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Properties

Label 40.0.22670745154...0000.1
Degree $40$
Signature $[0, 20]$
Discriminant $2^{60}\cdot 5^{30}\cdot 11^{32}$
Root discriminant $64.40$
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![1048576, 0, -7864320, 0, 49807360, 0, -308412416, 0, 1904214016, 0, -2363097088, 0, 2113028096, 0, -1663434752, 0, 1196122112, 0, -541724672, 0, 198264832, 0, -63672832, 0, 17928960, 0, -3915520, 0, 747840, 0, -127072, 0, 18496, 0, -2080, 0, 212, 0, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 18*x^38 + 212*x^36 - 2080*x^34 + 18496*x^32 - 127072*x^30 + 747840*x^28 - 3915520*x^26 + 17928960*x^24 - 63672832*x^22 + 198264832*x^20 - 541724672*x^18 + 1196122112*x^16 - 1663434752*x^14 + 2113028096*x^12 - 2363097088*x^10 + 1904214016*x^8 - 308412416*x^6 + 49807360*x^4 - 7864320*x^2 + 1048576)
 
gp: K = bnfinit(x^40 - 18*x^38 + 212*x^36 - 2080*x^34 + 18496*x^32 - 127072*x^30 + 747840*x^28 - 3915520*x^26 + 17928960*x^24 - 63672832*x^22 + 198264832*x^20 - 541724672*x^18 + 1196122112*x^16 - 1663434752*x^14 + 2113028096*x^12 - 2363097088*x^10 + 1904214016*x^8 - 308412416*x^6 + 49807360*x^4 - 7864320*x^2 + 1048576, 1)
 

Normalized defining polynomial

\( x^{40} - 18 x^{38} + 212 x^{36} - 2080 x^{34} + 18496 x^{32} - 127072 x^{30} + 747840 x^{28} - 3915520 x^{26} + 17928960 x^{24} - 63672832 x^{22} + 198264832 x^{20} - 541724672 x^{18} + 1196122112 x^{16} - 1663434752 x^{14} + 2113028096 x^{12} - 2363097088 x^{10} + 1904214016 x^{8} - 308412416 x^{6} + 49807360 x^{4} - 7864320 x^{2} + 1048576 \)

