↑

Properties

Label 44.0.31907961917...0000.1
Degree $44$
Signature $[0, 22]$
Discriminant $2^{44}\cdot 5^{33}\cdot 23^{42}$
Root discriminant $133.38$
Ramified primes $2, 5, 23$
Class number Not computed
Class group Not computed
Galois group $C_{44}$ (as 44T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25830078125, 0, 568261718750, 0, 4716572265625, 0, 20684726562500, 0, 55553265625000, 0, 99649341796875, 0, 126351650000000, 0, 117775857421875, 0, 82991007812500, 0, 45106755468750, 0, 19182837703125, 0, 6445987906250, 0, 1721656375000, 0, 366389353125, 0, 62064479375, 0, 8329148125, 0, 877527625, 0, 71512750, 0, 4405075, 0, 197800, 0, 6095, 0, 115, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^44 + 115*x^42 + 6095*x^40 + 197800*x^38 + 4405075*x^36 + 71512750*x^34 + 877527625*x^32 + 8329148125*x^30 + 62064479375*x^28 + 366389353125*x^26 + 1721656375000*x^24 + 6445987906250*x^22 + 19182837703125*x^20 + 45106755468750*x^18 + 82991007812500*x^16 + 117775857421875*x^14 + 126351650000000*x^12 + 99649341796875*x^10 + 55553265625000*x^8 + 20684726562500*x^6 + 4716572265625*x^4 + 568261718750*x^2 + 25830078125)
 
gp: K = bnfinit(x^44 + 115*x^42 + 6095*x^40 + 197800*x^38 + 4405075*x^36 + 71512750*x^34 + 877527625*x^32 + 8329148125*x^30 + 62064479375*x^28 + 366389353125*x^26 + 1721656375000*x^24 + 6445987906250*x^22 + 19182837703125*x^20 + 45106755468750*x^18 + 82991007812500*x^16 + 117775857421875*x^14 + 126351650000000*x^12 + 99649341796875*x^10 + 55553265625000*x^8 + 20684726562500*x^6 + 4716572265625*x^4 + 568261718750*x^2 + 25830078125, 1)
 

Normalized defining polynomial

\( x^{44} + 115 x^{42} + 6095 x^{40} + 197800 x^{38} + 4405075 x^{36} + 71512750 x^{34} + 877527625 x^{32} + 8329148125 x^{30} + 62064479375 x^{28} + 366389353125 x^{26} + 1721656375000 x^{24} + 6445987906250 x^{22} + 19182837703125 x^{20} + 45106755468750 x^{18} + 82991007812500 x^{16} + 117775857421875 x^{14} + 126351650000000 x^{12} + 99649341796875 x^{10} + 55553265625000 x^{8} + 20684726562500 x^{6} + 4716572265625 x^{4} + 568261718750 x^{2} + 25830078125 \)

