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Properties

Label 40.0.77195313049...0000.1
Degree $40$
Signature $[0, 20]$
Discriminant $2^{40}\cdot 3^{20}\cdot 5^{20}\cdot 11^{32}$
Root discriminant $52.75$
Ramified primes $2, 3, 5, 11$
Class number $341$ (GRH)
Class group $[341]$ (GRH)
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -45, 0, 1555, 0, -16992, 0, 122757, 0, -569703, 0, 1926084, 0, -4542648, 0, 8018080, 0, -10506063, 0, 10602647, 0, -8161689, 0, 4873258, 0, -2202717, 0, 758623, 0, -189939, 0, 35613, 0, -4707, 0, 452, 0, -27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 27*x^38 + 452*x^36 - 4707*x^34 + 35613*x^32 - 189939*x^30 + 758623*x^28 - 2202717*x^26 + 4873258*x^24 - 8161689*x^22 + 10602647*x^20 - 10506063*x^18 + 8018080*x^16 - 4542648*x^14 + 1926084*x^12 - 569703*x^10 + 122757*x^8 - 16992*x^6 + 1555*x^4 - 45*x^2 + 1)
 
gp: K = bnfinit(x^40 - 27*x^38 + 452*x^36 - 4707*x^34 + 35613*x^32 - 189939*x^30 + 758623*x^28 - 2202717*x^26 + 4873258*x^24 - 8161689*x^22 + 10602647*x^20 - 10506063*x^18 + 8018080*x^16 - 4542648*x^14 + 1926084*x^12 - 569703*x^10 + 122757*x^8 - 16992*x^6 + 1555*x^4 - 45*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{40} - 27 x^{38} + 452 x^{36} - 4707 x^{34} + 35613 x^{32} - 189939 x^{30} + 758623 x^{28} - 2202717 x^{26} + 4873258 x^{24} - 8161689 x^{22} + 10602647 x^{20} - 10506063 x^{18} + 8018080 x^{16} - 4542648 x^{14} + 1926084 x^{12} - 569703 x^{10} + 122757 x^{8} - 16992 x^{6} + 1555 x^{4} - 45 x^{2} + 1 \)

