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Properties

Label 40.0.10240000000...0000.2
Degree $40$
Signature $[0, 20]$
Discriminant $2^{80}\cdot 5^{70}$
Root discriminant $66.87$
Ramified primes $2, 5$
Class number $66110$ (GRH)
Class group $[66110]$ (GRH)
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![25, 0, 0, 0, 6875, 0, 0, 0, 139875, 0, 0, 0, 609125, 0, 0, 0, 846200, 0, 0, 0, 462515, 0, 0, 0, 124375, 0, 0, 0, 18100, 0, 0, 0, 1450, 0, 0, 0, 60, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 60*x^36 + 1450*x^32 + 18100*x^28 + 124375*x^24 + 462515*x^20 + 846200*x^16 + 609125*x^12 + 139875*x^8 + 6875*x^4 + 25)
 
gp: K = bnfinit(x^40 + 60*x^36 + 1450*x^32 + 18100*x^28 + 124375*x^24 + 462515*x^20 + 846200*x^16 + 609125*x^12 + 139875*x^8 + 6875*x^4 + 25, 1)
 

Normalized defining polynomial

\( x^{40} + 60 x^{36} + 1450 x^{32} + 18100 x^{28} + 124375 x^{24} + 462515 x^{20} + 846200 x^{16} + 609125 x^{12} + 139875 x^{8} + 6875 x^{4} + 25 \)

