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Properties

Label 40.40.3954109942...9648.1
Degree $40$
Signature $[40, 0]$
Discriminant $2^{40}\cdot 11^{36}\cdot 17^{35}$
Root discriminant $206.50$
Ramified primes $2, 11, 17$
Class number Not computed
Class group Not computed
Galois group $C_{40}$ (as 40T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20788126337, 0, -519703158425, 0, 4552599667803, 0, -20621821326304, 0, 56373730133061, 0, -100614642637531, 0, 122647277393476, 0, -105031396902212, 0, 64435213817040, 0, -28741268505675, 0, 9425882445199, 0, -2289732248573, 0, 413297142302, 0, -55335959701, 0, 5454569527, 0, -390041047, 0, 19741777, 0, -680119, 0, 14960, 0, -187, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 187*x^38 + 14960*x^36 - 680119*x^34 + 19741777*x^32 - 390041047*x^30 + 5454569527*x^28 - 55335959701*x^26 + 413297142302*x^24 - 2289732248573*x^22 + 9425882445199*x^20 - 28741268505675*x^18 + 64435213817040*x^16 - 105031396902212*x^14 + 122647277393476*x^12 - 100614642637531*x^10 + 56373730133061*x^8 - 20621821326304*x^6 + 4552599667803*x^4 - 519703158425*x^2 + 20788126337)
 
gp: K = bnfinit(x^40 - 187*x^38 + 14960*x^36 - 680119*x^34 + 19741777*x^32 - 390041047*x^30 + 5454569527*x^28 - 55335959701*x^26 + 413297142302*x^24 - 2289732248573*x^22 + 9425882445199*x^20 - 28741268505675*x^18 + 64435213817040*x^16 - 105031396902212*x^14 + 122647277393476*x^12 - 100614642637531*x^10 + 56373730133061*x^8 - 20621821326304*x^6 + 4552599667803*x^4 - 519703158425*x^2 + 20788126337, 1)
 

