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Properties

Label 44.44.133...000.1
Degree $44$
Signature $[44, 0]$
Discriminant $1.338\times 10^{100}$
Root discriminant \(188.63\)
Ramified primes $2,5,23$
Class number not computed
Class group not computed
Galois group $C_{44}$ (as 44T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - 230*x^42 + 24380*x^40 - 1582400*x^38 + 70481200*x^36 - 2288408000*x^34 + 56161768000*x^32 - 1066130960000*x^30 + 15888506720000*x^28 - 187591348800000*x^26 + 1762976128000000*x^24 - 13201383232000000*x^22 + 78572903232000000*x^20 - 369514540800000000*x^18 + 1359724672000000000*x^16 - 3859279296000000000*x^14 + 8280581734400000000*x^12 - 13061238528000000000*x^10 + 14562955264000000000*x^8 - 10844753920000000000*x^6 + 4945684480000000000*x^4 - 1191731200000000000*x^2 + 108339200000000000)
 
gp: K = bnfinit(y^44 - 230*y^42 + 24380*y^40 - 1582400*y^38 + 70481200*y^36 - 2288408000*y^34 + 56161768000*y^32 - 1066130960000*y^30 + 15888506720000*y^28 - 187591348800000*y^26 + 1762976128000000*y^24 - 13201383232000000*y^22 + 78572903232000000*y^20 - 369514540800000000*y^18 + 1359724672000000000*y^16 - 3859279296000000000*y^14 + 8280581734400000000*y^12 - 13061238528000000000*y^10 + 14562955264000000000*y^8 - 10844753920000000000*y^6 + 4945684480000000000*y^4 - 1191731200000000000*y^2 + 108339200000000000, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - 230*x^42 + 24380*x^40 - 1582400*x^38 + 70481200*x^36 - 2288408000*x^34 + 56161768000*x^32 - 1066130960000*x^30 + 15888506720000*x^28 - 187591348800000*x^26 + 1762976128000000*x^24 - 13201383232000000*x^22 + 78572903232000000*x^20 - 369514540800000000*x^18 + 1359724672000000000*x^16 - 3859279296000000000*x^14 + 8280581734400000000*x^12 - 13061238528000000000*x^10 + 14562955264000000000*x^8 - 10844753920000000000*x^6 + 4945684480000000000*x^4 - 1191731200000000000*x^2 + 108339200000000000);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - 230*x^42 + 24380*x^40 - 1582400*x^38 + 70481200*x^36 - 2288408000*x^34 + 56161768000*x^32 - 1066130960000*x^30 + 15888506720000*x^28 - 187591348800000*x^26 + 1762976128000000*x^24 - 13201383232000000*x^22 + 78572903232000000*x^20 - 369514540800000000*x^18 + 1359724672000000000*x^16 - 3859279296000000000*x^14 + 8280581734400000000*x^12 - 13061238528000000000*x^10 + 14562955264000000000*x^8 - 10844753920000000000*x^6 + 4945684480000000000*x^4 - 1191731200000000000*x^2 + 108339200000000000)
 

