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Properties

Label 40.0.31654584865...0000.2
Degree $40$
Signature $[0, 20]$
Discriminant $2^{40}\cdot 5^{30}\cdot 11^{36}$
Root discriminant $57.88$
Ramified primes $2, 5, 11$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{20}$ (as 40T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14641, 0, 73205, 0, 263538, 0, 863819, 0, 2752508, 0, 4152720, 0, 4694437, 0, 4573316, 0, 3942422, 0, 2574154, 0, 1416910, 0, 688611, 0, 294635, 0, 104786, 0, 33033, 0, 9273, 0, 2244, 0, 440, 0, 77, 0, 11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 11*x^38 + 77*x^36 + 440*x^34 + 2244*x^32 + 9273*x^30 + 33033*x^28 + 104786*x^26 + 294635*x^24 + 688611*x^22 + 1416910*x^20 + 2574154*x^18 + 3942422*x^16 + 4573316*x^14 + 4694437*x^12 + 4152720*x^10 + 2752508*x^8 + 863819*x^6 + 263538*x^4 + 73205*x^2 + 14641)
 
gp: K = bnfinit(x^40 + 11*x^38 + 77*x^36 + 440*x^34 + 2244*x^32 + 9273*x^30 + 33033*x^28 + 104786*x^26 + 294635*x^24 + 688611*x^22 + 1416910*x^20 + 2574154*x^18 + 3942422*x^16 + 4573316*x^14 + 4694437*x^12 + 4152720*x^10 + 2752508*x^8 + 863819*x^6 + 263538*x^4 + 73205*x^2 + 14641, 1)
 