Normalized defining polynomial

\( x^{40} - 187 x^{38} + 14960 x^{36} - 680119 x^{34} + 19741777 x^{32} - 390041047 x^{30} + 5454569527 x^{28} - 55335959701 x^{26} + 413297142302 x^{24} - 2289732248573 x^{22} + 9425882445199 x^{20} - 28741268505675 x^{18} + 64435213817040 x^{16} - 105031396902212 x^{14} + 122647277393476 x^{12} - 100614642637531 x^{10} + 56373730133061 x^{8} - 20621821326304 x^{6} + 4552599667803 x^{4} - 519703158425 x^{2} + 20788126337 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[40, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(395410994212712488583535125652348307396350159891122576225102881206574771716175037399614619648=2^{40}\cdot 11^{36}\cdot 17^{35}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $206.50$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(748=2^{2}\cdot 11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{748}(1,·)$, $\chi_{748}(263,·)$, $\chi_{748}(137,·)$, $\chi_{748}(535,·)$, $\chi_{748}(655,·)$, $\chi_{748}(273,·)$, $\chi_{748}(19,·)$, $\chi_{748}(151,·)$, $\chi_{748}(157,·)$, $\chi_{748}(421,·)$, $\chi_{748}(169,·)$, $\chi_{748}(43,·)$, $\chi_{748}(563,·)$, $\chi_{748}(565,·)$, $\chi_{748}(695,·)$, $\chi_{748}(441,·)$, $\chi_{748}(699,·)$, $\chi_{748}(577,·)$, $\chi_{748}(195,·)$, $\chi_{748}(69,·)$, $\chi_{748}(353,·)$, $\chi_{748}(713,·)$, $\chi_{748}(81,·)$, $\chi_{748}(83,·)$, $\chi_{748}(87,·)$, $\chi_{748}(89,·)$, $\chi_{748}(219,·)$, $\chi_{748}(477,·)$, $\chi_{748}(225,·)$, $\chi_{748}(739,·)$, $\chi_{748}(359,·)$, $\chi_{748}(519,·)$, $\chi_{748}(361,·)$, $\chi_{748}(491,·)$, $\chi_{748}(625,·)$, $\chi_{748}(723,·)$, $\chi_{748}(489,·)$, $\chi_{748}(315,·)$, $\chi_{748}(509,·)$, $\chi_{748}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{17} a^{8}$, $\frac{1}{17} a^{9}$, $\frac{1}{187} a^{10}$, $\frac{1}{187} a^{11}$, $\frac{1}{187} a^{12}$, $\frac{1}{187} a^{13}$, $\frac{1}{187} a^{14}$, $\frac{1}{187} a^{15}$, $\frac{1}{3179} a^{16}$, $\frac{1}{3179} a^{17}$, $\frac{1}{3179} a^{18}$, $\frac{1}{3179} a^{19}$, $\frac{1}{34969} a^{20}$, $\frac{1}{34969} a^{21}$, $\frac{1}{34969} a^{22}$, $\frac{1}{34969} a^{23}$, $\frac{1}{594473} a^{24}$, $\frac{1}{594473} a^{25}$, $\frac{1}{594473} a^{26}$, $\frac{1}{594473} a^{27}$, $\frac{1}{594473} a^{28}$, $\frac{1}{594473} a^{29}$, $\frac{1}{340038556} a^{30} + \frac{4}{7728149} a^{28} + \frac{1}{7728149} a^{26} + \frac{5}{15456298} a^{24} - \frac{6}{454597} a^{22} + \frac{1}{454597} a^{20} + \frac{25}{165308} a^{18} + \frac{2}{41327} a^{16} + \frac{5}{2431} a^{14} + \frac{1}{2431} a^{12} - \frac{1}{2431} a^{10} - \frac{3}{221} a^{8} + \frac{4}{13} a^{4} - \frac{4}{13} a^{2} - \frac{5}{52}$, $\frac{1}{340038556} a^{31} + \frac{4}{7728149} a^{29} + \frac{1}{7728149} a^{27} + \frac{5}{15456298} a^{25} - \frac{6}{454597} a^{23} + \frac{1}{454597} a^{21} + \frac{25}{165308} a^{19} + \frac{2}{41327} a^{17} + \frac{5}{2431} a^{15} + \frac{1}{2431} a^{13} - \frac{1}{2431} a^{11} - \frac{3}{221} a^{9} + \frac{4}{13} a^{5} - \frac{4}{13} a^{3} - \frac{5}{52} a$, $\frac{1}{5780655452} a^{32} + \frac{3}{7728149} a^{28} + \frac{1}{15456298} a^{26} + \frac{1}{7728149} a^{24} + \frac{1}{454597} a^{22} - \frac{1}{139876} a^{20} + \frac{5}{41327} a^{18} - \frac{5}{41327} a^{16} - \frac{2}{2431} a^{14} - \frac{2}{2431} a^{12} + \frac{2}{221} a^{8} + \frac{1}{13} a^{6} + \frac{5}{13} a^{4} - \frac{9}{52} a^{2} + \frac{3}{13}$, $\frac{1}{5780655452} a^{33} + \frac{3}{7728149} a^{29} + \frac{1}{15456298} a^{27} + \frac{1}{7728149} a^{25} + \frac{1}{454597} a^{23} - \frac{1}{139876} a^{21} + \frac{5}{41327} a^{19} - \frac{5}{41327} a^{17} - \frac{2}{2431} a^{15} - \frac{2}{2431} a^{13} + \frac{2}{221} a^{9} + \frac{1}{13} a^{7} + \frac{5}{13} a^{5} - \frac{9}{52} a^{3} + \frac{3}{13} a$, $\frac{1}{34470048460276} a^{34} + \frac{431}{17235024230138} a^{32} - \frac{2953}{2027649909428} a^{30} - \frac{601}{1035571966} a^{28} + \frac{32018}{46082952487} a^{26} + \frac{2537}{8378718634} a^{24} + \frac{89435}{10843047644} a^{22} + \frac{66701}{5421523822} a^{20} - \frac{43837}{985731604} a^{18} - \frac{10189}{246432901} a^{16} - \frac{300}{852709} a^{14} - \frac{2048}{1317823} a^{12} - \frac{500}{1115081} a^{10} + \frac{30028}{1317823} a^{8} + \frac{38489}{77519} a^{6} + \frac{13631}{310076} a^{4} + \frac{36641}{155038} a^{2} - \frac{20927}{310076}$, $\frac{1}{34470048460276} a^{35} + \frac{431}{17235024230138} a^{33} - \frac{2953}{2027649909428} a^{31} - \frac{601}{1035571966} a^{29} + \frac{32018}{46082952487} a^{27} + \frac{2537}{8378718634} a^{25} + \frac{89435}{10843047644} a^{23} + \frac{66701}{5421523822} a^{21} - \frac{43837}{985731604} a^{19} - \frac{10189}{246432901} a^{17} - \frac{300}{852709} a^{15} - \frac{2048}{1317823} a^{13} - \frac{500}{1115081} a^{11} + \frac{30028}{1317823} a^{9} + \frac{38489}{77519} a^{7} + \frac{13631}{310076} a^{5} + \frac{36641}{155038} a^{3} - \frac{20927}{310076} a$, $\frac{1}{34470048460276} a^{36} - \frac{83}{17235024230138} a^{32} - \frac{1381}{1013824954714} a^{30} - \frac{20848}{46082952487} a^{28} - \frac{33521}{46082952487} a^{26} - \frac