\( x^{44} - 230 x^{42} + 24380 x^{40} - 1582400 x^{38} + 70481200 x^{36} - 2288408000 x^{34} + \cdots + 10\!\cdots\!00 \) 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:  $[44, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(133\!\cdots\!000\) \(\medspace = 2^{66}\cdot 5^{33}\cdot 23^{42}\) 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:  \(188.63\)
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:  $2^{3/2}5^{3/4}23^{21/22}\approx 188.62689373777334$
Ramified primes:   \(2\), \(5\), \(23\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:  $44$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(920=2^{3}\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{920}(1,·)$, $\chi_{920}(773,·)$, $\chi_{920}(9,·)$, $\chi_{920}(209,·)$, $\chi_{920}(493,·)$, $\chi_{920}(917,·)$, $\chi_{920}(601,·)$, $\chi_{920}(409,·)$, $\chi_{920}(797,·)$, $\chi_{920}(517,·)$, $\chi_{920}(289,·)$, $\chi_{920}(37,·)$, $\chi_{920}(561,·)$, $\chi_{920}(809,·)$, $\chi_{920}(557,·)$, $\chi_{920}(157,·)$, $\chi_{920}(49,·)$, $\chi_{920}(53,·)$, $\chi_{920}(441,·)$, $\chi_{920}(573,·)$, $\chi_{920}(169,·)$, $\chi_{920}(373,·)$, $\chi_{920}(449,·)$, $\chi_{920}(841,·)$, $\chi_{920}(333,·)$, $\chi_{920}(721,·)$, $\chi_{920}(597,·)$, $\chi_{920}(121,·)$, $\chi_{920}(729,·)$, $\chi_{920}(361,·)$, $\chi_{920}(733,·)$, $\chi_{920}(677,·)$, $\chi_{920}(293,·)$, $\chi_{920}(613,·)$, $\chi_{920}(81,·)$, $\chi_{920}(489,·)$, $\chi_{920}(237,·)$, $\chi_{920}(369,·)$, $\chi_{920}(757,·)$, $\chi_{920}(41,·)$, $\chi_{920}(413,·)$, $\chi_{920}(761,·)$, $\chi_{920}(477,·)$, $\chi_{920}(893,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{20}a^{4}$, $\frac{1}{20}a^{5}$, $\frac{1}{40}a^{6}$, $\frac{1}{40}a^{7}$, $\frac{1}{400}a^{8}$, $\frac{1}{400}a^{9}$, $\frac{1}{800}a^{10}$, $\frac{1}{800}a^{11}$, $\frac{1}{8000}a^{12}$, $\frac{1}{8000}a^{13}$, $\frac{1}{16000}a^{14}$, $\frac{1}{16000}a^{15}$, $\frac{1}{160000}a^{16}$, $\frac{1}{160000}a^{17}$, $\frac{1}{320000}a^{18}$, $\frac{1}{320000}a^{19}$, $\frac{1}{3200000}a^{20}$, $\frac{1}{3200000}a^{21}$, $\frac{1}{147200000}a^{22}$, $\frac{1}{147200000}a^{23}$, $\frac{1}{1472000000}a^{24}$, $\frac{1}{1472000000}a^{25}$, $\frac{1}{2944000000}a^{26}$, $\frac{1}{2944000000}a^{27}$, $\frac{1}{29440000000}a^{28}$, $\frac{1}{29440000000}a^{29}$, $\frac{1}{58880000000}a^{30}$, $\frac{1}{58880000000}a^{31}$, $\frac{1}{588800000000}a^{32}$, $\frac{1}{588800000000}a^{33}$, $\frac{1}{1177600000000}a^{34}$, $\frac{1}{1177600000000}a^{35}$, $\frac{1}{11776000000000}a^{36}$, $\frac{1}{11776000000000}a^{37}$, $\frac{1}{23552000000000}a^{38}$, $\frac{1}{23552000000000}a^{39}$, $\frac{1}{32\!\cdots\!00}a^{40}-\frac{19}{16\!\cdots\!00}a^{38}+\frac{57}{16\!\cdots\!00}a^{36}-\frac{37}{163686400000000}a^{34}+\frac{7}{10230400000000}a^{32}+\frac{1}{1023040000000}a^{30}-\frac{33}{2046080000000}a^{28}+\frac{9}{204608000000}a^{26}-\frac{1}{3197000000}a^{24}+\frac{3}{5115200000}a^{22}-\frac{13}{88960000}a^{20}+\frac{61}{44480000}a^{18}-\frac{1}{5560000}a^{16}+\frac{61}{2224000}a^{14}+\frac{11}{1112000}a^{12}+\frac{7}{111200}a^{10}-\frac{7}{27800}a^{8}-\frac{27}{2780}a^{6}+\frac{3}{695}a^{4}-\frac{29}{278}a^{2}+\frac{50}{139}$, $\frac{1}{32\!\cdots\!00}a^{41}-\frac{19}{16\!\cdots\!00}a^{39}+\frac{57}{16\!\cdots\!00}a^{37}-\frac{37}{163686400000000}a^{35}+\frac{7}{10230400000000}a^{33}+\frac{1}{1023040000000}a^{31}-\frac{33}{2046080000000}a^{29}+\frac{9}{204608000000}a^{27}-\frac{1}{3197000000}a^{25}+\frac{3}{5115200000}a^{23}-\frac{13}{88960000}a^{21}+\frac{61}{44480000}a^{19}-\frac{1}{5560000}a^{17}+\frac{61}{2224000}a^{15}+\frac{11}{1112000}a^{13}+\frac{7}{111200}a^{11}-\frac{7}{27800}a^{9}-\frac{27}{2780}a^{7}+\frac{3}{695}a^{5}-\frac{29}{278}a^{3}+\frac{50}{139}a$, $\frac{1}{27\!\cdots\!00}a^{42}-\frac{4101983}{34\!\cdots\!00}a^{40}+\frac{18964067483}{13\!\cdots\!00}a^{38}+\frac{2059442837}{34\!\cdots\!00}a^{36}-\frac{115769181}{34\!\cdots\!00}a^{34}-\frac{300910469}{17\!\cdots\!00}a^{32}-\frac{5740952073}{17\!\cdots\!00}a^{30}+\frac{82769191}{38\!\cdots\!00}a^{28}+\frac{9668509}{27\!\cdots\!00}a^{26}+\frac{17097125271}{87\!\cdots\!00}a^{24}-\frac{13064830797}{87\!\cdots\!00}a^{22}+\frac{17927745913}{19\!\cdots\!00}a^{20}-\frac{6433289177}{95\!\cdots\!00}a^{18}-\frac{200687567}{11\!\cdots\!00}a^{16}+\frac{11825438567}{475323155672000}a^{14}+\frac{609999133}{11883078891800}a^{12}+\frac{8842120783}{47532315567200}a^{10}-\frac{596365133}{594153944590}a^{8}+\frac{3923259251}{594153944590}a^{6}-\frac{22412302837}{1188307889180}a^{4}+\frac{7428755939}{118830788918}a^{2}+\frac{17364519196}{59415394459}$, $\frac{1}{27\!\cdots\!00}a^{43}-\frac{4101983}{34\!\cdots\!00}a^{41}+\frac{18964067483}{13\!\cdots\!00}a^{39}+\frac{2059442837}{34\!\cdots\!00}a^{37}-\frac{115769181}{34\!\cdots\!00}a^{35}-\frac{300910469}{17\!\cdots\!00}a^{33}-\frac{5740952073}{17\!\cdots\!00}a^{31}+\frac{82769191}{38\!\cdots\!00}a^{29}+\frac{9668509}{27\!\cdots\!00}a^{27}+\frac{17097125271}{87\!\cdots\!00}a^{25}-\frac{13064830797}{87\!\cdots\!00}a^{23}+\frac{17927745913}{19\!\cdots\!00}a^{21}-\frac{6433289177}{95\!\cdots\!00}a^{19}-\frac{200687567}{11\!\cdots\!00}a^{17}+\frac{11825438567}{475323155672000}a^{15}+\frac{609999133}{11883078891800}a^{13}+\frac{8842120783}{47532315567200}a^{11}-\frac{596365133}{594153944590}a^{9}+\frac{3923259251}{594153944590}a^{7}-\frac{22412302837}{1188307889180}a^{5}+\frac{7428755939}{118830788918}a^{3}+\frac{17364519196}{59415394459}a$ 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:  $43$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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 - 230*x^42 + 24380*x^40 - 1582400*x^38 + 70481200*x^36 - 2288408000*x^34 + 56161768000*x^32 - 1066130960000*x^30 + 15888506720000*x^28 - 187591348800000*x^26 + 1762976128000000*x^24 - 13201383232000000*x^22 + 78572903232000000*x^20 - 369514540800000000*x^18 + 1359724672000000000*x^16 - 3859279296000000000*x^14 + 8280581734400000000*x^12 - 13061238528000000000*x^10 + 14562955264000000000*x^8 - 10844753920000000000*x^6 + 4945684480000000000*x^4 - 1191731200000000000*x^2 + 108339200000000000)
 
