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Properties

Label 44.44.653...000.1
Degree $44$
Signature $[44, 0]$
Discriminant $6.535\times 10^{85}$
Root discriminant $89.20$
Ramified primes $2, 5, 23$
Class number $1$ (GRH)
Class group trivial (GRH)
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 - 69*x^42 + 2139*x^40 - 39468*x^38 + 484495*x^36 - 4193682*x^34 + 26499197*x^32 - 125010267*x^30 + 447055439*x^28 - 1224773115*x^26 + 2588447416*x^24 - 4235761650*x^22 + 5369828745*x^20 - 5258413410*x^18 + 3950328660*x^16 - 2250421545*x^14 + 955257620*x^12 - 294507525*x^10 + 63543480*x^8 - 9077640*x^6 + 785565*x^4 - 34914*x^2 + 529)
 
gp: K = bnfinit(x^44 - 69*x^42 + 2139*x^40 - 39468*x^38 + 484495*x^36 - 4193682*x^34 + 26499197*x^32 - 125010267*x^30 + 447055439*x^28 - 1224773115*x^26 + 2588447416*x^24 - 4235761650*x^22 + 5369828745*x^20 - 5258413410*x^18 + 3950328660*x^16 - 2250421545*x^14 + 955257620*x^12 - 294507525*x^10 + 63543480*x^8 - 9077640*x^6 + 785565*x^4 - 34914*x^2 + 529, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![529, 0, -34914, 0, 785565, 0, -9077640, 0, 63543480, 0, -294507525, 0, 955257620, 0, -2250421545, 0, 3950328660, 0, -5258413410, 0, 5369828745, 0, -4235761650, 0, 2588447416, 0, -1224773115, 0, 447055439, 0, -125010267, 0, 26499197, 0, -4193682, 0, 484495, 0, -39468, 0, 2139, 0, -69, 0, 1]);
 

\( x^{44} - 69 x^{42} + 2139 x^{40} - 39468 x^{38} + 484495 x^{36} - 4193682 x^{34} + 26499197 x^{32} - 125010267 x^{30} + 447055439 x^{28} - 1224773115 x^{26} + 2588447416 x^{24} - 4235761650 x^{22} + 5369828745 x^{20} - 5258413410 x^{18} + 3950328660 x^{16} - 2250421545 x^{14} + 955257620 x^{12} - 294507525 x^{10} + 63543480 x^{8} - 9077640 x^{6} + 785565 x^{4} - 34914 x^{2} + 529 \)

