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Properties

Label 44.0.371...625.1
Degree $44$
Signature $[0, 22]$
Discriminant $3.715\times 10^{72}$
Root discriminant \(44.60\)
Ramified primes $5,23$
Class number not computed
Class group not computed
Galois group $C_2\times C_{22}$ (as 44T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 + 2*x^42 - 3*x^41 + 5*x^40 - 8*x^39 + 13*x^38 - 21*x^37 + 34*x^36 - 55*x^35 + 89*x^34 - 144*x^33 + 233*x^32 - 377*x^31 + 610*x^30 - 987*x^29 + 1597*x^28 - 2584*x^27 + 4181*x^26 - 6765*x^25 + 10946*x^24 - 17711*x^23 + 28657*x^22 + 17711*x^21 + 10946*x^20 + 6765*x^19 + 4181*x^18 + 2584*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1)
 
gp: K = bnfinit(y^44 - y^43 + 2*y^42 - 3*y^41 + 5*y^40 - 8*y^39 + 13*y^38 - 21*y^37 + 34*y^36 - 55*y^35 + 89*y^34 - 144*y^33 + 233*y^32 - 377*y^31 + 610*y^30 - 987*y^29 + 1597*y^28 - 2584*y^27 + 4181*y^26 - 6765*y^25 + 10946*y^24 - 17711*y^23 + 28657*y^22 + 17711*y^21 + 10946*y^20 + 6765*y^19 + 4181*y^18 + 2584*y^17 + 1597*y^16 + 987*y^15 + 610*y^14 + 377*y^13 + 233*y^12 + 144*y^11 + 89*y^10 + 55*y^9 + 34*y^8 + 21*y^7 + 13*y^6 + 8*y^5 + 5*y^4 + 3*y^3 + 2*y^2 + y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - x^43 + 2*x^42 - 3*x^41 + 5*x^40 - 8*x^39 + 13*x^38 - 21*x^37 + 34*x^36 - 55*x^35 + 89*x^34 - 144*x^33 + 233*x^32 - 377*x^31 + 610*x^30 - 987*x^29 + 1597*x^28 - 2584*x^27 + 4181*x^26 - 6765*x^25 + 10946*x^24 - 17711*x^23 + 28657*x^22 + 17711*x^21 + 10946*x^20 + 6765*x^19 + 4181*x^18 + 2584*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - x^43 + 2*x^42 - 3*x^41 + 5*x^40 - 8*x^39 + 13*x^38 - 21*x^37 + 34*x^36 - 55*x^35 + 89*x^34 - 144*x^33 + 233*x^32 - 377*x^31 + 610*x^30 - 987*x^29 + 1597*x^28 - 2584*x^27 + 4181*x^26 - 6765*x^25 + 10946*x^24 - 17711*x^23 + 28657*x^22 + 17711*x^21 + 10946*x^20 + 6765*x^19 + 4181*x^18 + 2584*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1)
 

