Properties

 Label 42.0.506...267.1 Degree $42$ Signature $[0, 21]$ Discriminant $-5.070\times 10^{82}$ Root discriminant $93.15$ Ramified primes $3, 29$ Class number not computed Class group not computed Galois group $C_{42}$ (as 42T1)

Related objects

Show commands for: SageMath / Pari/GP / Magma

Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 - 16*x^39 + 1066*x^36 + 13384*x^33 + 639480*x^30 + 581064*x^27 + 10535858*x^24 - 25402305*x^21 + 171805922*x^18 - 184669454*x^15 + 198189968*x^12 + 3751640*x^9 + 57337*x^6 + 267*x^3 + 1)

gp: K = bnfinit(x^42 - 16*x^39 + 1066*x^36 + 13384*x^33 + 639480*x^30 + 581064*x^27 + 10535858*x^24 - 25402305*x^21 + 171805922*x^18 - 184669454*x^15 + 198189968*x^12 + 3751640*x^9 + 57337*x^6 + 267*x^3 + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 267, 0, 0, 57337, 0, 0, 3751640, 0, 0, 198189968, 0, 0, -184669454, 0, 0, 171805922, 0, 0, -25402305, 0, 0, 10535858, 0, 0, 581064, 0, 0, 639480, 0, 0, 13384, 0, 0, 1066, 0, 0, -16, 0, 0, 1]);

$$x^{42} - 16 x^{39} + 1066 x^{36} + 13384 x^{33} + 639480 x^{30} + 581064 x^{27} + 10535858 x^{24} - 25402305 x^{21} + 171805922 x^{18} - 184669454 x^{15} + 198189968 x^{12} + 3751640 x^{9} + 57337 x^{6} + 267 x^{3} + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