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 - 230*x^42 + 24380*x^40 - 1582400*x^38 + 70481200*x^36 - 2288408000*x^34 + 56161768000*x^32 - 1066130960000*x^30 + 15888506720000*x^28 - 187591348800000*x^26 + 1762976128000000*x^24 - 13201383232000000*x^22 + 78572903232000000*x^20 - 369514540800000000*x^18 + 1359724672000000000*x^16 - 3859279296000000000*x^14 + 8280581734400000000*x^12 - 13061238528000000000*x^10 + 14562955264000000000*x^8 - 10844753920000000000*x^6 + 4945684480000000000*x^4 - 1191731200000000000*x^2 + 108339200000000000, 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 - 230*x^42 + 24380*x^40 - 1582400*x^38 + 70481200*x^36 - 2288408000*x^34 + 56161768000*x^32 - 1066130960000*x^30 + 15888506720000*x^28 - 187591348800000*x^26 + 1762976128000000*x^24 - 13201383232000000*x^22 + 78572903232000000*x^20 - 369514540800000000*x^18 + 1359724672000000000*x^16 - 3859279296000000000*x^14 + 8280581734400000000*x^12 - 13061238528000000000*x^10 + 14562955264000000000*x^8 - 10844753920000000000*x^6 + 4945684480000000000*x^4 - 1191731200000000000*x^2 + 108339200000000000);
 
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 - 230*x^42 + 24380*x^40 - 1582400*x^38 + 70481200*x^36 - 2288408000*x^34 + 56161768000*x^32 - 1066130960000*x^30 + 15888506720000*x^28 - 187591348800000*x^26 + 1762976128000000*x^24 - 13201383232000000*x^22 + 78572903232000000*x^20 - 369514540800000000*x^18 + 1359724672000000000*x^16 - 3859279296000000000*x^14 + 8280581734400000000*x^12 - 13061238528000000000*x^10 + 14562955264000000000*x^8 - 10844753920000000000*x^6 + 4945684480000000000*x^4 - 1191731200000000000*x^2 + 108339200000000000);
 
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

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

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)
 
A cyclic group of order 44
The 44 conjugacy class representatives for $C_{44}$
Character table for $C_{44}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.4232000.1, \(\Q(\zeta_{23})^+\), 22.22.83796671451884098775580820361328125.1

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

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 R $44$ R $44$ ${\href{/padicField/11.11.0.1}{11} }^{4}$ $44$ $44$ $22^{2}$ R ${\href{/padicField/29.11.0.1}{11} }^{4}$ ${\href{/padicField/31.11.0.1}{11} }^{4}$ $44$ ${\href{/padicField/41.11.0.1}{11} }^{4}$ $44$ ${\href{/padicField/47.4.0.1}{4} }^{11}$ $44$ ${\href{/padicField/59.11.0.1}{11} }^{4}$

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
\(2\) Copy content Toggle raw display Deg $44$$2$$22$$66$
\(5\) Copy content Toggle raw display Deg $44$$4$$11$$33$
\(23\) Copy content Toggle raw display Deg $44$$22$$2$$42$