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:  \(771953130491261281023374345353486886213236398489600000000000000000000=2^{40}\cdot 3^{20}\cdot 5^{20}\cdot 11^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.75$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(660=2^{2}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{660}(1,·)$, $\chi_{660}(389,·)$, $\chi_{660}(641,·)$, $\chi_{660}(521,·)$, $\chi_{660}(631,·)$, $\chi_{660}(269,·)$, $\chi_{660}(529,·)$, $\chi_{660}(31,·)$, $\chi_{660}(289,·)$, $\chi_{660}(419,·)$, $\chi_{660}(421,·)$, $\chi_{660}(551,·)$, $\chi_{660}(169,·)$, $\chi_{660}(71,·)$, $\chi_{660}(301,·)$, $\chi_{660}(559,·)$, $\chi_{660}(49,·)$, $\chi_{660}(179,·)$, $\chi_{660}(181,·)$, $\chi_{660}(311,·)$, $\chi_{660}(59,·)$, $\chi_{660}(191,·)$, $\chi_{660}(449,·)$, $\chi_{660}(581,·)$, $\chi_{660}(199,·)$, $\chi_{660}(331,·)$, $\chi_{660}(599,·)$, $\chi_{660}(89,·)$, $\chi_{660}(91,·)$, $\chi_{660}(221,·)$, $\chi_{660}(379,·)$, $\chi_{660}(229,·)$, $\chi_{660}(401,·)$, $\chi_{660}(361,·)$, $\chi_{660}(619,·)$, $\chi_{660}(499,·)$, $\chi_{660}(119,·)$, $\chi_{660}(251,·)$, $\chi_{660}(509,·)$, $\chi_{660}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{8} a^{30} - \frac{1}{4} a^{24} - \frac{3}{8} a^{18} - \frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{3}{8}$, $\frac{1}{8} a^{31} - \frac{1}{4} a^{25} - \frac{3}{8} a^{19} - \frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{3}{8} a$, $\frac{1}{2648} a^{32} - \frac{3}{2648} a^{30} - \frac{36}{331} a^{28} - \frac{425}{1324} a^{26} - \frac{521}{1324} a^{24} - \frac{110}{331} a^{22} - \frac{499}{2648} a^{20} + \frac{1153}{2648} a^{18} - \frac{71}{331} a^{16} - \frac{151}{662} a^{14} - \frac{21}{662} a^{12} - \frac{61}{331} a^{10} - \frac{305}{662} a^{8} - \frac{289}{662} a^{6} + \frac{61}{331} a^{4} - \frac{467}{2648} a^{2} + \frac{161}{2648}$, $\frac{1}{2648} a^{33} - \frac{3}{2648} a^{31} - \frac{36}{331} a^{29} - \frac{425}{1324} a^{27} - \frac{521}{1324} a^{25} - \frac{110}{331} a^{23} - \frac{499}{2648} a^{21} + \frac{1153}{2648} a^{19} - \frac{71}{331} a^{17} - \frac{151}{662} a^{15} - \frac{21}{662} a^{13} - \frac{61}{331} a^{11} - \frac{305}{662} a^{9} - \frac{289}{662} a^{7} + \frac{61}{331} a^{5} - \frac{467}{2648} a^{3} + \frac{161}{2648} a$, $\frac{1}{2648} a^{34} + \frac{17}{1324} a^{30} + \frac{467}{1324} a^{28} - \frac{118}{331} a^{26} + \frac{157}{662} a^{24} - \frac{491}{2648} a^{22} - \frac{43}{331} a^{20} - \frac{375}{1324} a^{18} + \frac{85}{662} a^{16} + \frac{94}{331} a^{14} + \frac{73}{331} a^{12} - \frac{9}{662} a^{10} + \frac{60}{331} a^{8} + \frac{124}{331} a^{6} + \frac{997}{2648} a^{4} - \frac{155}{331} a^{2} - \frac{255}{1324}$, $\frac{1}{2648} a^{35} + \frac{17}{1324} a^{31} + \frac{467}{1324} a^{29} - \frac{118}{331} a^{27} + \frac{157}{662} a^{25} - \frac{491}{2648} a^{23} - \frac{43}{331} a^{21} - \frac{375}{1324} a^{19} + \frac{85}{662} a^{17} + \frac{94}{331} a^{15} + \frac{73}{331} a^{13} - \frac{9}{662} a^{11} + \frac{60}{331} a^{9} + \frac{124}{331} a^{7} + \frac{997}{2648} a^{5} - \frac{155}{331} a^{3} - \frac{255}{1324} a$, $\frac{1}{1757670105895448} a^{36} + \frac{125745432187}{1757670105895448} a^{34} + \frac{53611515429}{439417526473862} a^{32} - \frac{48695506214033}{1757670105895448} a^{30} - \frac{181109982530143}{878835052947724} a^{28} + \frac{13685539502581}{219708763236931} a^{26} + \frac{167333300054771}{1757670105895448} a^{24} - \frac{831415846775529}{1757670105895448} a^{22} + \frac{161388495202393}{439417526473862} a^{20} + \frac{262987820129721}{1757670105895448} a^{18} - \frac{213541105864217}{439417526473862} a^{16} + \frac{70833740732344}{219708763236931} a^{14} - \frac{81468512342141}{219708763236931} a^{12} - \frac{18802528569891}{439417526473862} a^{10} - \frac{35013202274552}{219708763236931} a^{8} - \frac{487696764048079}{1757670105895448} a^{6} - \frac{399124144934985}{1757670105895448} a^{4} - \frac{70135783645755}{439417526473862} a^{2} + \frac{362063534391965}{1757670105895448}$, $\frac{1}{1757670105895448} a^{37} + \frac{125745432187}{1757670105895448} a^{35} + \frac{53611515429}{439417526473862} a^{33} - \frac{48695506214033}{1757670105895448} a^{31} - \frac{181109982530143}{878835052947724} a^{29} + \frac{13685539502581}{219708763236931} a^