{79873}{184331809948} a^{24} + \frac{14635}{2710761911} a^{22} - \frac{4935}{492865802} a^{20} + \frac{109}{1115081} a^{18} - \frac{2127}{18956377} a^{16} + \frac{2967}{1115081} a^{14} + \frac{20960}{14496053} a^{12} + \frac{18012}{14496053} a^{10} + \frac{329}{77519} a^{8} + \frac{87643}{310076} a^{6} - \frac{21142}{77519} a^{4} - \frac{6013}{155038} a^{2} - \frac{1241}{77519}$, $\frac{1}{34470048460276} a^{37} - \frac{83}{17235024230138} a^{33} - \frac{1381}{1013824954714} a^{31} - \frac{20848}{46082952487} a^{29} - \frac{33521}{46082952487} a^{27} - \frac{79873}{184331809948} a^{25} + \frac{14635}{2710761911} a^{23} - \frac{4935}{492865802} a^{21} + \frac{109}{1115081} a^{19} - \frac{2127}{18956377} a^{17} + \frac{2967}{1115081} a^{15} + \frac{20960}{14496053} a^{13} + \frac{18012}{14496053} a^{11} + \frac{329}{77519} a^{9} + \frac{87643}{310076} a^{7} - \frac{21142}{77519} a^{5} - \frac{6013}{155038} a^{3} - \frac{1241}{77519} a$, $\frac{1}{4484539813932127119572697843960609478996} a^{38} - \frac{24995166823751829520736029}{4484539813932127119572697843960609478996} a^{36} - \frac{19605476159586316298096297}{4484539813932127119572697843960609478996} a^{34} - \frac{23359469919010911280734888169}{4484539813932127119572697843960609478996} a^{32} - \frac{149175836110756650248135645553}{131898229821533150575667583645900278794} a^{30} + \frac{7825379983630036578751428355947}{11990748165593922779606143967809116254} a^{28} + \frac{4422514943591019482791761177201}{23981496331187845559212287935618232508} a^{26} + \frac{1761387892034528235467616610387}{2180136030107985959928389812328930228} a^{24} + \frac{9249432598862216927117752472525}{1410676254775755621130134584448131324} a^{22} + \frac{16797386992347880824571475442135}{1410676254775755621130134584448131324} a^{20} - \frac{4470884999512796577498117559204}{32060823972176264116593967828366621} a^{18} - \frac{371285137410589839381198279237}{2914620361106933101508542529851511} a^{16} - \frac{2081930667068594218936508584579}{1885930821892721418623174578139213} a^{14} - \frac{4859546773033670420846746908528}{1885930821892721418623174578139213} a^{12} + \frac{3402599421557854997530640402860}{1885930821892721418623174578139213} a^{10} - \frac{561138175135536942184679557205}{685793026142807788590245301141532} a^{8} + \frac{5183737359567758263626583639209}{40340766243694575799426194184796} a^{6} + \frac{8609168483631995714243731976233}{40340766243694575799426194184796} a^{4} + \frac{17167558226612650992979367513255}{40340766243694575799426194184796} a^{2} + \frac{4421903488907584006490327320628}{10085191560923643949856548546199}$, $\frac{1}{4484539813932127119572697843960609478996} a^{39} - \frac{24995166823751829520736029}{4484539813932127119572697843960609478996} a^{37} - \frac{19605476159586316298096297}{4484539813932127119572697843960609478996} a^{35} - \frac{23359469919010911280734888169}{4484539813932127119572697843960609478996} a^{33} - \frac{149175836110756650248135645553}{131898229821533150575667583645900278794} a^{31} + \frac{7825379983630036578751428355947}{11990748165593922779606143967809116254} a^{29} + \frac{4422514943591019482791761177201}{23981496331187845559212287935618232508} a^{27} + \frac{1761387892034528235467616610387}{2180136030107985959928389812328930228} a^{25} + \frac{9249432598862216927117752472525}{1410676254775755621130134584448131324} a^{23} + \frac{16797386992347880824571475442135}{1410676254775755621130134584448131324} a^{21} - \frac{4470884999512796577498117559204}{32060823972176264116593967828366621} a^{19} - \frac{371285137410589839381198279237}{2914620361106933101508542529851511} a^{17} - \frac{2081930667068594218936508584579}{1885930821892721418623174578139213} a^{15} - \frac{4859546773033670420846746908528}{1885930821892721418623174578139213} a^{13} + \frac{3402599421557854997530640402860}{1885930821892721418623174578139213} a^{11} - \frac{561138175135536942184679557205}{685793026142807788590245301141532} a^{9} + \frac{5183737359567758263626583639209}{40340766243694575799426194184796} a^{7} + \frac{8609168483631995714243731976233}{40340766243694575799426194184796} a^{5} + \frac{17167558226612650992979367513255}{40340766243694575799426194184796} a^{3} + \frac{4421903488907584006490327320628}{10085191560923643949856548546199} a$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $39$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{40}$ (as 40T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 40
The 40 conjugacy class representatives for $C_{40}$
Character table for $C_{40}$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{11})^+\), 8.8.1537988738916608.1, 10.10.304358957700017.1, 20.20.131527565972137936816816034072938673.1

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 $40$ $40$ $40$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{8}$ R $20^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{5}$ $40$ $40$ $40$ $40$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{8}$ $20^{2}$ $20^{2}$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
11Data not computed
17Data not computed