Invariants

 Degree: $42$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 21]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-50\!\cdots\!267$$$$\medspace = -\,3^{63}\cdot 29^{36}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $93.15$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $3, 29$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Gal(K/\Q)|$: $42$ This field is Galois and abelian over $\Q$. Conductor: $$261=3^{2}\cdot 29$$ Dirichlet character group: $\lbrace$$\chi_{261}(256,·), \chi_{261}(1,·), \chi_{261}(7,·), \chi_{261}(136,·), \chi_{261}(139,·), \chi_{261}(140,·), \chi_{261}(16,·), \chi_{261}(146,·), \chi_{261}(20,·), \chi_{261}(23,·), \chi_{261}(152,·), \chi_{261}(25,·), \chi_{261}(161,·), \chi_{261}(169,·), \chi_{261}(170,·), \chi_{261}(257,·), \chi_{261}(175,·), \chi_{261}(49,·), \chi_{261}(52,·), \chi_{261}(53,·), \chi_{261}(59,·), \chi_{261}(190,·), \chi_{261}(181,·), \chi_{261}(65,·), \chi_{261}(194,·), \chi_{261}(197,·), \chi_{261}(199,·), \chi_{261}(74,·), \chi_{261}(82,·), \chi_{261}(83,·), \chi_{261}(88,·), \chi_{261}(94,·), \chi_{261}(223,·), \chi_{261}(226,·), \chi_{261}(227,·), \chi_{261}(103,·), \chi_{261}(233,·), \chi_{261}(107,·), \chi_{261}(110,·), \chi_{261}(239,·), \chi_{261}(112,·)$$\chi_{261}(248,·)$$\rbrace$ This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{171721640178611} a^{36} - \frac{83285078158257}{171721640178611} a^{33} - \frac{41211870895966}{171721640178611} a^{30} - \frac{55415557325085}{171721640178611} a^{27} + \frac{67310111522949}{171721640178611} a^{24} - \frac{34080177541187}{171721640178611} a^{21} + \frac{48573519142526}{171721640178611} a^{18} + \frac{77963249104841}{171721640178611} a^{15} - \frac{48617152457367}{171721640178611} a^{12} - \frac{22823869911138}{171721640178611} a^{9} + \frac{34054536605198}{171721640178611} a^{6} - \frac{54873007010306}{171721640178611} a^{3} + \frac{17212569281587}{171721640178611}$, $\frac{1}{171721640178611} a^{37} - \frac{83285078158257}{171721640178611} a^{34} - \frac{41211870895966}{171721640178611} a^{31} - \frac{55415557325085}{171721640178611} a^{28} + \frac{67310111522949}{171721640178611} a^{25} - \frac{34080177541187}{171721640178611} a^{22} + \frac{48573519142526}{171721640178611} a^{19} + \frac{77963249104841}{171721640178611} a^{16} - \frac{48617152457367}{171721640178611} a^{13} - \frac{22823869911138}{171721640178611} a^{10} + \frac{34054536605198}{171721640178611} a^{7} - \frac{54873007010306}{171721640178611} a^{4} + \frac{17212569281587}{171721640178611} a$, $\frac{1}{171721640178611} a^{38} - \frac{83285078158257}{171721640178611} a^{35} - \frac{41211870895966}{171721640178611} a^{32} - \frac{55415557325085}{171721640178611} a^{29} + \frac{67310111522949}{171721640178611} a^{26} - \frac{34080177541187}{171721640178611} a^{23} + \frac{48573519142526}{171721640178611} a^{20} + \frac{77963249104841}{171721640178611} a^{17} - \frac{48617152457367}{171721640178611} a^{14} - \frac{22823869911138}{171721640178611} a^{11} + \frac{34054536605198}{171721640178611} a^{8} - \frac{54873007010306}{171721640178611} a^{5} + \frac{17212569281587}{171721640178611} a^{2}$, $\frac{1}{2676106971224551374446759534886634335460916386021997} a^{39} - \frac{2733600406176698800082013841193351995}{2676106971224551374446759534886634335460916386021997} a^{36} + \frac{697481714756157566936323746025851258017886698456933}{2676106971224551374446759534886634335460916386021997} a^{33} + \frac{540510634217631627085731956652564434292061826463848}{2676106971224551374446759534886634335460916386021997} a^{30} + \frac{596618160069694753052241832560344202265855799802264}{2676106971224551374446759534886634335460916386021997} a^{27} + \frac{664305371317999207334906519793259490916118455337018}{2676106971224551374446759534886634335460916386021997} a^{24} + \frac{174831254994100485180084623317309736643308061807121}{2676106971224551374446759534886634335460916386021997} a^{21} + \frac{190323854239706666721449514804126997978700975373557}{2676106971224551374446759534886634335460916386021997} a^{18} - \frac{1063887725464180880181211218507818742216832958127012}{2676106971224551374446759534886634335460916386021997} a^{15} + \frac{646676432853899666234832341348358426472506778396256}{2676106971224551374446759534886634335460916386021997} a^{12} + \frac{26863433736037598637849655274943817258327057688034}{157418057130855963202750560875684372674171552118941} a^{9} - \frac{436339076642108033816185817726200765408402804815822}{2676106971224551374446759534886634335460916386021997} a^{6} + \frac{1269362859785224150078525564452420187918037269915645}{2676106971224551374446759534886634335460916386021997} a^{3} + \frac{152220266972783025633111445818642166975900797034703}{2676106971224551374446759534886634335460916386021997}$, $\frac{1}{2676106971224551374446759534886634335460916386021997} a^{40} - \frac{2733600406176698800082013841193351995}{2676106971224551374446759534886634335460916386021997} a^{37} + \frac{697481714756157566936323746025851258017886698456933}{2676106971224551374446759534886634335460916386021997} a^{34} + \frac{540510634217631627085731956652564434292061826463848}{2676106971224551374446759534886634335460916386021997} a^{31} + \frac{596618160069694753052241832560344202265855799802264}{2676106971224551374446759534886634335460916386021997} a^{28} + \frac{664305371317999207334906519793259490916118455337018}{2676106971224551374446759534886634335460916386021997} a^{25} + \frac{174831254994100485180084623317309736643308061807121}{2676106971224551374446759534886634335460916386021997} a^{22} + \frac{190323854239706666721449514804126997978700975373557}{2676106971224551374446759534886634335460916386021997} a^{19} - \frac{1063887725464180880181211218507818742216832958127012}{2676106971224551374446759534886634335460916386021997} a^{16} + \frac{646676432853899666234832341348358426472506778396256}{2676106971224551374446759534886634335460916386021997} a^{13} + \frac{26863433736037598637849655274943817258327057688034}{157418057130855963202750560875684372674171552118941} a^{10} - \frac{436339076642108033816185817726200765408402804815822}{2676106971224551374446759534886634335460916386021997} a^{7} + \frac{1269362859785224150078525564452420187918037269915645}{2676106971224551374446759534886634335460916386021997} a^{4} + \frac{152220266972783025633111445818642166975900797034703}{2676106971224551374446759534886634335460916386021997} a$, $\frac{1}{2676106971224551374446759534886634335460916386021997} a^{41} - \frac{2733600406176698800082013841193351995}{2676106971224551374446759534886634335460916386021997} a^{38} + \frac{697481714756157566936323746025851258017886698456933}{2676106971224551374446759534886634335460916386021997} a^{35} + \frac{540510634217631627085731956652564434292061826463848}{2676106971224551374446759534886634335460916386021997} a^{32} + \frac{596618160069694753052241832560344202265855799802264}{2676106971224551374446759534886634335460916386021997} a^{29} + \frac{664305371317999207334906519793259490916118455337018}{2676106971224551374446759534886634335460916386021997} a^{26} + \frac{174831254994100485180084623317309736643308061807121}{2676106971224551374446759534886634335460916386021997} a^{23} + \frac{190323854239706666721449514804126997978700975373557}{2676106971224551374446759534886634335460916386021997} a^{20} - \frac{1063887725464180880181211218507818742216832958127012}{2676106971224551374446759534886634335460916386021997} a^{17} + \frac{646676432853899666234832341348358426472506778396256}{2676106971224551374446759534886634335460916386021997} a^{14} + \frac{26863433736037598637849655274943817258327057688034}{157418057130855963202750560875684372674171552118941} a^{11} - \frac{436339076642108033816185817726200765408402804815822}{2676106971224551374446759534886634335460916386021997} a^{8} + \frac{1269362859785224150078525564452420187918037269915645}{2676106971224551374446759534886634335460916386021997} a^{5} + \frac{152220266972783025633111445818642166975900797034703}{2676106971224551374446759534886634335460916386021997} a^{2}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

