Properties

Label 44.44.666...569.1
Degree $44$
Signature $[44, 0]$
Discriminant $6.665\times 10^{83}$
Root discriminant $80.37$
Ramified prime $89$
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 - 43*x^42 + 42*x^41 + 861*x^40 - 820*x^39 - 10660*x^38 + 9880*x^37 + 91390*x^36 - 82251*x^35 - 575757*x^34 + 501942*x^33 + 2760681*x^32 - 2324784*x^31 - 10295472*x^30 + 8347680*x^29 + 30260340*x^28 - 23535820*x^27 - 70607460*x^26 + 52451256*x^25 + 131128140*x^24 - 92561040*x^23 - 193536720*x^22 + 129024480*x^21 + 225792840*x^20 - 141120525*x^19 - 206253075*x^18 + 119759850*x^17 + 145422675*x^16 - 77558760*x^15 - 77558760*x^14 + 37442160*x^13 + 30421755*x^12 - 13037895*x^11 - 8436285*x^10 + 3124550*x^9 + 1562275*x^8 - 480700*x^7 - 177100*x^6 + 42504*x^5 + 10626*x^4 - 1771*x^3 - 253*x^2 + 22*x + 1)
 
gp: K = bnfinit(x^44 - x^43 - 43*x^42 + 42*x^41 + 861*x^40 - 820*x^39 - 10660*x^38 + 9880*x^37 + 91390*x^36 - 82251*x^35 - 575757*x^34 + 501942*x^33 + 2760681*x^32 - 2324784*x^31 - 10295472*x^30 + 8347680*x^29 + 30260340*x^28 - 23535820*x^27 - 70607460*x^26 + 52451256*x^25 + 131128140*x^24 - 92561040*x^23 - 193536720*x^22 + 129024480*x^21 + 225792840*x^20 - 141120525*x^19 - 206253075*x^18 + 119759850*x^17 + 145422675*x^16 - 77558760*x^15 - 77558760*x^14 + 37442160*x^13 + 30421755*x^12 - 13037895*x^11 - 8436285*x^10 + 3124550*x^9 + 1562275*x^8 - 480700*x^7 - 177100*x^6 + 42504*x^5 + 10626*x^4 - 1771*x^3 - 253*x^2 + 22*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 22, -253, -1771, 10626, 42504, -177100, -480700, 1562275, 3124550, -8436285, -13037895, 30421755, 37442160, -77558760, -77558760, 145422675, 119759850, -206253075, -141120525, 225792840, 129024480, -193536720, -92561040, 131128140, 52451256, -70607460, -23535820, 30260340, 8347680, -10295472, -2324784, 2760681, 501942, -575757, -82251, 91390, 9880, -10660, -820, 861, 42, -43, -1, 1]);
 

\( x^{44} - x^{43} - 43 x^{42} + 42 x^{41} + 861 x^{40} - 820 x^{39} - 10660 x^{38} + 9880 x^{37} + 91390 x^{36} - 82251 x^{35} - 575757 x^{34} + 501942 x^{33} + 2760681 x^{32} - 2324784 x^{31} - 10295472 x^{30} + 8347680 x^{29} + 30260340 x^{28} - 23535820 x^{27} - 70607460 x^{26} + 52451256 x^{25} + 131128140 x^{24} - 92561040 x^{23} - 193536720 x^{22} + 129024480 x^{21} + 225792840 x^{20} - 141120525 x^{19} - 206253075 x^{18} + 119759850 x^{17} + 145422675 x^{16} - 77558760 x^{15} - 77558760 x^{14} + 37442160 x^{13} + 30421755 x^{12} - 13037895 x^{11} - 8436285 x^{10} + 3124550 x^{9} + 1562275 x^{8} - 480700 x^{7} - 177100 x^{6} + 42504 x^{5} + 10626 x^{4} - 1771 x^{3} - 253 x^{2} + 22 x + 1 \)

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

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[44, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(666\!\cdots\!569\)\(\medspace = 89^{43}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $80.37$
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:  $89$
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:  \(89\)
Dirichlet character group:    $\lbrace$$\chi_{89}(1,·)$, $\chi_{89}(2,·)$, $\chi_{89}(4,·)$, $\chi_{89}(5,·)$, $\chi_{89}(8,·)$, $\chi_{89}(9,·)$, $\chi_{89}(10,·)$, $\chi_{89}(11,·)$, $\chi_{89}(16,·)$, $\chi_{89}(17,·)$, $\chi_{89}(18,·)$, $\chi_{89}(20,·)$, $\chi_{89}(21,·)$, $\chi_{89}(22,·)$, $\chi_{89}(25,·)$, $\chi_{89}(32,·)$, $\chi_{89}(34,·)$, $\chi_{89}(36,·)$, $\chi_{89}(39,·)$, $\chi_{89}(40,·)$, $\chi_{89}(42,·)$, $\chi_{89}(44,·)$, $\chi_{89}(45,·)$, $\chi_{89}(47,·)$, $\chi_{89}(49,·)$, $\chi_{89}(50,·)$, $\chi_{89}(53,·)$, $\chi_{89}(55,·)$, $\chi_{89}(57,·)$, $\chi_{89}(64,·)$, $\chi_{89}(67,·)$, $\chi_{89}(68,·)$, $\chi_{89}(69,·)$, $\chi_{89}(71,·)$, $\chi_{89}(72,·)$, $\chi_{89}(73,·)$, $\chi_{89}(78,·)$, $\chi_{89}(79,·)$, $\chi_{89}(80,·)$, $\chi_{89}(81,·)$, $\chi_{89}(84,·)$, $\chi_{89}(85,·)$, $\chi_{89}(87,·)$, $\chi_{89}(88,·)$$\rbrace$
This is not a CM field.

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

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

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:  \( 9928065388501928000000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{44}\cdot(2\pi)^{0}\cdot 9928065388501928000000000000 \cdot 1}{2\sqrt{666454163935483494165986073535521413339908119119439689887653437787720225729135378569}}\approx 0.106971798982240$ (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

\(\Q(\sqrt{89}) \), 4.4.704969.1, 11.11.31181719929966183601.1, 22.22.86534669543385676516186776267386878120889.1

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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{4}$ $44$ $22^{2}$ $44$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{4}$ $44$ $22^{2}$ $44$ $44$ $44$ $44$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{11}$ $44$ $44$ $22^{2}$ $22^{2}$ $44$

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
89Data not computed