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Properties

Label 40.0.31839946004...5625.1
Degree $40$
Signature $[0, 20]$
Discriminant $5^{70}\cdot 19^{20}$
Root discriminant $72.87$
Ramified primes $5, 19$
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![95367431640625, 0, 0, 0, 0, -3082275390625, 0, 0, 0, 0, 69101562500, 0, 0, 0, 0, -1247034375, 0, 0, 0, 0, 18191651, 0, 0, 0, 0, -399051, 0, 0, 0, 0, 7076, 0, 0, 0, 0, -101, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 101*x^35 + 7076*x^30 - 399051*x^25 + 18191651*x^20 - 1247034375*x^15 + 69101562500*x^10 - 3082275390625*x^5 + 95367431640625)
 
gp: K = bnfinit(x^40 - 101*x^35 + 7076*x^30 - 399051*x^25 + 18191651*x^20 - 1247034375*x^15 + 69101562500*x^10 - 3082275390625*x^5 + 95367431640625, 1)
 

Normalized defining polynomial

\( x^{40} - 101 x^{35} + 7076 x^{30} - 399051 x^{25} + 18191651 x^{20} - 1247034375 x^{15} + 69101562500 x^{10} - 3082275390625 x^{5} + 95367431640625 \)

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:  \(318399460049560753522799707977372109246605447196998284198343753814697265625=5^{70}\cdot 19^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.87$
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:  $5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(475=5^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{475}(1,·)$, $\chi_{475}(132,·)$, $\chi_{475}(134,·)$, $\chi_{475}(267,·)$, $\chi_{475}(398,·)$, $\chi_{475}(18,·)$, $\chi_{475}(151,·)$, $\chi_{475}(153,·)$, $\chi_{475}(284,·)$, $\chi_{475}(286,·)$, $\chi_{475}(417,·)$, $\chi_{475}(419,·)$, $\chi_{475}(37,·)$, $\chi_{475}(39,·)$, $\chi_{475}(172,·)$, $\chi_{475}(303,·)$, $\chi_{475}(436,·)$, $\chi_{475}(438,·)$, $\chi_{475}(56,·)$, $\chi_{475}(58,·)$, $\chi_{475}(189,·)$, $\chi_{475}(191,·)$, $\chi_{475}(322,·)$, $\chi_{475}(324,·)$, $\chi_{475}(457,·)$, $\chi_{475}(77,·)$, $\chi_{475}(208,·)$, $\chi_{475}(341,·)$, $\chi_{475}(343,·)$, $\chi_{475}(474,·)$, $\chi_{475}(94,·)$, $\chi_{475}(96,·)$, $\chi_{475}(227,·)$, $\chi_{475}(229,·)$, $\chi_{475}(362,·)$, $\chi_{475}(113,·)$, $\chi_{475}(246,·)$, $\chi_{475}(248,·)$, $\chi_{475}(379,·)$, $\chi_{475}(381,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{11} a^{20} - \frac{1}{11} a^{15} + \frac{1}{11} a^{10} - \frac{1}{11} a^{5} + \frac{1}{11}$, $\frac{1}{55} a^{21} - \frac{1}{55} a^{16} + \frac{1}{55} a^{11} - \frac{1}{55} a^{6} + \frac{1}{55} a$, $\frac{1}{275} a^{22} - \frac{1}{275} a^{17} + \frac{1}{275} a^{12} - \frac{1}{275} a^{7} + \frac{1}{275} a^{2}$, $\frac{1}{1375} a^{23} + \frac{274}{1375} a^{18} - \frac{549}{1375} a^{13} - \frac{551}{1375} a^{8} + \frac{276}{1375} a^{3}$, $\frac{1}{6875} a^{24} + \frac{3024}{6875} a^{19} + \frac{826}{6875} a^{14} + \frac{2199}{6875} a^{9} + \frac{1651}{6875} a^{4}$, $\frac{1}{625338003125} a^{25} - \frac{326}{34375} a^{20} + \frac{11051}{34375} a^{15} + \frac{5724}{34375} a^{10} - \frac{15624}{34375} a^{5} + \frac{90559204}{200108161}$, $\frac{1}{3126690015625} a^{26} - \frac{326}{171875} a^{21} + \frac{45426}{171875} a^{16} + \frac{40099}{171875} a^{11} - \frac{84374}{171875} a^{6} + \frac{490775526}{1000540805} a$, $\frac{1}{15633450078125} a^{27} - \frac{326}{859375} a^{22} - \frac{126449}{859375} a^{17} + \frac{40099}{859375} a^{12} + \frac{87501}{859375} a^{7} - \frac{509765279}{5002704025} a^{2}$, $\frac{1}{78167250390625} a^{28} - \frac{326}{4296875} a^{23} - \frac{1845199}{4296875} a^{18} - \frac{1678651}{4296875} a^{13} + \frac{1806251}{4296875} a^{8} + \frac{4492938746}{25013520125} a^{3}$, $\frac{1}{390836251953125} a^{29} - \frac{326}{21484375} a^{24} - \frac{10438949}{21484375} a^{19} + \frac{2618224}{21484375} a^{14} + \frac{1806251}{21484375} a^{9} + \frac{29506458871}{125067600625} a^{4}$, $\frac{1}{1954181259765625} a^{30} - \frac{101}{1954181259765625