}{903797337968266405995539176965434889261299} a^{32} + \frac{434324421684235595982463372570948519167017}{903797337968266405995539176965434889261299} a^{30} + \frac{64488271171369193552193798792419153903851}{903797337968266405995539176965434889261299} a^{28} - \frac{136118574919356883203766843615480783007646}{903797337968266405995539176965434889261299} a^{26} + \frac{25684985262706259522704035798228708790782}{903797337968266405995539176965434889261299} a^{24} - \frac{172738058963559142833814800806180982354053}{903797337968266405995539176965434889261299} a^{22} + \frac{34540688792776578338007341838722238008896}{903797337968266405995539176965434889261299} a^{20} + \frac{133504439791512061099760523876135413873000}{903797337968266405995539176965434889261299} a^{18} + \frac{330682891645019761201373867414243647711589}{903797337968266405995539176965434889261299} a^{16} + \frac{287186121623732381928443141302586426709898}{903797337968266405995539176965434889261299} a^{14} + \frac{232948205096083511224887496985959973242831}{903797337968266405995539176965434889261299} a^{12} + \frac{149821569836974546160834695685307324964894}{903797337968266405995539176965434889261299} a^{10} - \frac{232291130789377550800283608790411538854601}{903797337968266405995539176965434889261299} a^{8} + \frac{170086229709210106584174425345470656263400}{903797337968266405995539176965434889261299} a^{6} - \frac{185601737904770903928142946690797135264135}{903797337968266405995539176965434889261299} a^{4} + \frac{72667242694983196693549637721719047775499}{903797337968266405995539176965434889261299} a^{2} - \frac{47806379124997534888933378856771150573911}{903797337968266405995539176965434889261299}$, $\frac{1}{903797337968266405995539176965434889261299} a^{43} + \frac{130177666064561621313887748338781075730}{903797337968266405995539176965434889261299} a^{41} + \frac{99058032285335090257918538339087825780952}{903797337968266405995539176965434889261299} a^{39} + \frac{338186811088381392638801068780181294361127}{903797337968266405995539176965434889261299} a^{37} - \frac{7385849353393373698483025322163291875960}{903797337968266405995539176965434889261299} a^{35} + \frac{32513950234371015415844648090714104490715}{903797337968266405995539176965434889261299} a^{33} + \frac{434324421684235595982463372570948519167017}{903797337968266405995539176965434889261299} a^{31} + \frac{64488271171369193552193798792419153903851}{903797337968266405995539176965434889261299} a^{29} - \frac{136118574919356883203766843615480783007646}{903797337968266405995539176965434889261299} a^{27} + \frac{25684985262706259522704035798228708790782}{903797337968266405995539176965434889261299} a^{25} - \frac{172738058963559142833814800806180982354053}{903797337968266405995539176965434889261299} a^{23} + \frac{34540688792776578338007341838722238008896}{903797337968266405995539176965434889261299} a^{21} + \frac{133504439791512061099760523876135413873000}{903797337968266405995539176965434889261299} a^{19} + \frac{330682891645019761201373867414243647711589}{903797337968266405995539176965434889261299} a^{17} + \frac{287186121623732381928443141302586426709898}{903797337968266405995539176965434889261299} a^{15} + \frac{232948205096083511224887496985959973242831}{903797337968266405995539176965434889261299} a^{13} + \frac{149821569836974546160834695685307324964894}{903797337968266405995539176965434889261299} a^{11} - \frac{232291130789377550800283608790411538854601}{903797337968266405995539176965434889261299} a^{9} + \frac{170086229709210106584174425345470656263400}{903797337968266405995539176965434889261299} a^{7} - \frac{185601737904770903928142946690797135264135}{903797337968266405995539176965434889261299} a^{5} + \frac{72667242694983196693549637721719047775499}{903797337968266405995539176965434889261299} a^{3} - \frac{47806379124997534888933378856771150573911}{903797337968266405995539176965434889261299} a$

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:  \( \frac{209157351019064485193720}{2226090219617027257916699} a^{43} + \frac{12995488274359730586935290}{2226090219617027257916699} a^{41} + \frac{359143284978099066457720105}{2226090219617027257916699} a^{39} + \frac{5839972721540257020635425985}{2226090219617027257916699} a^{37} + \frac{62348418235602803927625805125}{2226090219617027257916699} a^{35} + \frac{462315626899471636311969926105}{2226090219617027257916699} a^{33} + \frac{2459676882912455947235289455830}{2226090219617027257916699} a^{31} + \frac{9577529956515647388992261040325}{2226090219617027257916699} a^{29} + \frac{27616587656070514976128092203875}{2226090219617027257916699} a^{27} + \frac{59278266630197454362918907321225}{2226090219617027257916699} a^{25} + \frac{94532141338580093200503304010726}{2226090219617027257916699} a^{23} + \frac{110623091364929624646755757921827}{2226090219617027257916699} a^{21} + \frac{92010792085643524841366328007535}{2226090219617027257916699} a^{19} + \frac{50081888735077478032214559584484}{2226090219617027257916699} a^{17} + \frac{12880295636412651288925218674382}{2226090219617027257916699} a^{15} - \frac{3673808100033226154684241754493}{2226090219617027257916699} a^{13} - \frac{4851804509885062421451654632776}{2226090219617027257916699} a^{11} - \frac{2053398889912630968727299729775}{2226090219617027257916699} a^{9} - \frac{460153217780433676988443046173}{2226090219617027257916699} a^{7} - \frac{54333828788409951784086810349}{2226090219617027257916699} a^{5} - \frac{2882529697730366774413556719}{2226090219617027257916699} a^{3} - \frac{44310276008005938920339643}{2226090219617027257916699} a \) (order $4$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(i, \sqrt{5})\), \(\Q(\zeta_{23})^+\), 22.0.7198079267989980836471065337135104.1, 22.22.83796671451884098775580820361328125.1, 22.0.351468714257323283030813737164800000000000.1

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 $22^{2}$ R $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$ $22^{2}$ $22^{2}$

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
5Data not computed
23Data not computed