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

Invariants

Degree:  $44$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 22]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3190796191738142789235043789002363949895144644980550209800192000000000000000000000000000000000=2^{44}\cdot 5^{33}\cdot 23^{42}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $133.38$
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:  $2, 5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(460=2^{2}\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{460}(1,·)$, $\chi_{460}(387,·)$, $\chi_{460}(261,·)$, $\chi_{460}(7,·)$, $\chi_{460}(9,·)$, $\chi_{460}(267,·)$, $\chi_{460}(269,·)$, $\chi_{460}(143,·)$, $\chi_{460}(107,·)$, $\chi_{460}(409,·)$, $\chi_{460}(283,·)$, $\chi_{460}(29,·)$, $\chi_{460}(287,·)$, $\chi_{460}(289,·)$, $\chi_{460}(41,·)$, $\chi_{460}(43,·)$, $\chi_{460}(301,·)$, $\chi_{460}(49,·)$, $\chi_{460}(183,·)$, $\chi_{460}(441,·)$, $\chi_{460}(63,·)$, $\chi_{460}(449,·)$, $\chi_{460}(67,·)$, $\chi_{460}(327,·)$, $\chi_{460}(203,·)$, $\chi_{460}(141,·)$, $\chi_{460}(209,·)$, $\chi_{460}(83,·)$, $\chi_{460}(343,·)$, $\chi_{460}(349,·)$, $\chi_{460}(263,·)$, $\chi_{460}(227,·)$, $\chi_{460}(101,·)$, $\chi_{460}(81,·)$, $\chi_{460}(361,·)$, $\chi_{460}(103,·)$, $\chi_{460}(367,·)$, $\chi_{460}(369,·)$, $\chi_{460}(169,·)$, $\chi_{460}(121,·)$, $\chi_{460}(447,·)$, $\chi_{460}(247,·)$, $\chi_{460}(381,·)$, $\chi_{460}(383,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{125} a^{14}$, $\frac{1}{125} a^{15}$, $\frac{1}{625} a^{16}$, $\frac{1}{625} a^{17}$, $\frac{1}{625} a^{18}$, $\frac{1}{625} a^{19}$, $\frac{1}{3125} a^{20}$, $\frac{1}{3125} a^{21}$, $\frac{1}{71875} a^{22}$, $\frac{1}{71875} a^{23}$, $\frac{1}{359375} a^{24}$, $\frac{1}{359375} a^{25}$, $\frac{1}{359375} a^{26}$, $\frac{1}{359375} a^{27}$, $\frac{1}{1796875} a^{28}$, $\frac{1}{1796875} a^{29}$, $\frac{1}{1796875} a^{30}$, $\frac{1}{1796875} a^{31}$, $\frac{1}{8984375} a^{32}$, $\frac{1}{8984375} a^{33}$, $\frac{1}{8984375} a^{34}$, $\frac{1}{8984375} a^{35}$, $\frac{1}{44921875} a^{36}$, $\frac{1}{44921875} a^{37}$, $\frac{1}{44921875} a^{38}$, $\frac{1}{44921875} a^{39}$, $\frac{1}{31220703125} a^{40} + \frac{38}{6244140625} a^{38} + \frac{57}{6244140625} a^{36} + \frac{37}{1248828125} a^{34} + \frac{56}{1248828125} a^{32} - \frac{8}{249765625} a^{30} - \frac{66}{249765625} a^{28} - \frac{18}{49953125} a^{26} - \frac{64}{49953125} a^{24} - \frac{12}{9990625} a^{22} - \frac{13}{86875} a^{20} - \frac{61}{86875} a^{18} - \frac{4}{86875} a^{16} - \frac{61}{17375} a^{14} + \frac{11}{17375} a^{12} - \frac{7}{3475} a^{10} - \frac{14}{3475} a^{8} + \frac{54}{695} a^{6} + \frac{12}{695} a^{4} + \frac{29}{139} a^{2} + \frac{50}{139}$, $\frac{1}{31220703125} a^{41} + \frac{38}{6244140625} a^{39} + \frac{57}{6244140625} a^{37} + \frac{37}{1248828125} a^{35} + \frac{56}{1248828125} a^{33} - \frac{8}{249765625} a^{31} - \frac{66}{249765625} a^{29} - \frac{18}{49953125} a^{27} - \frac{64}{49953125} a^{25} - \frac{12}{9990625} a^{23} - \frac{13}{86875} a^{21} - \frac{61}{86875} a^{19} - \frac{4}{86875} a^{17} - \frac{61}{17375} a^{15} + \frac{11}{17375} a^{13} - \frac{7}{3475} a^{11} - \frac{14}{3475} a^{9} + \frac{54}{695} a^{7} + \frac{12}{695} a^{5} + \frac{29}{139} a^{3} + \frac{50}{139} a$, $\frac{1}{13345254614814453125} a^{42} + \frac{16407932}{13345254614814453125} a^{40} + \frac{18964067483}{2669050922962890625} a^{38} - \frac{4118885674}{2669050922962890625} a^{36} - \frac{926153448}{21352407383703125} a^{34} + \frac{601820938}{533810184592578125} a^{32} - \frac{11481904146}{106762036918515625} a^{30} - \frac{165538382}{4641827692109375} a^{28} + \frac{1237569152}{4270481476740625} a^{26} - \frac{17097125271}{21352407383703125} a^{24} - \frac{13064830797}{4270481476740625} a^{22} - \frac{17927745913}{185673107684375} a^{20} - \frac{12866578354}{37134621536875} a^{18} + \frac{1605500536}{37134621536875} a^{16} + \frac{23650877134}{7426924307375} a^{14} - \frac{4879993064}{1485384861475} a^{12} + \frac{8842120783}{1485384861475} a^{10} + \frac{4770921064}{297076972295} a^{8} + \frac{15693037004}{297076972295} a^{6} + \frac{22412302837}{297076972295} a^{4} + \frac{7428755939}{59415394459} a^{2} - \frac{17364519196}{59415394459}$, $\frac{1}{13345254614814453125} a^{43} + \frac{16407932}{13345254614814453125} a^{41} + \frac{18964067483}{2669050922962890625} a^{39} - \frac{4118885674}{2669050922962890625} a^{37} - \frac{926153448}{21352407383703125} a^{35} + \frac{601820938}{533810184592578125} a^{33} - \frac{11481904146}{106762036918515625} a^{31} - \frac{165538382}{4641827692109375} a^{29} + \frac{1237569152}{4270481476740625} a^{27} - \frac{17097125271}{21352407383703125} a^{25} - \frac{13064830797}{4270481476740625} a^{23} - \frac{17927745913}{185673107684375} a^{21} - \frac{12866578354}{37134621536875} a^{19} + \frac{1605500536}{37134621536875} a^{17} + \frac{23650877134}{7426924307375} a^{15} - \frac{4879993064}{1485384861475} a^{13} + \frac{8842120783}{1485384861475} a^{11} + \frac{4770921064}{297076972295} a^{9} + \frac{15693037004}{297076972295} a^{7} + \frac{22412302837}{297076972295} a^{5} + \frac{7428755939}{59415394459} a^{3} - \frac{17364519196}{59415394459} a$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $21$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{5}) \), 4.0.1058000.1, \(\Q(\zeta_{23})^+\), 22.22.83796671451884098775580820361328125.1

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 $44$ R $44$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{4}$ $44$ $44$ $22^{2}$ R $22^{2}$ $22^{2}$ $44$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{4}$ $44$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{11}$ $44$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{4}$

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
5Data not computed
23Data not computed