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:  \(2267074515408363362607389758257584253435904000000000000000000000000000000=2^{60}\cdot 5^{30}\cdot 11^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.40$
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:  \(440=2^{3}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{440}(1,·)$, $\chi_{440}(3,·)$, $\chi_{440}(257,·)$, $\chi_{440}(9,·)$, $\chi_{440}(267,·)$, $\chi_{440}(273,·)$, $\chi_{440}(147,·)$, $\chi_{440}(27,·)$, $\chi_{440}(289,·)$, $\chi_{440}(291,·)$, $\chi_{440}(243,·)$, $\chi_{440}(49,·)$, $\chi_{440}(169,·)$, $\chi_{440}(427,·)$, $\chi_{440}(419,·)$, $\chi_{440}(433,·)$, $\chi_{440}(179,·)$, $\chi_{440}(137,·)$, $\chi_{440}(313,·)$, $\chi_{440}(59,·)$, $\chi_{440}(411,·)$, $\chi_{440}(67,·)$, $\chi_{440}(97,·)$, $\chi_{440}(201,·)$, $\chi_{440}(331,·)$, $\chi_{440}(81,·)$, $\chi_{440}(163,·)$, $\chi_{440}(203,·)$, $\chi_{440}(89,·)$, $\chi_{440}(91,·)$, $\chi_{440}(353,·)$, $\chi_{440}(379,·)$, $\chi_{440}(401,·)$, $\chi_{440}(361,·)$, $\chi_{440}(323,·)$, $\chi_{440}(177,·)$, $\chi_{440}(113,·)$, $\chi_{440}(339,·)$, $\chi_{440}(377,·)$, $\chi_{440}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{4096} a^{24}$, $\frac{1}{4096} a^{25}$, $\frac{1}{8192} a^{26}$, $\frac{1}{8192} a^{27}$, $\frac{1}{16384} a^{28}$, $\frac{1}{16384} a^{29}$, $\frac{1}{32768} a^{30}$, $\frac{1}{32768} a^{31}$, $\frac{1}{65536} a^{32}$, $\frac{1}{65536} a^{33}$, $\frac{1}{30726607481960847245312} a^{34} + \frac{6759100026384703}{7681651870490211811328} a^{32} - \frac{65833524192462789}{7681651870490211811328} a^{30} - \frac{20433294832115723}{3840825935245105905664} a^{28} - \frac{86876548576091573}{1920412967622552952832} a^{26} + \frac{50672309448483447}{480103241905638238208} a^{24} - \frac{87435839101689013}{480103241905638238208} a^{22} - \frac{4187296534272345}{240051620952819119104} a^{20} + \frac{6995903662163519}{30006452619102389888} a^{18} + \frac{21450234570504517}{15003226309551194944} a^{16} + \frac{42104619363901185}{30006452619102389888} a^{14} - \frac{22360797328965165}{3750806577387798736} a^{12} + \frac{28627908967594925}{1875403288693899368} a^{10} + \frac{91697677915639135}{3750806577387798736} a^{8} + \frac{22514369310365389}{1875403288693899368} a^{6} + \frac{94466978223430401}{937701644346949684} a^{4} - \frac{39955723921154417}{468850822173474842} a^{2} + \frac{73884285512829285}{234425411086737421}$, $\frac{1}{30726607481960847245312} a^{35} + \frac{6759100026384703}{7681651870490211811328} a^{33} - \frac{65833524192462789}{7681651870490211811328} a^{31} - \frac{20433294832115723}{3840825935245105905664} a^{29} - \frac{86876548576091573}{1920412967622552952832} a^{27} + \frac{50672309448483447}{480103241905638238208} a^{25} - \frac{87435839101689013}{480103241905638238208} a^{23} - \frac{4187296534272345}{240051620952819119104} a^{21} + \frac{6995903662163519}{30006452619102389888} a^{19} + \frac{21450234570504517}{15003226309551194944} a^{17} + \frac{42104619363901185}{30006452619102389888} a^{15} - \frac{22360797328965165}{3750806577387798736} a^{13} + \frac{28627908967594925}{1875403288693899368} a^{11} + \frac{91697677915639135}{3750806577387798736} a^{9} + \frac{22514369310365389}{1875403288693899368} a^{7} + \frac{94466978223430401}{937701644346949684} a^{5} - \frac{39955723921154417}{468850822173474842} a^{3} + \frac{73884285512829285}{234425411086737421} a$, $\frac{1}{61453214963921694490624} a^{36} - \frac{68754752360449093}{15363303740980423622656} a^{32} + \frac{77128489216892857}{7681651870490211811328} a^{30} + \frac{58849607706851771}{3840825935245105905664} a^{28} + \frac{1779715984954641}{30006452619102389888} a^{26} - \frac{16809526586405449}{240051620952819119104} a^{24} + \frac{47267567493151067}{480103241905638238208} a^{22} + \frac{25615840045361013}{120025810476409559552} a^{20} - \frac{6623296589396319}{30006452619102389888} a^{18} - \frac{73659650091367177}{60012905238204779776} a^{16} - \frac{16155923098805981}{15003226309551194944} a^{14} - \frac{53504726083231895}{15003226309551194944} a^{12} + \frac{38860356124084795}{7501613154775597472} a^{10} - \frac{105689486442819809}{3750806577387798736} a^{8} - \frac{51675247862754947}{937701644346949684} a^{6} + \frac{72194495950593475}{937701644346949684} a^{4} - \frac{48637259237983121}{468850822173474842} a^{2} - \frac{75682692523940347}{234425411086737421}$, $\frac{1}{61453214963921694490624} a^{37} - \frac{68754752360449093}{15363303740980423622656} a^{33} + \frac{77128489216892857}{7681651870490211811328} a^{31} + \frac{58849607706851771}{3840825935245105905664} a^{29} + \frac{1779715984954641}{30006452619102389888} a^{27} - \frac{16809526586405449}{240051620952819119104} a^{25} + \frac{47267567493151067}{480103241905638238208} a^{23} + \frac{25615840045361013}{120025810476409559552} a^{21} - \frac{6623296589396319}{30006452619102389888} a^{19} - \frac{73659650091367177}{60012905238204779776} a^{17} - \frac{16155923098805981}{15003226309551194944} a^{15} - \frac{53504726083231895}{15003226309551194944} a^{13} + \frac{38860356124084795}{7501613154775597472} a^{11} - \frac{105689486442819809}{3750806577387798736} a^{9} - \frac{51675247862754947}{937701644346949684} a^{7} + \frac{72194495950593475}{937701644346949684} a^{5} - \frac{48637259237983121}{468850822173474842} a^{3} - \frac{75682692523940347}{234425411086737421} a$, $\frac{1}{122906429927843388981248} a^{38} + \frac{90192232013913741}{15363303740980423622656} a^{32} + \frac{107979966750978021}{7681651870490211811328} a^{30} - \frac{3308887197797235}{240051620952819119104} a^{28} - \frac{23291812546524549}{1920412967622552952832} a^{26} - \frac{33891342427068891}{480103241905638238208} a^{24} + \frac{23840458070988403}{120025810476409559552} a^{22} - \frac{22722263288480555}{120025810476409559552} a^{20} + \frac{51412482975660505}{120025810476409559552} a^{18} - \frac{331570794339577}{937701644346949684} a^{16} + \frac{26026248887311983}{7501613154775597472} a^{14} - \frac{37711569894111983}{7501613154775597472} a^{12} - \frac{2901180300692853}{1875403288693899368} a^{10} - \frac{41576451642245981}{3750806577387798736} a^{8} - \frac{10863455489497767}{234425411086737421} a^{6} + \frac{7153085222732233}{468850822173474842} a^{4} - \frac{34922295513083563}{234425411086737421} a^{2} + \frac{73505267368585198}{234425411086737421}$, $\frac{1}{122906429927843388981248} a^{39} + \frac{90192232013913741}{15363303740980423622656} a^{33} + \frac{107979966750978021}{7681651870490211811328} a^{31} - \frac{3308887197797235}{240051620952819119104} a^{29} - \frac{23291812546524549}{1920412967622552952832} a^{27} - \frac{33891342427068891}{480103241905638238208} a^{25} + \frac{23840458070988403}{120025810476409559552} a^{23} - \frac{22722263288480555}{120025810476409559552} a^{21} + \frac{51412482975660505}{120025810476409559552} a^{19} - \frac{331570794339577}{937701644346949684} a^{17} + \frac{26026248887311983}{7501613154775597472} a^{15} - \frac{37711569894111983}{7501613154775597472} a^{13} - \frac{2901180300692853}{1875403288693899368} a^{11} - \frac{41576451642245981}{3750806577387798736} a^{9} - \frac{10863455489497767}{234425411086737421} a^{7} + \frac{7153085222732233}{468850822173474842} a^{5} - \frac{34922295513083563}{234425411086737421} a^{3} + \frac{73505267368585198}{234425411086737421} a$

