Properties

Label 42.42.496...469.1
Degree $42$
Signature $[42, 0]$
Discriminant $4.969\times 10^{80}$
Root discriminant $83.43$
Ramified primes $7, 29$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{42}$ (as 42T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 - x^41 - 67*x^40 + 62*x^39 + 1993*x^38 - 1704*x^37 - 34847*x^36 + 27519*x^35 + 399768*x^34 - 291838*x^33 - 3181854*x^32 + 2151119*x^31 + 18125561*x^30 - 11376768*x^29 - 75199314*x^28 + 43948028*x^27 + 229267037*x^26 - 125109956*x^25 - 515411960*x^24 + 263200585*x^23 + 854114535*x^22 - 408541162*x^21 - 1040280967*x^20 + 465602383*x^19 + 926307827*x^18 - 386475878*x^17 - 597992464*x^16 + 230741151*x^15 + 276295754*x^14 - 97190872*x^13 - 89602671*x^12 + 28039534*x^11 + 19807180*x^10 - 5297092*x^9 - 2853616*x^8 + 611802*x^7 + 248962*x^6 - 38836*x^5 - 11508*x^4 + 1155*x^3 + 217*x^2 - 14*x - 1)
 
gp: K = bnfinit(x^42 - x^41 - 67*x^40 + 62*x^39 + 1993*x^38 - 1704*x^37 - 34847*x^36 + 27519*x^35 + 399768*x^34 - 291838*x^33 - 3181854*x^32 + 2151119*x^31 + 18125561*x^30 - 11376768*x^29 - 75199314*x^28 + 43948028*x^27 + 229267037*x^26 - 125109956*x^25 - 515411960*x^24 + 263200585*x^23 + 854114535*x^22 - 408541162*x^21 - 1040280967*x^20 + 465602383*x^19 + 926307827*x^18 - 386475878*x^17 - 597992464*x^16 + 230741151*x^15 + 276295754*x^14 - 97190872*x^13 - 89602671*x^12 + 28039534*x^11 + 19807180*x^10 - 5297092*x^9 - 2853616*x^8 + 611802*x^7 + 248962*x^6 - 38836*x^5 - 11508*x^4 + 1155*x^3 + 217*x^2 - 14*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -14, 217, 1155, -11508, -38836, 248962, 611802, -2853616, -5297092, 19807180, 28039534, -89602671, -97190872, 276295754, 230741151, -597992464, -386475878, 926307827, 465602383, -1040280967, -408541162, 854114535, 263200585, -515411960, -125109956, 229267037, 43948028, -75199314, -11376768, 18125561, 2151119, -3181854, -291838, 399768, 27519, -34847, -1704, 1993, 62, -67, -1, 1]);
 

