Properties

Label 44.44.181...125.1
Degree $44$
Signature $[44, 0]$
Discriminant $1.814\times 10^{80}$
Root discriminant $66.69$
Ramified primes $5, 23$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{44}$ (as 44T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 - 44*x^42 + 43*x^41 + 902*x^40 - 859*x^39 - 11441*x^38 + 10582*x^37 + 100567*x^36 - 89985*x^35 - 650236*x^34 + 560251*x^33 + 3203650*x^32 - 2643399*x^31 - 12294569*x^30 + 9651170*x^29 + 37253225*x^28 - 27602055*x^27 - 89808680*x^26 + 62206625*x^25 + 172779011*x^24 - 110572386*x^23 - 265002643*x^22 + 154430258*x^21 + 322457859*x^20 - 168027624*x^19 - 308473797*x^18 + 140446403*x^17 + 228768475*x^16 - 88323383*x^15 - 128871012*x^14 + 40552321*x^13 + 53558102*x^12 - 13016729*x^11 - 15742859*x^10 + 2742874*x^9 + 3073662*x^8 - 347233*x^7 - 361455*x^6 + 24089*x^5 + 21786*x^4 - 986*x^3 - 504*x^2 + 24*x + 1)
 
gp: K = bnfinit(x^44 - x^43 - 44*x^42 + 43*x^41 + 902*x^40 - 859*x^39 - 11441*x^38 + 10582*x^37 + 100567*x^36 - 89985*x^35 - 650236*x^34 + 560251*x^33 + 3203650*x^32 - 2643399*x^31 - 12294569*x^30 + 9651170*x^29 + 37253225*x^28 - 27602055*x^27 - 89808680*x^26 + 62206625*x^25 + 172779011*x^24 - 110572386*x^23 - 265002643*x^22 + 154430258*x^21 + 322457859*x^20 - 168027624*x^19 - 308473797*x^18 + 140446403*x^17 + 228768475*x^16 - 88323383*x^15 - 128871012*x^14 + 40552321*x^13 + 53558102*x^12 - 13016729*x^11 - 15742859*x^10 + 2742874*x^9 + 3073662*x^8 - 347233*x^7 - 361455*x^6 + 24089*x^5 + 21786*x^4 - 986*x^3 - 504*x^2 + 24*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 24, -504, -986, 21786, 24089, -361455, -347233, 3073662, 2742874, -15742859, -13016729, 53558102, 40552321, -128871012, -88323383, 228768475, 140446403, -308473797, -168027624, 322457859, 154430258, -265002643, -110572386, 172779011, 62206625, -89808680, -27602055, 37253225, 9651170, -12294569, -2643399, 3203650, 560251, -650236, -89985, 100567, 10582, -11441, -859, 902, 43, -44, -1, 1]);
 

\(x^{44} - x^{43} - 44 x^{42} + 43 x^{41} + 902 x^{40} - 859 x^{39} - 11441 x^{38} + 10582 x^{37} + 100567 x^{36} - 89985 x^{35} - 650236 x^{34} + 560251 x^{33} + 3203650 x^{32} - 2643399 x^{31} - 12294569 x^{30} + 9651170 x^{29} + 37253225 x^{28} - 27602055 x^{27} - 89808680 x^{26} + 62206625 x^{25} + 172779011 x^{24} - 110572386 x^{23} - 265002643 x^{22} + 154430258 x^{21} + 322457859 x^{20} - 168027624 x^{19} - 308473797 x^{18} + 140446403 x^{17} + 228768475 x^{16} - 88323383 x^{15} - 128871012 x^{14} + 40552321 x^{13} + 53558102 x^{12} - 13016729 x^{11} - 15742859 x^{10} + 2742874 x^{9} + 3073662 x^{8} - 347233 x^{7} - 361455 x^{6} + 24089 x^{5} + 21786 x^{4} - 986 x^{3} - 504 x^{2} + 24 x + 1\)  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:  $[44, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(181\!\cdots\!125\)\(\medspace = 5^{33}\cdot 23^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $66.69$
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:  $5, 23$
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:  \(115=5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{115}(1,·)$, $\chi_{115}(4,·)$, $\chi_{115}(6,·)$, $\chi_{115}(7,·)$, $\chi_{115}(9,·)$, $\chi_{115}(16,·)$, $\chi_{115}(17,·)$, $\chi_{115}(22,·)$, $\chi_{115}(24,·)$, $\chi_{115}(26,·)$, $\chi_{115}(28,·)$, $\chi_{115}(29,·)$, $\chi_{115}(31,·)$, $\chi_{115}(33,·)$, $\chi_{115}(36,·)$, $\chi_{115}(37,·)$, $\chi_{115}(38,·)$, $\chi_{115}(39,·)$, $\chi_{115}(41,·)$, $\chi_{115}(42,·)$, $\chi_{115}(43,·)$, $\chi_{115}(49,·)$, $\chi_{115}(53,·)$, $\chi_{115}(54,·)$, $\chi_{115}(57,·)$, $\chi_{115}(59,·)$, $\chi_{115}(63,·)$, $\chi_{115}(64,·)$, $\chi_{115}(67,·)$, $\chi_{115}(68,·)$, $\chi_{115}(71,·)$, $\chi_{115}(81,·)$, $\chi_{115}(83,·)$, $\chi_{115}(88,·)$, $\chi_{115}(94,·)$, $\chi_{115}(96,·)$, $\chi_{115}(97,·)$, $\chi_{115}(101,·)$, $\chi_{115}(102,·)$, $\chi_{115}(103,·)$, $\chi_{115}(104,·)$, $\chi_{115}(107,·)$, $\chi_{115}(112,·)$, $\chi_{115}(113,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$  Toggle raw display

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 $2$)  Toggle raw display
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:  \( 183282479706502550000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{44}\cdot(2\pi)^{0}\cdot 183282479706502550000000000 \cdot 1}{2\sqrt{181375764426442332776050749828434842919201892356144879261148162186145782470703125}}\approx 0.119707438429760$ (assuming GRH)

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.66125.1, \(\Q(\zeta_{23})^+\), 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 $44$ $44$ R $44$ $22^{2}$ $44$ $44$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{4}$ R $22^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{4}$ $44$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{4}$ $44$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{11}$ $44$ $22^{2}$

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
5Data not computed
23Data not computed