Normalized defining polynomial

\( x^{40} + 11 x^{38} + 77 x^{36} + 440 x^{34} + 2244 x^{32} + 9273 x^{30} + 33033 x^{28} + 104786 x^{26} + 294635 x^{24} + 688611 x^{22} + 1416910 x^{20} + 2574154 x^{18} + 3942422 x^{16} + 4573316 x^{14} + 4694437 x^{12} + 4152720 x^{10} + 2752508 x^{8} + 863819 x^{6} + 263538 x^{4} + 73205 x^{2} + 14641 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(31654584865659568778929513407372752241664000000000000000000000000000000=2^{40}\cdot 5^{30}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.88$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(220=2^{2}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{220}(1,·)$, $\chi_{220}(131,·)$, $\chi_{220}(133,·)$, $\chi_{220}(7,·)$, $\chi_{220}(9,·)$, $\chi_{220}(139,·)$, $\chi_{220}(141,·)$, $\chi_{220}(43,·)$, $\chi_{220}(19,·)$, $\chi_{220}(151,·)$, $\chi_{220}(157,·)$, $\chi_{220}(37,·)$, $\chi_{220}(39,·)$, $\chi_{220}(169,·)$, $\chi_{220}(171,·)$, $\chi_{220}(49,·)$, $\chi_{220}(51,·)$, $\chi_{220}(53,·)$, $\chi_{220}(137,·)$, $\chi_{220}(63,·)$, $\chi_{220}(181,·)$, $\chi_{220}(69,·)$, $\chi_{220}(201,·)$, $\chi_{220}(183,·)$, $\chi_{220}(79,·)$, $\chi_{220}(81,·)$, $\chi_{220}(83,·)$, $\chi_{220}(213,·)$, $\chi_{220}(87,·)$, $\chi_{220}(89,·)$, $\chi_{220}(219,·)$, $\chi_{220}(93,·)$, $\chi_{220}(107,·)$, $\chi_{220}(97,·)$, $\chi_{220}(167,·)$, $\chi_{220}(177,·)$, $\chi_{220}(113,·)$, $\chi_{220}(211,·)$, $\chi_{220}(123,·)$, $\chi_{220}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10}$, $\frac{1}{11} a^{11}$, $\frac{1}{11} a^{12}$, $\frac{1}{11} a^{13}$, $\frac{1}{11} a^{14}$, $\frac{1}{11} a^{15}$, $\frac{1}{11} a^{16}$, $\frac{1}{11} a^{17}$, $\frac{1}{121} a^{18} - \frac{1}{11} a^{8}$, $\frac{1}{121} a^{19} - \frac{1}{11} a^{9}$, $\frac{1}{121} a^{20}$, $\frac{1}{121} a^{21}$, $\frac{1}{121} a^{22}$, $\frac{1}{121} a^{23}$, $\frac{1}{121} a^{24}$, $\frac{1}{121} a^{25}$, $\frac{1}{1331} a^{26} - \frac{2}{121} a^{16} + \frac{1}{11} a^{6}$, $\frac{1}{1331} a^{27} - \frac{2}{121} a^{17} + \frac{1}{11} a^{7}$, $\frac{1}{1331} a^{28} - \frac{1}{11} a^{8}$, $\frac{1}{1331} a^{29} - \frac{1}{11} a^{9}$, $\frac{1}{1331} a^{30}$, $\frac{1}{1331} a^{31}$, $\frac{1}{1331} a^{32}$, $\frac{1}{1331} a^{33}$, $\frac{1}{774464896196321} a^{34} - \frac{3190709248}{70405899654211} a^{32} - \frac{1618603772}{70405899654211} a^{30} + \frac{16792576465}{70405899654211} a^{28} + \frac{10252294982}{70405899654211} a^{26} - \frac{17977069773}{70405899654211} a^{24} + \frac{2237044627}{6400536332201} a^{22} + \frac{14502562927}{6400536332201} a^{20} + \frac{598530065}{581866939291} a^{18} + \frac{288566388494}{6400536332201} a^{16} - \frac{286816002206}{6400536332201} a^{14} - \frac{18532336046}{581866939291} a^{12} + \frac{3055556333}{581866939291} a^{10} - \frac{161664475384}{581866939291} a^{8} + \frac{128503633979}{581866939291} a^{6} + \frac{127185291727}{581866939291} a^{4} - \frac{11798061945}{52896994481} a^{2} + \frac{7493800533}{52896994481}$, $\frac{1}{774464896196321} a^{35} - \frac{3190709248}{70405899654211} a^{33} - \frac{1618603772}{70405899654211} a^{31} + \frac{16792576465}{70405899654211} a^{29} + \frac{10252294982}{70405899654211} a^{27} - \frac{17977069773}{70405899654211} a^{25} + \frac{2237044627}{6400536332201} a^{23} + \frac{14502562927}{6400536332201} a^{21} + \frac{598530065}{581866939291} a^{19} + \frac{288566388494}{6400536332201} a^{17} - \frac{286816002206}{6400536332201} a^{15} - \frac{18532336046}{581866939291} a^{13} + \frac{3055556333}{581866939291} a^{11} - \frac{161664475384}{581866939291} a^{9} + \frac{128503633979}{581866939291} a^{7} + \frac{127185291727}{581866939291} a^{5} - \frac{11798061945}{52896994481} a^{3} + \frac{7493800533}{52896994481} a$, $\frac{1}{774464896196321} a^{36} + \frac{16008996310}{70405899654211} a^{32} - \frac{4958158505}{70405899654211} a^{30} - \frac{18377658466}{70405899654211} a^{28} + \frac{7464166299}{70405899654211} a^{26} - \frac{9174883319}{6400536332201} a^{24} + \frac{18298437958}{6400536332201} a^{22} + \frac{15603799148}{6400536332201} a^{20} - \frac{1848375322}{581866939291} a^{18} - \frac{68172970267}{6400536332201} a^{16} - \frac{10106953870}{581866939291} a^{14} - \frac{2098182813}{52896994481} a^{12} + \frac{13017979267}{581866939291} a^{10} - \frac{198233377945}{581866939291} a^{8} - \frac{185892891824}{581866939291} a^{6} + \frac{14568552221}{52896994481} a^{4} + \frac{20239880052}{52896994481} a^{2} + \frac{23106908469}{52896994481}$, $\frac{1}{774464896196321} a^{37} + \frac{16008996310}{70405899654211} a^{33} - \frac{4958158505}{70405899654211} a^{31} - \frac{18377658466}{70405899654211} a^{29} + \frac{7464166299}{70405899654211} a^{27} - \frac{9174883319}{6400536332201} a^{25} + \frac{18298437958}{6400536332201} a^{23} + \frac{15603799148}{6400536332201} a^{21} - \frac{1848375322}{581866939291} a^{19} - \frac{68172970267}{6400536332201} a^{17} - \frac{10106953870}{581866939291} a^{15} - \frac{2098182813}{52896994481} a^{13} + \frac{13017979267}{581866939291} a^{11} - \frac{198233377945}{581866939291} a^{9} - \frac{185892891824}{581866939291} a^{7} + \frac{14568552221}{52896994481} a^{5} + \frac{20239880052}{52896994481} a^{3} + \frac{23106908469}{52896994481} a$, $\frac{1}{774464896196321} a^{38} - \frac{13287510312}{70405899654211} a^{32} - \frac{4106145961}{70405899654211} a^{30} - \frac{14089504255}{70405899654211} a^{28} + \frac{13731987822}{70405899654211} a^{26} + \frac{9374904263}{6400536332201} a^{24} - \frac{25218143065}{6400536332201} a^{22} + \frac{5056542771}{6400536332201} a^{20} - \frac{1384939810}{581866939291} a^{18} - \frac{168663881843}{6400536332201} a^{16} - \frac{21165014350}{581866939291} a^{14} + \frac{22839883305}{581866939291} a^{12} + \frac{3463587132}{581866939291} a^{10} - \frac{239027494430}{581866939291} a^{8} - \frac{83208508182}{581866939291} a^{6} + \frac{2968650789}{52896994481} a^{4} - \frac{11182737850}{52896994481} a^{2} - \frac{21109201013}{52896994481}$, $\frac{1}{774464896196321} a^{39} - \frac{13287510312}{70405899654211} a^{33} - \frac{4106145961}{70405899654211} a^{31} - \frac{14089504255}{70405899654211} a^{29} + \frac{13731987822}{70405899654211} a^{27} + \frac{9374904263}{6400536332201} a^{25} - \frac{25218143065}{6400536332201} a^{23} + \frac{5056542771}{6400536332201} a^{21} - \frac{1384939810}{581866939291} a^{19} - \frac{168663881843}{6400536332201} a^{17} - \frac{21165014350}{581866939291} a^{15} + \frac{22839883305}{581866939291} a^{13} + \frac{3463587132}{581866939291} a^{11} - \frac{239027494430}{581866939291} a^{9} - \frac{83208508182}{581866939291} a^{7} + \frac{2968650789}{52896994481} a^{5} - \frac{11182737850}{52896994481} a^{3} - \frac{21109201013}{52896994481} a$

