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Properties

Label 40.0.31654584865...0000.3
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![9765625, 0, 9765625, 0, 7812500, 0, 5859375, 0, 4296875, 0, 3125000, 0, 2265625, 0, 1640625, 0, 1187500, 0, 859375, 0, 621875, 0, 171875, 0, 47500, 0, 13125, 0, 3625, 0, 1000, 0, 275, 0, 75, 0, 20, 0, 5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 5*x^38 + 20*x^36 + 75*x^34 + 275*x^32 + 1000*x^30 + 3625*x^28 + 13125*x^26 + 47500*x^24 + 171875*x^22 + 621875*x^20 + 859375*x^18 + 1187500*x^16 + 1640625*x^14 + 2265625*x^12 + 3125000*x^10 + 4296875*x^8 + 5859375*x^6 + 7812500*x^4 + 9765625*x^2 + 9765625)
 
gp: K = bnfinit(x^40 + 5*x^38 + 20*x^36 + 75*x^34 + 275*x^32 + 1000*x^30 + 3625*x^28 + 13125*x^26 + 47500*x^24 + 171875*x^22 + 621875*x^20 + 859375*x^18 + 1187500*x^16 + 1640625*x^14 + 2265625*x^12 + 3125000*x^10 + 4296875*x^8 + 5859375*x^6 + 7812500*x^4 + 9765625*x^2 + 9765625, 1)
 

Normalized defining polynomial

\( x^{40} + 5 x^{38} + 20 x^{36} + 75 x^{34} + 275 x^{32} + 1000 x^{30} + 3625 x^{28} + 13125 x^{26} + 47500 x^{24} + 171875 x^{22} + 621875 x^{20} + 859375 x^{18} + 1187500 x^{16} + 1640625 x^{14} + 2265625 x^{12} + 3125000 x^{10} + 4296875 x^{8} + 5859375 x^{6} + 7812500 x^{4} + 9765625 x^{2} + 9765625 \)

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$$\chi_{220}(1,·)$, $\chi_{220}(3,·)$, $\chi_{220}(7,·)$, $\chi_{220}(9,·)$, $\chi_{220}(141,·)$, $\chi_{220}(147,·)$, $\chi_{220}(149,·)$, $\chi_{220}(23,·)$, $\chi_{220}(127,·)$, $\chi_{220}(27,·)$, $\chi_{220}(29,·)$, $\chi_{220}(161,·)$, $\chi_{220}(163,·)$, $\chi_{220}(167,·)$, $\chi_{220}(129,·)$, $\chi_{220}(41,·)$, $\chi_{220}(43,·)$, $\chi_{220}(47,·)$, $\chi_{220}(49,·)$, $\chi_{220}(181,·)$, $\chi_{220}(183,·)$, $\chi_{220}(61,·)$, $\chi_{220}(63,·)$, $\chi_{220}(67,·)$, $\chi_{220}(69,·)$, $\chi_{220}(201,·)$, $\chi_{220}(203,·)$, $\chi_{220}(207,·)$, $\chi_{220}(81,·)$, $\chi_{220}(83,·)$, $\chi_{220}(87,·)$, $\chi_{220}(89,·)$, $\chi_{220}(101,·)$, $\chi_{220}(103,·)$, $\chi_{220}(107,·)$, $\chi_{220}(109,·)$, $\chi_{220}(189,·)$, $\chi_{220}(169,·)$, $\chi_{220}(123,·)$, $\chi_{220}(21,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{125} a^{14}$, $\frac{1}{125} a^{15}$, $\frac{1}{625} a^{16}$, $\frac{1}{625} a^{17}$, $\frac{1}{625} a^{18}$, $\frac{1}{625} a^{19}$, $\frac{1}{3125} a^{20}$, $\frac{1}{3125} a^{21}$, $\frac{1}{621875} a^{22} + \frac{76}{199}$, $\frac{1}{621875} a^{23} + \frac{76}{199} a$, $\frac{1}{3109375} a^{24} + \frac{55}{199} a^{2}$, $\frac{1}{3109375} a^{25} + \frac{55}{199} a^{3}$, $\frac{1}{3109375} a^{26} + \frac{76}{995} a^{4}$, $\frac{1}{3109375} a^{27} + \frac{76}{995} a^{5}$, $\frac{1}{15546875} a^{28} + \frac{11}{199} a^{6}$, $\frac{1}{15546875} a^{29} + \frac{11}{199} a^{7}$, $\frac{1}{15546875} a^{30} + \frac{76}{4975} a^{8}$, $\frac{1}{15546875} a^{31} + \frac{76}{4975} a^{9}$, $\frac{1}{77734375} a^{32} + \frac{11}{995} a^{10}$, $\frac{1}{77734375} a^{33} + \frac{11}{995} a^{11}$, $\frac{1}{77734375} a^{34} + \frac{76}{24875} a^{12}$, $\frac{1}{77734375} a^{35} + \frac{76}{24875} a^{13}$, $\frac{1}{388671875} a^{36} + \frac{11}{4975} a^{14}$, $\frac{1}{388671875} a^{37} + \frac{11}{4975} a^{15}$, $\frac{1}{388671875} a^{38} + \frac{76}{124375} a^{16}$, $\frac{1}{388671875} a^{39} + \frac{76}{124375} a^{17}$

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:  \( -\frac{76}{388671875} a^{38} - \frac{304}{388671875} a^{36} - \frac{228}{77734375} a^{34} - \frac{836}{77734375} a^{32} - \frac{608}{15546875} a^{30} - \frac{2204}{15546875} a^{28} - \frac{1596}{3109375} a^{26} - \frac{5776}{3109375} a^{24} - \frac{836}{124375} a^{22} - \frac{76}{3125} a^{20} - \frac{11}{125} a^{18} - \frac{5776}{124375} a^{16} - \frac{1596}{24875} a^{14} - \frac{2204}{24875} a^{12} - \frac{608}{4975} a^{10} - \frac{836}{4975} a^{8} - \frac{228}{995} a^{6} - \frac{304}{995} a^{4} - \frac{76}{199} a^{2} - \frac{76}{199} \) (order $22$)
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

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), 4.0.242000.2, \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{11})^+\), 8.0.58564000000.1, \(\Q(\zeta_{11})\), 10.10.669871503125.1, 10.0.7368586534375.1, 20.0.54296067514572573056640625.1, 20.0.177917354031751407392000000000000000.1, 20.20.1470391355634309152000000000000000.1

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 $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{8}$

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
11Data not computed