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:  \( \frac{62334716573591}{30726607481960847245312} a^{38} - \frac{1932376213781321}{61453214963921694490624} a^{36} + \frac{329483501888981}{960206483811276476416} a^{34} - \frac{24751280451700561}{7681651870490211811328} a^{32} + \frac{107099948032939851}{3840825935245105905664} a^{30} - \frac{664087355496506975}{3840825935245105905664} a^{28} + \frac{1827965563520556075}{1920412967622552952832} a^{26} - \frac{2252749942091047201}{480103241905638238208} a^{24} + \frac{4726951738064208495}{240051620952819119104} a^{22} - \frac{411311174831084957}{7501613154775597472} a^{20} + \frac{4649733513373873463}{30006452619102389888} a^{18} - \frac{10797895658622258923}{30006452619102389888} a^{16} + \frac{15125385402189198455}{30006452619102389888} a^{14} + \frac{1698218026825336555}{3750806577387798736} a^{12} + \frac{1380242009250983461}{1875403288693899368} a^{10} - \frac{2307729162108954463}{3750806577387798736} a^{8} + \frac{186888385247136331}{1875403288693899368} a^{6} - \frac{15093906370319535}{937701644346949684} a^{4} + \frac{1447444828934605402}{234425411086737421} a^{2} - \frac{80144635594617}{234425411086737421} \) (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

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\zeta_{5})\), 4.4.8000.1, \(\Q(\zeta_{11})^+\), 8.0.64000000.1, 10.0.7024111812608.1, 10.10.669871503125.1, 10.0.21950349414400000.4, 20.0.481817839414250422927360000000000.3, 20.0.1402274470934209014892578125.1, 20.20.1505680748169532571648000000000000000.2

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 $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$