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:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1555976772}{774464896196321} a^{38} - \frac{14392785141}{774464896196321} a^{36} - \frac{91802629548}{774464896196321} a^{34} - \frac{4098794166}{6400536332201} a^{32} - \frac{221726690010}{70405899654211} a^{30} - \frac{831280590441}{70405899654211} a^{28} - \frac{2758357822563}{70405899654211} a^{26} - \frac{8205443507142}{70405899654211} a^{24} - \frac{175640306498}{581866939291} a^{22} - \frac{3802807230768}{6400536332201} a^{20} - \frac{7123261662216}{6400536332201} a^{18} - \frac{11337624749178}{6400536332201} a^{16} - \frac{13330441999917}{6400536332201} a^{14} - \frac{623510550556}{581866939291} a^{12} - \frac{1140530973876}{581866939291} a^{10} - \frac{781489333737}{581866939291} a^{8} - \frac{245455335783}{581866939291} a^{6} - \frac{75075879249}{581866939291} a^{4} - \frac{168705162564}{52896994481} a^{2} - \frac{388994193}{52896994481} \) (order $10$)
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

$C_2\times C_{20}$ (as 40T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2\times C_{20}$
Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{5}, \sqrt{11})\), \(\Q(\zeta_{5})\), 4.0.242000.2, \(\Q(\zeta_{11})^+\), 8.0.58564000000.3, \(\Q(\zeta_{44})^+\), 10.10.669871503125.1, 10.10.7545432611200000.1, 20.20.56933553290160450365440000000000.1, 20.0.1402274470934209014892578125.1, 20.0.177917354031751407392000000000000000.1

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 $20^{2}$ R $20^{2}$ R $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
5Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$