\( x^{42} - x^{41} - 67 x^{40} + 62 x^{39} + 1993 x^{38} - 1704 x^{37} - 34847 x^{36} + 27519 x^{35} + 399768 x^{34} - 291838 x^{33} - 3181854 x^{32} + 2151119 x^{31} + 18125561 x^{30} - 11376768 x^{29} - 75199314 x^{28} + 43948028 x^{27} + 229267037 x^{26} - 125109956 x^{25} - 515411960 x^{24} + 263200585 x^{23} + 854114535 x^{22} - 408541162 x^{21} - 1040280967 x^{20} + 465602383 x^{19} + 926307827 x^{18} - 386475878 x^{17} - 597992464 x^{16} + 230741151 x^{15} + 276295754 x^{14} - 97190872 x^{13} - 89602671 x^{12} + 28039534 x^{11} + 19807180 x^{10} - 5297092 x^{9} - 2853616 x^{8} + 611802 x^{7} + 248962 x^{6} - 38836 x^{5} - 11508 x^{4} + 1155 x^{3} + 217 x^{2} - 14 x - 1 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[42, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(496\!\cdots\!469\)\(\medspace = 7^{28}\cdot 29^{39}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $83.43$
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:  $7, 29$
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:  \(203=7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{203}(1,·)$, $\chi_{203}(4,·)$, $\chi_{203}(51,·)$, $\chi_{203}(9,·)$, $\chi_{203}(23,·)$, $\chi_{203}(141,·)$, $\chi_{203}(16,·)$, $\chi_{203}(149,·)$, $\chi_{203}(22,·)$, $\chi_{203}(151,·)$, $\chi_{203}(25,·)$, $\chi_{203}(158,·)$, $\chi_{203}(36,·)$, $\chi_{203}(165,·)$, $\chi_{203}(169,·)$, $\chi_{203}(170,·)$, $\chi_{203}(179,·)$, $\chi_{203}(30,·)$, $\chi_{203}(183,·)$, $\chi_{203}(57,·)$, $\chi_{203}(190,·)$, $\chi_{203}(53,·)$, $\chi_{203}(64,·)$, $\chi_{203}(65,·)$, $\chi_{203}(67,·)$, $\chi_{203}(197,·)$, $\chi_{203}(198,·)$, $\chi_{203}(71,·)$, $\chi_{203}(74,·)$, $\chi_{203}(78,·)$, $\chi_{203}(81,·)$, $\chi_{203}(86,·)$, $\chi_{203}(88,·)$, $\chi_{203}(92,·)$, $\chi_{203}(93,·)$, $\chi_{203}(144,·)$, $\chi_{203}(100,·)$, $\chi_{203}(107,·)$, $\chi_{203}(109,·)$, $\chi_{203}(120,·)$, $\chi_{203}(121,·)$$\chi_{203}(123,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $\frac{1}{161587} a^{40} + \frac{14993}{161587} a^{39} - \frac{20872}{161587} a^{38} + \frac{6467}{161587} a^{37} - \frac{78294}{161587} a^{36} + \frac{73150}{161587} a^{35} + \frac{40923}{161587} a^{34} - \frac{53623}{161587} a^{33} + \frac{34818}{161587} a^{32} - \frac{5138}{161587} a^{31} - \frac{20811}{161587} a^{30} - \frac{9494}{161587} a^{29} - \frac{64573}{161587} a^{28} - \frac{17565}{161587} a^{27} + \frac{67548}{161587} a^{26} - \frac{72957}{161587} a^{25} + \frac{64384}{161587} a^{24} + \frac{37918}{161587} a^{23} + \frac{54743}{161587} a^{22} - \frac{24140}{161587} a^{21} - \frac{73110}{161587} a^{20} - \frac{37276}{161587} a^{19} - \frac{68576}{161587} a^{18} + \frac{14181}{161587} a^{17} + \frac{24460}{161587} a^{16} - \frac{22901}{161587} a^{15} - \frac{64373}{161587} a^{14} - \frac{79095}{161587} a^{13} - \frac{1482}{161587} a^{12} - \frac{19861}{161587} a^{11} - \frac{78451}{161587} a^{10} + \frac{57389}{161587} a^{9} + \frac{36447}{161587} a^{8} - \frac{29715}{161587} a^{7} + \frac{50248}{161587} a^{6} - \frac{2427}{161587} a^{5} - \frac{46822}{161587} a^{4} + \frac{24309}{161587} a^{3} - \frac{78848}{161587} a^{2} + \frac{68277}{161587} a + \frac{44969}{161587}$, $\frac{1}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{41} - \frac{63523844410928982431213022042258852950848595439517498150008975664866511810398068473121072171}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{40} - \frac{9409388652117697079862136863846737407839229300376354312552932256362041643381302110027100136117179}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{39} - \frac{8329609683028909868799189971946592649712621800178366219273288135605285545419986669847246862906249}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{38} - \frac{13103569031887411233242263457825791578563146909318718568366280867896707965454226505746506147944279}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{37} - \frac{1405396844920389159480515833181225470730870185224852340440601891905237027621324802010713006024828}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{36} + \frac{5317052274234269202111144527063378700204753720433491978509682743912590280081360815038665515950157}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{35} + \frac{205780738600070835532434053678843981949628702352940265953813150667814510826080635791351597999050}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{34} - \frac{6464132664831052698694288361419417523819459364695680171711481139929098971219720451672387086150142}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{33} + \frac{3385696621423724157726813001705066468201860226681447418595162798011584578356897585034336667280826}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{32} + \frac{5314551020886437468872717351601966356608869029903905857990543242229318394925957853946412119942066}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{31} - \frac{9350706539562107679889160246135122917164846554602926549820116230353893533680245632327975030222921}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{30} - \frac{12089626498535271801624905505829884393324906756244083115689262067505800098963992332475130486762151}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{29} + \frac{4222221395955314615084682994196550659188395219055922310436349224412296105826606721280507402741557}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{28} - \frac{4452473033343320440525489302784499449592886031196909797066870796843135589422030472839742977067537}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