\( x^{44} - x^{43} + 2 x^{42} - 3 x^{41} + 5 x^{40} - 8 x^{39} + 13 x^{38} - 21 x^{37} + 34 x^{36} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 22]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(3714575655453538975253519356486345582985254755453847127268314361572265625\) \(\medspace = 5^{22}\cdot 23^{42}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(44.60\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $5^{1/2}23^{21/22}\approx 44.59807549620821$
Ramified primes:   \(5\), \(23\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $44$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(115=5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{115}(1,·)$, $\chi_{115}(4,·)$, $\chi_{115}(6,·)$, $\chi_{115}(9,·)$, $\chi_{115}(11,·)$, $\chi_{115}(14,·)$, $\chi_{115}(16,·)$, $\chi_{115}(19,·)$, $\chi_{115}(21,·)$, $\chi_{115}(24,·)$, $\chi_{115}(26,·)$, $\chi_{115}(29,·)$, $\chi_{115}(31,·)$, $\chi_{115}(34,·)$, $\chi_{115}(36,·)$, $\chi_{115}(39,·)$, $\chi_{115}(41,·)$, $\chi_{115}(44,·)$, $\chi_{115}(49,·)$, $\chi_{115}(51,·)$, $\chi_{115}(54,·)$, $\chi_{115}(56,·)$, $\chi_{115}(59,·)$, $\chi_{115}(61,·)$, $\chi_{115}(64,·)$, $\chi_{115}(66,·)$, $\chi_{115}(71,·)$, $\chi_{115}(74,·)$, $\chi_{115}(76,·)$, $\chi_{115}(79,·)$, $\chi_{115}(81,·)$, $\chi_{115}(84,·)$, $\chi_{115}(86,·)$, $\chi_{115}(89,·)$, $\chi_{115}(91,·)$, $\chi_{115}(94,·)$, $\chi_{115}(96,·)$, $\chi_{115}(99,·)$, $\chi_{115}(101,·)$, $\chi_{115}(104,·)$, $\chi_{115}(106,·)$, $\chi_{115}(109,·)$, $\chi_{115}(111,·)$, $\chi_{115}(114,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{28657}a^{23}-\frac{10946}{28657}$, $\frac{1}{28657}a^{24}-\frac{10946}{28657}a$, $\frac{1}{28657}a^{25}-\frac{10946}{28657}a^{2}$, $\frac{1}{28657}a^{26}-\frac{10946}{28657}a^{3}$, $\frac{1}{28657}a^{27}-\frac{10946}{28657}a^{4}$, $\frac{1}{28657}a^{28}-\frac{10946}{28657}a^{5}$, $\frac{1}{28657}a^{29}-\frac{10946}{28657}a^{6}$, $\frac{1}{28657}a^{30}-\frac{10946}{28657}a^{7}$, $\frac{1}{28657}a^{31}-\frac{10946}{28657}a^{8}$, $\frac{1}{28657}a^{32}-\frac{10946}{28657}a^{9}$, $\frac{1}{28657}a^{33}-\frac{10946}{28657}a^{10}$, $\frac{1}{28657}a^{34}-\frac{10946}{28657}a^{11}$, $\frac{1}{28657}a^{35}-\frac{10946}{28657}a^{12}$, $\frac{1}{28657}a^{36}-\frac{10946}{28657}a^{13}$, $\frac{1}{28657}a^{37}-\frac{10946}{28657}a^{14}$, $\frac{1}{28657}a^{38}-\frac{10946}{28657}a^{15}$, $\frac{1}{28657}a^{39}-\frac{10946}{28657}a^{16}$, $\frac{1}{28657}a^{40}-\frac{10946}{28657}a^{17}$, $\frac{1}{28657}a^{41}-\frac{10946}{28657}a^{18}$, $\frac{1}{28657}a^{42}-\frac{10946}{28657}a^{19}$, $\frac{1}{28657}a^{43}-\frac{10946}{28657}a^{20}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $21$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{610}{28657} a^{38} - \frac{39088169}{28657} a^{15} \)  (order $46$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 + 2*x^42 - 3*x^41 + 5*x^40 - 8*x^39 + 13*x^38 - 21*x^37 + 34*x^36 - 55*x^35 + 89*x^34 - 144*x^33 + 233*x^32 - 377*x^31 + 610*x^30 - 987*x^29 + 1597*x^28 - 2584*x^27 + 4181*x^26 - 6765*x^25 + 10946*x^24 - 17711*x^23 + 28657*x^22 + 17711*x^21 + 10946*x^20 + 6765*x^19 + 4181*x^18 + 2584*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^44 - x^43 + 2*x^42 - 3*x^41 + 5*x^40 - 8*x^39 + 13*x^38 - 21*x^37 + 34*x^36 - 55*x^35 + 89*x^34 - 144*x^33 + 233*x^32 - 377*x^31 + 610*x^30 - 987*x^29 + 1597*x^28 - 2584*x^27 + 4181*x^26 - 6765*x^25 + 10946*x^24 - 17711*x^23 + 28657*x^22 + 17711*x^21 + 10946*x^20 + 6765*x^19 + 4181*x^18 + 2584*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^44 - x^43 + 2*x^42 - 3*x^41 + 5*x^40 - 8*x^39 + 13*x^38 - 21*x^37 + 34*x^36 - 55*x^35 + 89*x^34 - 144*x^33 + 233*x^32 - 377*x^31 + 610*x^30 - 987*x^29 + 1597*x^28 - 2584*x^27 + 4181*x^26 - 6765*x^25 + 10946*x^24 - 17711*x^23 + 28657*x^22 + 17711*x^21 + 10946*x^20 + 6765*x^19 + 4181*x^18 + 2584*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - x^43 + 2*x^42 - 3*x^41 + 5*x^40 - 8*x^39 + 13*x^38 - 21*x^37 + 34*x^36 - 55*x^35 + 89*x^34 - 144*x^33 + 233*x^32 - 377*x^31 + 610*x^30 - 987*x^29 + 1597*x^28 - 2584*x^27 + 4181*x^26 - 6765*x^25 + 10946*x^24 - 17711*x^23 + 28657*x^22 + 17711*x^21 + 10946*x^20 + 6765*x^19 + 4181*x^18 + 2584*x^17 + 1597*x^16 + 987*x^15 + 610*x^14 + 377*x^13 + 233*x^12 + 144*x^11 + 89*x^10 + 55*x^9 + 34*x^8 + 21*x^7 + 13*x^6 + 8*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2\times C_{22}$ (as 44T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
An abelian group of order 44
The 44 conjugacy class representatives for $C_2\times C_{22}$
Character table for $C_2\times C_{22}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-115}) \), \(\Q(\sqrt{5}, \sqrt{-23})\), \(\Q(\zeta_{23})^+\), 22.22.83796671451884098775580820361328125.1, \(\Q(\zeta_{23})\), 22.0.1927323443393334271838358868310546875.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $22^{2}$ $22^{2}$ R $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ R ${\href{/padicField/29.11.0.1}{11} }^{4}$ ${\href{/padicField/31.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/padicField/41.11.0.1}{11} }^{4}$ $22^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{22}$ $22^{2}$ ${\href{/padicField/59.11.0.1}{11} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display Deg $44$$2$$22$$22$
\(23\) Copy content Toggle raw display Deg $44$$22$$2$$42$