{27} + \frac{167333300054771}{1757670105895448} a^{25} - \frac{831415846775529}{1757670105895448} a^{23} + \frac{161388495202393}{439417526473862} a^{21} + \frac{262987820129721}{1757670105895448} a^{19} - \frac{213541105864217}{439417526473862} a^{17} + \frac{70833740732344}{219708763236931} a^{15} - \frac{81468512342141}{219708763236931} a^{13} - \frac{18802528569891}{439417526473862} a^{11} - \frac{35013202274552}{219708763236931} a^{9} - \frac{487696764048079}{1757670105895448} a^{7} - \frac{399124144934985}{1757670105895448} a^{5} - \frac{70135783645755}{439417526473862} a^{3} + \frac{362063534391965}{1757670105895448} a$, $\frac{1}{190464294096175871036769228047358678728} a^{38} - \frac{12419385469845324074091}{190464294096175871036769228047358678728} a^{36} - \frac{2626553875050354705426570913118893}{190464294096175871036769228047358678728} a^{34} - \frac{15071263560068018848555524440096911}{190464294096175871036769228047358678728} a^{32} + \frac{1990972507113331760226977946658621663}{47616073524043967759192307011839669682} a^{30} + \frac{20044986708948107901293027956923789613}{95232147048087935518384614023679339364} a^{28} + \frac{79731729239473220507171632599672089831}{190464294096175871036769228047358678728} a^{26} - \frac{39747562924451927285449177822031175571}{190464294096175871036769228047358678728} a^{24} - \frac{39361251771117294806077253772510935145}{190464294096175871036769228047358678728} a^{22} - \frac{59853549823964304167271614987235813}{790308274257991166127673145424724808} a^{20} + \frac{3219125190743403549407993980436057613}{95232147048087935518384614023679339364} a^{18} + \frac{7680545230823462716762344296445445269}{47616073524043967759192307011839669682} a^{16} + \frac{2505286755342207888832040408823481949}{23808036762021983879596153505919834841} a^{14} + \frac{18670488324485711289587245478539939469}{47616073524043967759192307011839669682} a^{12} - \frac{1409430317026641595902145617367705555}{47616073524043967759192307011839669682} a^{10} + \frac{2022860239520441059331771826177264081}{190464294096175871036769228047358678728} a^{8} - \frac{66961405942924845060844912800926535}{135176929805660660778402574909409992} a^{6} + \frac{68082084537433385004465934399368879023}{190464294096175871036769228047358678728} a^{4} - \frac{11320708239066942685475853954834872113}{190464294096175871036769228047358678728} a^{2} + \frac{30780413146575889301252993814540343119}{95232147048087935518384614023679339364}$, $\frac{1}{190464294096175871036769228047358678728} a^{39} - \frac{12419385469845324074091}{190464294096175871036769228047358678728} a^{37} - \frac{2626553875050354705426570913118893}{190464294096175871036769228047358678728} a^{35} - \frac{15071263560068018848555524440096911}{190464294096175871036769228047358678728} a^{33} + \frac{1990972507113331760226977946658621663}{47616073524043967759192307011839669682} a^{31} + \frac{20044986708948107901293027956923789613}{95232147048087935518384614023679339364} a^{29} + \frac{79731729239473220507171632599672089831}{190464294096175871036769228047358678728} a^{27} - \frac{39747562924451927285449177822031175571}{190464294096175871036769228047358678728} a^{25} - \frac{39361251771117294806077253772510935145}{190464294096175871036769228047358678728} a^{23} - \frac{59853549823964304167271614987235813}{790308274257991166127673145424724808} a^{21} + \frac{3219125190743403549407993980436057613}{95232147048087935518384614023679339364} a^{19} + \frac{7680545230823462716762344296445445269}{47616073524043967759192307011839669682} a^{17} + \frac{2505286755342207888832040408823481949}{23808036762021983879596153505919834841} a^{15} + \frac{18670488324485711289587245478539939469}{47616073524043967759192307011839669682} a^{13} - \frac{1409430317026641595902145617367705555}{47616073524043967759192307011839669682} a^{11} + \frac{2022860239520441059331771826177264081}{190464294096175871036769228047358678728} a^{9} - \frac{66961405942924845060844912800926535}{135176929805660660778402574909409992} a^{7} + \frac{68082084537433385004465934399368879023}{190464294096175871036769228047358678728} a^{5} - \frac{11320708239066942685475853954834872113}{190464294096175871036769228047358678728} a^{3} + \frac{30780413146575889301252993814540343119}{95232147048087935518384614023679339364} a$