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:  \(10240000000000000000000000000000000000000000000000000000000000000000000000=2^{80}\cdot 5^{70}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.87$
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:  $2, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(200=2^{3}\cdot 5^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{200}(1,·)$, $\chi_{200}(3,·)$, $\chi_{200}(133,·)$, $\chi_{200}(129,·)$, $\chi_{200}(9,·)$, $\chi_{200}(13,·)$, $\chi_{200}(147,·)$, $\chi_{200}(151,·)$, $\chi_{200}(27,·)$, $\chi_{200}(157,·)$, $\chi_{200}(159,·)$, $\chi_{200}(161,·)$, $\chi_{200}(163,·)$, $\chi_{200}(37,·)$, $\chi_{200}(39,·)$, $\chi_{200}(41,·)$, $\chi_{200}(43,·)$, $\chi_{200}(173,·)$, $\chi_{200}(49,·)$, $\chi_{200}(53,·)$, $\chi_{200}(31,·)$, $\chi_{200}(191,·)$, $\chi_{200}(67,·)$, $\chi_{200}(197,·)$, $\chi_{200}(71,·)$, $\chi_{200}(119,·)$, $\chi_{200}(77,·)$, $\chi_{200}(79,·)$, $\chi_{200}(81,·)$, $\chi_{200}(83,·)$, $\chi_{200}(89,·)$, $\chi_{200}(93,·)$, $\chi_{200}(199,·)$, $\chi_{200}(187,·)$, $\chi_{200}(107,·)$, $\chi_{200}(111,·)$, $\chi_{200}(117,·)$, $\chi_{200}(169,·)$, $\chi_{200}(121,·)$, $\chi_{200}(123,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{5} a^{20}$, $\frac{1}{5} a^{21}$, $\frac{1}{5} a^{22}$, $\frac{1}{5} a^{23}$, $\frac{1}{5} a^{24}$, $\frac{1}{5} a^{25}$, $\frac{1}{5} a^{26}$, $\frac{1}{5} a^{27}$, $\frac{1}{5} a^{28}$, $\frac{1}{5} a^{29}$, $\frac{1}{25} a^{30} - \frac{1}{5} a^{10}$, $\frac{1}{25} a^{31} - \frac{1}{5} a^{11}$, $\frac{1}{26075} a^{32} + \frac{69}{5215} a^{28} - \frac{274}{5215} a^{24} + \frac{334}{5215} a^{20} + \frac{268}{1043} a^{16} + \frac{2264}{5215} a^{12} - \frac{145}{1043} a^{8} + \frac{445}{1043} a^{4} + \frac{361}{1043}$, $\frac{1}{26075} a^{33} + \frac{69}{5215} a^{29} - \frac{274}{5215} a^{25} + \frac{334}{5215} a^{21} + \frac{268}{1043} a^{17} + \frac{2264}{5215} a^{13} - \frac{145}{1043} a^{9} + \frac{445}{1043} a^{5} + \frac{361}{1043} a$, $\frac{1}{26075} a^{34} + \frac{69}{5215} a^{30} - \frac{274}{5215} a^{26} + \frac{334}{5215} a^{22} + \frac{268}{1043} a^{18} + \frac{2264}{5215} a^{14} - \frac{145}{1043} a^{10} + \frac{445}{1043} a^{6} + \frac{361}{1043} a^{2}$, $\frac{1}{26075} a^{35} + \frac{69}{5215} a^{31} - \frac{274}{5215} a^{27} + \frac{334}{5215} a^{23} + \frac{268}{1043} a^{19} + \frac{2264}{5215} a^{15} - \frac{145}{1043} a^{11} + \frac{445}{1043} a^{7} + \frac{361}{1043} a^{3}$, $\frac{1}{175097715821980175} a^{36} + \frac{862771630747}{175097715821980175} a^{32} + \frac{2941707114408821}{35019543164396035} a^{28} - \frac{263750122213856}{35019543164396035} a^{24} + \frac{1635973729726536}{35019543164396035} a^{20} - \frac{10947004026169311}{35019543164396035} a^{16} + \frac{13454956493402128}{35019543164396035} a^{12} - \frac{1966885865585092}{7003908632879207} a^{8} - \frac{3380005680980668}{7003908632879207} a^{4} + \frac{2960908872308735}{7003908632879207}$, $\frac{1}{175097715821980175} a^{37} + \frac{862771630747}{175097715821980175} a^{33} + \frac{2941707114408821}{35019543164396035} a^{29} - \frac{263750122213856}{35019543164396035} a^{25} + \frac{1635973729726536}{35019543164396035} a^{21} - \frac{10947004026169311}{35019543164396035} a^{17} + \frac{13454956493402128}{35019543164396035} a^{13} - \frac{1966885865585092}{7003908632879207} a^{9} - \frac{3380005680980668}{7003908632879207} a^{5} + \frac{2960908872308735}{7003908632879207} a$, $\frac{1}{175097715821980175} a^{38} + \frac{862771630747}{175097715821980175} a^{34} + \frac{700718306285691}{175097715821980175} a^{30} - \frac{263750122213856}{35019543164396035} a^{26} + \frac{1635973729726536}{35019543164396035} a^{22} - \frac{10947004026169311}{35019543164396035} a^{18} + \frac{13454956493402128}{35019543164396035} a^{14} + \frac{4173387937832954}{35019543164396035} a^{10} - \frac{3380005680980668}{7003908632879207} a^{6} + \frac{2960908872308735}{7003908632879207} a^{2}$, $\frac{1}{175097715821980175} a^{39} + \frac{862771630747}{175097715821980175} a^{35} + \frac{700718306285691}{175097715821980175} a^{31} - \frac{263750122213856}{35019543164396035} a^{27} + \frac{1635973729726536}{35019543164396035} a^{23} - \frac{10947004026169311}{35019543164396035} a^{19} + \frac{13454956493402128}{35019543164396035} a^{15} + \frac{4173387937832954}{35019543164396035} a^{11} - \frac{3380005680980668}{7003908632879207} a^{7} + \frac{2960908872308735}{7003908632879207} a^{3}$

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

Class group and class number

$C_{66110}$, which has order $66110$ (assuming GRH)

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:  \( -\frac{333704774931502}{175097715821980175} a^{38} - \frac{4005018972799788}{35019543164396035} a^{34} - \frac{69148663425264151}{25013959403140025} a^{30} - \frac{241765669003499887}{7003908632879207} a^{26} - \frac{1662227344020705884}{7003908632879207} a^{22} - \frac{4419908901484720599}{5002791880628005} a^{18} - \frac{1621190985026069482}{1000558376125601} a^{14} - \frac{41151113414639059803}{35019543164396035} a^{10} - \frac{277814755719135911}{1000558376125601} a^{6} - \frac{119017667899350380}{7003908632879207} a^{2} \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4715770945035671.0 \) (assuming GRH)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(i, \sqrt{5})\), 4.4.8000.1, 4.0.8000.2, 5.5.390625.1, 8.0.1024000000.2, 10.0.156250000000000.1, \(\Q(\zeta_{25})^+\), 10.0.781250000000000.1, 20.0.610351562500000000000000000000.1, 20.20.3125000000000000000000000000000000.1, 20.0.3125000000000000000000000000000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{4}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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