{27} - \frac{11138407266752602927866366587084616099935788379165330207613372779570591063433651433557854496563635}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{26} - \frac{2882705386373059151527822612581514707353157699641758584264730309270840822574919245714270059910342}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{25} - \frac{7506536157793489198211255704883703029554161305819946541287531245402016832435661796055825170608846}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{24} - \frac{4052235761174456639935270609129635521034848731440219356449365483871570203016451960592937587930853}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{23} - \frac{9787460313221486811966179523881486022787731260701409036734759035009856199219469509953699620743588}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{22} - \frac{249850186862415922599164049511175686267530462705827933119250284227187287699605722696741428526424}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{21} - \frac{10625887964566455118893346730478288262260585692116371060550295998976524040254229355677698271616899}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{20} + \frac{7688415165586736688660033226784072375747091392284272451317526862018180816680418391494347883836097}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{19} + \frac{8431190433902142236163269269227593573577642445741278007225291477648215126200226126529231222829237}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{18} + \frac{8177736659445694489112521107090765401963133161736682478308371738641747177577013587515525865525173}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{17} - \frac{12026337406178981346908266034377621370337508467197299121703257971608309981021171800005171395964315}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{16} + \frac{1852084775047534739371615612370547930739441212294391272587260325436492593914316394559007855274910}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{15} + \frac{11121220440291383083068100960777190988053773642346910306684707159632403755567337935857093965617746}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{14} + \frac{246624371098978216564288346299553627773761749346886667599000641684934423912528486573976885540997}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{13} + \frac{9515962533521679196482718712416250562572455445058686849278790661593617663342142843445879958409642}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{12} + \frac{4821359618343689651971879291572218042876274659242134247159566416747694011856524128059963407351285}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{11} + \frac{8286020219251350698536710165753835973403195722000700376917309045298224844429934946670390023346777}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{10} - \frac{8462793990987760770530859783121198754466335148908297544404299976223746579897954459202291573171356}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{9} - \frac{9000244859428609843472352960674640088503283537998997593660702614947759764711774397301634676828981}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{8} + \frac{2171563590513233561785136597846704898150037287005587820208214331617078205681979705461600755599294}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{7} - \frac{2828721674566339196234654624843978620702711758511310659783686713455244660176614472785952455400932}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{6} + \frac{11057276046798034276623891262161411486974570613448597380770904242242487349772989906851288388258823}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{5} + \frac{11692092720809740940468258055822021069210454961962187541243896620982850653124719385453655576894276}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{4} + \frac{6428799841265556815660180463283838652380232882431038097111243709886063373553544467687935498515124}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{3} + \frac{12274235086900834744759788222921716512427512887345383194596716557993918383652455100711832083932255}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a^{2} + \frac{9650707139731862520264073343788792126218517162000604352663950272231085729711310051881270746348981}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151} a + \frac{2120438473819317273780823928439315274261603824566554148112979243005739272349844026490428383317827}{26440770602838375959086217965893686237934016604065386758846139632731499428698372589516698880908151}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $41$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1205750664686969300000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{42}\cdot(2\pi)^{0}\cdot 1205750664686969300000000000 \cdot 1}{2\sqrt{496897759422042196258605771077406782550407598249513303021389442457964675897236469}}\approx 0.118947087634194$ (assuming GRH)

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

\(\Q(\sqrt{29}) \), \(\Q(\zeta_{7})^+\), 6.6.58557989.1, 7.7.594823321.1, \(\Q(\zeta_{29})^+\), 21.21.142736986105602839685204351151303673689.1

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 $42$ $42$ $21^{2}$ R $42$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{7}$ $42$ $21^{2}$ R $42$ $42$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{21}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{3}$ $42$ $21^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{14}$

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
7Data not computed
$29$29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$