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

Class group and class number

$C_{341}$, which has order $341$ (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:  \( -\frac{27316684355959916833092637517528873027}{190464294096175871036769228047358678728} a^{39} + \frac{184786157913599474017454827572781642377}{47616073524043967759192307011839669682} a^{37} - \frac{184925026152781696588747275062874615211}{2842750658151878672190585493244159384} a^{35} + \frac{64647315661418559540151101414199990632063}{95232147048087935518384614023679339364} a^{33} - \frac{980239929196757703403305316205883316441685}{190464294096175871036769228047358678728} a^{31} + \frac{2622156757649396795370564131466026052406637}{95232147048087935518384614023679339364} a^{29} - \frac{21018603182167478807582811611463321395513007}{190464294096175871036769228047358678728} a^{27} + \frac{30670925038352686297381961920571351569550571}{95232147048087935518384614023679339364} a^{25} - \frac{136480382534396773319898606663283980600718213}{190464294096175871036769228047358678728} a^{23} + \frac{238881564242539777515945593236944320144517}{197577068564497791531918286356181202} a^{21} - \frac{301686455636751478653712570025779821008056249}{190464294096175871036769228047358678728} a^{19} + \frac{75581490226939698786760573599790549057375565}{47616073524043967759192307011839669682} a^{17} - \frac{58446617004531023945757994856826333473082225}{47616073524043967759192307011839669682} a^{15} + \frac{33740394388690060158903629494688515166346657}{47616073524043967759192307011839669682} a^{13} - \frac{14612998976656374992952461211789402745508809}{47616073524043967759192307011839669682} a^{11} + \frac{17877224119426929500843934407316000783482505}{190464294096175871036769228047358678728} a^{9} - \frac{494799451533960849617512891601807511701572}{23808036762021983879596153505919834841} a^{7} + \frac{576125839469863601959393448442508266697507}{190464294096175871036769228047358678728} a^{5} - \frac{6542627175778891895214920494175257391033}{23808036762021983879596153505919834841} a^{3} + \frac{1518659142800836072755025387903128043839}{190464294096175871036769228047358678728} a \) (order $12$)
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:  \( 17492090815064632 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_{10}$ (as 40T7):

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 $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{15}) \), \(\Q(i, \sqrt{5})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\zeta_{11})^+\), 8.0.12960000.1, 10.0.219503494144.1, 10.10.669871503125.1, 10.0.685948419200000.1, 10.0.52089208083.1, 10.10.53339349076992.1, 10.0.162778775259375.1, 10.10.166685465865600000.1, 20.0.470525233802978928640000000000.1, 20.0.2845086159957207322343768064.1, 20.0.27784044530832102757263360000000000.3, 20.0.26496929674942114598525390625.1, 20.20.27784044530832102757263360000000000.1, 20.0.27784044530832102757263360000000000.2, 20.0.27784044530832102757263360000000000.1

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 R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
3Data not computed
5Data not computed
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$