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

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[44, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(653\!\cdots\!000\)\(\medspace = 2^{44}\cdot 5^{22}\cdot 23^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $89.20$
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$$\chi_{460}(1,·)$, $\chi_{460}(261,·)$, $\chi_{460}(9,·)$, $\chi_{460}(11,·)$, $\chi_{460}(141,·)$, $\chi_{460}(19,·)$, $\chi_{460}(409,·)$, $\chi_{460}(411,·)$, $\chi_{460}(29,·)$, $\chi_{460}(159,·)$, $\chi_{460}(289,·)$, $\chi_{460}(291,·)$, $\chi_{460}(41,·)$, $\chi_{460}(171,·)$, $\chi_{460}(301,·)$, $\chi_{460}(431,·)$, $\chi_{460}(49,·)$, $\chi_{460}(91,·)$, $\chi_{460}(51,·)$, $\chi_{460}(441,·)$, $\chi_{460}(319,·)$, $\chi_{460}(449,·)$, $\chi_{460}(451,·)$, $\chi_{460}(199,·)$, $\chi_{460}(459,·)$, $\chi_{460}(269,·)$, $\chi_{460}(209,·)$, $\chi_{460}(419,·)$, $\chi_{460}(79,·)$, $\chi_{460}(349,·)$, $\chi_{460}(99,·)$, $\chi_{460}(101,·)$, $\chi_{460}(81,·)$, $\chi_{460}(379,·)$, $\chi_{460}(361,·)$, $\chi_{460}(359,·)$, $\chi_{460}(191,·)$, $\chi_{460}(111,·)$, $\chi_{460}(369,·)$, $\chi_{460}(339,·)$, $\chi_{460}(169,·)$, $\chi_{460}(121,·)$, $\chi_{460}(251,·)$, $\chi_{460}(381,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{23} a^{22}$, $\frac{1}{23} a^{23}$, $\frac{1}{23} a^{24}$, $\frac{1}{23} a^{25}$, $\frac{1}{23} a^{26}$, $\frac{1}{23} a^{27}$, $\frac{1}{23} a^{28}$, $\frac{1}{23} a^{29}$, $\frac{1}{23} a^{30}$, $\frac{1}{23} a^{31}$, $\frac{1}{23} a^{32}$, $\frac{1}{23} a^{33}$, $\frac{1}{23} a^{34}$, $\frac{1}{23} a^{35}$, $\frac{1}{23} a^{36}$, $\frac{1}{23} a^{37}$, $\frac{1}{23} a^{38}$, $\frac{1}{23} a^{39}$, $\frac{1}{23} a^{40}$, $\frac{1}{23} a^{41}$, $\frac{1}{119065662966459206469142305042008031773} a^{42} + \frac{2366306122525813891090010977660989726}{119065662966459206469142305042008031773} a^{40} + \frac{659241420377185617287538190701293598}{119065662966459206469142305042008031773} a^{38} - \frac{823722461331763048903311289996699497}{119065662966459206469142305042008031773} a^{36} - \frac{74633285535024869919254267388543196}{5176767955063443759527926306174262251} a^{34} - \frac{1102018233721431688549009340392226645}{119065662966459206469142305042008031773} a^{32} + \frac{692472151679804406039322617839847398}{119065662966459206469142305042008031773} a^{30} - \frac{854781014505855961290814810941202388}{119065662966459206469142305042008031773} a^{28} - \frac{699804487662845403224171832518458002}{119065662966459206469142305042008031773} a^{26} - \frac{1036494116831807342490339593844649112}{119065662966459206469142305042008031773} a^{24} - \frac{1395801635517096940389950280632949041}{119065662966459206469142305042008031773} a^{22} + \frac{1698051835025592547271696061539685296}{5176767955063443759527926306174262251} a^{20} - \frac{1318566500120216940814401688536628461}{5176767955063443759527926306174262251} a^{18} - \frac{199714160702359264876728874507936990}{5176767955063443759527926306174262251} a^{16} - \frac{1434525147276707909459488885610419622}{5176767955063443759527926306174262251} a^{14} - \frac{948912050754673992699344456400037232}{5176767955063443759527926306174262251} a^{12} - \frac{28562325843893428536399586216427882}{5176767955063443759527926306174262251} a^{10} + \frac{2536880362138947577469682819431071366}{5176767955063443759527926306174262251} a^{8} - \frac{2123925454680881557819778079015500742}{5176767955063443759527926306174262251} a^{6} + \frac{605187513823834534390531883007438827}{5176767955063443759527926306174262251} a^{4} + \frac{1044093066154803695826250555500293316}{5176767955063443759527926306174262251} a^{2} - \frac{1655220652593536351987183524758208257}{5176767955063443759527926306174262251}$, $\frac{1}{119065662966459206469142305042008031773} a^{43} + \frac{2366306122525813891090010977660989726}{119065662966459206469142305042008031773} a^{41} + \frac{659241420377185617287538190701293598}{119065662966459206469142305042008031773} a^{39} - \frac{823722461331763048903311289996699497}{119065662966459206469142305042008031773} a^{37} - \frac{74633285535024869919254267388543196}{5176767955063443759527926306174262251} a^{35} - \frac{1102018233721431688549009340392226645}{119065662966459206469142305042008031773} a^{33} + \frac{692472151679804406039322617839847398}{119065662966459206469142305042008031773} a^{31} - \frac{854781014505855961290814810941202388}{119065662966459206469142305042008031773} a^{29} - \frac{699804487662845403224171832518458002}{119065662966459206469142305042008031773} a^{27} - \frac{1036494116831807342490339593844649112}{119065662966459206469142305042008031773} a^{25} - \frac{1395801635517096940389950280632949041}{119065662966459206469142305042008031773} a^{23} + \frac{1698051835025592547271696061539685296}{5176767955063443759527926306174262251} a^{21} - \frac{1318566500120216940814401688536628461}{5176767955063443759527926306174262251} a^{19} - \frac{199714160702359264876728874507936990}{5176767955063443759527926306174262251} a^{17} - \frac{1434525147276707909459488885610419622}{5176767955063443759527926306174262251} a^{15} - \frac{948912050754673992699344456400037232}{5176767955063443759527926306174262251} a^{13} - \frac{28562325843893428536399586216427882}{5176767955063443759527926306174262251} a^{11} + \frac{2536880362138947577469682819431071366}{5176767955063443759527926306174262251} a^{9} - \frac{2123925454680881557819778079015500742}{5176767955063443759527926306174262251} a^{7} + \frac{605187513823834534390531883007438827}{5176767955063443759527926306174262251} a^{5} + \frac{1044093066154803695826250555500293316}{5176767955063443759527926306174262251} a^{3} - \frac{1655220652593536351987183524758208257}{5176767955063443759527926306174262251} a$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $43$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 90452671457773850000000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{44}\cdot(2\pi)^{0}\cdot 90452671457773850000000000000 \cdot 1}{2\sqrt{65347506006797164323533696798768413693852562329201668296707932160000000000000000000000}}\approx 0.0984230236514070$ (assuming GRH)

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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{23}) \), \(\Q(\sqrt{115}) \), \(\Q(\sqrt{5}, \sqrt{23})\), \(\Q(\zeta_{23})^+\), 22.22.83796671451884098775580820361328125.1, \(\Q(\zeta_{92})^+\), 22.22.8083780427918435509708715954790400000000000.1

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 $22^{2}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{4}$ 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