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)

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Normalized defining 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

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), 7.7.594823321.1, 14.0.773792930870360792667.1, 21.21.4814587615056751193058435502319478353721.1

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 $42$ 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