Class group and class number

not computed

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $20$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$\frac{506555599156652661449309472991747718983058900032}{157418057130855963202750560875684372674171552118941} a^{40} - \frac{8114583616242293050940009467933134571260805848818}{157418057130855963202750560875684372674171552118941} a^{37} + \frac{540143405824382084789333222256098190206354875785817}{157418057130855963202750560875684372674171552118941} a^{34} + \frac{6769405777152002546262368018472930814863807568729202}{157418057130855963202750560875684372674171552118941} a^{31} + \frac{323802464519563768341463442999828448581845183651136849}{157418057130855963202750560875684372674171552118941} a^{28} + \frac{288142517330108029904735240247623985318031699094195776}{157418057130855963202750560875684372674171552118941} a^{25} + \frac{5331385835500769726808039233660652606344991507052105504}{157418057130855963202750560875684372674171552118941} a^{22} - \frac{12969798271891141628500682868998519653854136096029691491}{157418057130855963202750560875684372674171552118941} a^{19} + \frac{87275844225049443301551336843866278720782741443613971589}{157418057130855963202750560875684372674171552118941} a^{16} - \frac{95211707577760126850777186038687512368139969173505476623}{157418057130855963202750560875684372674171552118941} a^{13} + \frac{102190137672910470361389908578234127631442163567919249487}{157418057130855963202750560875684372674171552118941} a^{10} - \frac{27539214918725399502213103022121123741017756427717838}{157418057130855963202750560875684372674171552118941} a^{7} - \frac{128239545942073126390259283430967160595130463488664}{157418057130855963202750560875684372674171552118941} a^{4} - \frac{927575793299550729533637408652827313433016575552919}{157418057130855963202750560875684372674171552118941} a$$ (order $18$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: not computed sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: not computed sage: K.regulator()  gp: K.reg  magma: Regulator(K);

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s)$ not computed

Galois group

$C_{42}$ (as 42T1):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A cyclic group of order 42 The 42 conjugacy class representatives for $C_{42}$ Character table for $C_{42}$ is not computed

Intermediate fields

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 $42$ R $42$ $21^{2}$ $42$ $21^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{21}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{6}$ $42$ R $21^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{7}$ $21^{2}$ $42$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{7}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])

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];

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
29Data not computed