} a^{25} - \frac{4510824}{107421875} a^{20} - \frac{31669276}{107421875} a^{15} + \frac{1}{107421875} a^{10} - \frac{399051}{625338003125} a^{5} + \frac{7076}{200108161}$, $\frac{1}{9770906298828125} a^{31} - \frac{101}{9770906298828125} a^{26} - \frac{4510824}{537109375} a^{21} + \frac{183174474}{537109375} a^{16} - \frac{107421874}{537109375} a^{11} + \frac{625337604074}{3126690015625} a^{6} - \frac{40020217}{200108161} a$, $\frac{1}{48854531494140625} a^{32} - \frac{101}{48854531494140625} a^{27} - \frac{4510824}{2685546875} a^{22} + \frac{183174474}{2685546875} a^{17} + \frac{966796876}{2685546875} a^{12} - \frac{5628042427176}{15633450078125} a^{7} + \frac{72039221}{200108161} a^{2}$, $\frac{1}{244272657470703125} a^{33} - \frac{101}{244272657470703125} a^{28} - \frac{4510824}{13427734375} a^{23} - \frac{2502372401}{13427734375} a^{18} - \frac{1718749999}{13427734375} a^{13} - \frac{5628042427176}{78167250390625} a^{8} + \frac{272147382}{1000540805} a^{3}$, $\frac{1}{1221363287353515625} a^{34} - \frac{101}{1221363287353515625} a^{29} - \frac{4510824}{67138671875} a^{24} - \frac{2502372401}{67138671875} a^{19} - \frac{1718749999}{67138671875} a^{14} - \frac{161962543208426}{390836251953125} a^{9} - \frac{728393423}{5002704025} a^{4}$, $\frac{1}{6106816436767578125} a^{35} - \frac{101}{6106816436767578125} a^{30} + \frac{7076}{6106816436767578125} a^{25} + \frac{2858955724}{335693359375} a^{20} + \frac{30517578126}{335693359375} a^{15} - \frac{177652842195926}{1954181259765625} a^{10} + \frac{56848916451}{625338003125} a^{5} - \frac{18191752}{200108161}$, $\frac{1}{30534082183837890625} a^{36} - \frac{101}{30534082183837890625} a^{31} + \frac{7076}{30534082183837890625} a^{26} + \frac{2858955724}{1678466796875} a^{21} - \frac{305175781249}{1678466796875} a^{16} + \frac{1776528417569699}{9770906298828125} a^{11} - \frac{568489086674}{3126690015625} a^{6} + \frac{181916409}{1000540805} a$, $\frac{1}{152670410919189453125} a^{37} - \frac{101}{152670410919189453125} a^{32} + \frac{7076}{152670410919189453125} a^{27} + \frac{2858955724}{8392333984375} a^{22} - \frac{305175781249}{8392333984375} a^{17} + \frac{1776528417569699}{48854531494140625} a^{12} - \frac{568489086674}{15633450078125} a^{7} + \frac{181916409}{5002704025} a^{2}$, $\frac{1}{763352054595947265625} a^{38} - \frac{101}{763352054595947265625} a^{33} + \frac{7076}{763352054595947265625} a^{28} + \frac{2858955724}{41961669921875} a^{23} + \frac{8087158203126}{41961669921875} a^{18} - \frac{47078003076570926}{244272657470703125} a^{13} + \frac{15064960991451}{78167250390625} a^{8} - \frac{4820787616}{25013520125} a^{3}$, $\frac{1}{3816760272979736328125} a^{39} - \frac{101}{3816760272979736328125} a^{34} + \frac{7076}{3816760272979736328125} a^{29} + \frac{2858955724}{209808349609375} a^{24} - \frac{75836181640624}{209808349609375} a^{19} - \frac{47078003076570926}{1221363287353515625} a^{14} + \frac{171399461772701}{390836251953125} a^{9} + \frac{20192732509}{125067600625} a^{4}$

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:  \( \frac{559}{1221363287353515625} a^{39} - \frac{15815100054316}{1221363287353515625} a^{14} \) (order $50$)
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

\(\Q(\sqrt{-95}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{5}, \sqrt{-19})\), \(\Q(\zeta_{5})\), 4.4.45125.1, 5.5.390625.1, 8.0.2036265625.1, 10.0.1889113616943359375.3, \(\Q(\zeta_{25})^+\), 10.0.377822723388671875.3, 20.0.3568750257720821537077426910400390625.2, \(\Q(\zeta_{25})\), 20.20.17843751288604107685387134552001953125.1

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 $20^{2}$ $20^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{8}$ $20^{2}$ $20^{2}$ R $20^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
5Data not computed
19Data not computed