Properties

Label 42.42.104...125.1
Degree $42$
Signature $[42, 0]$
Discriminant $1.040\times 10^{80}$
Root discriminant $80.38$
Ramified primes $5, 43$
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 - x^41 - 61*x^40 + 56*x^39 + 1654*x^38 - 1379*x^37 - 26380*x^36 + 19735*x^35 + 276302*x^34 - 183047*x^33 - 2012061*x^32 + 1164081*x^31 + 10536175*x^30 - 5248100*x^29 - 40536685*x^28 + 17139885*x^27 + 116229751*x^26 - 41141876*x^25 - 250648402*x^24 + 73243022*x^23 + 408521488*x^22 - 97102633*x^21 - 503598537*x^20 + 95768202*x^19 + 467783098*x^18 - 69790048*x^17 - 324502952*x^16 + 37071837*x^15 + 165565463*x^14 - 14037193*x^13 - 60679502*x^12 + 3661307*x^11 + 15419833*x^10 - 625768*x^9 - 2575782*x^8 + 66132*x^7 + 260062*x^6 - 4477*x^5 - 13750*x^4 + 275*x^3 + 286*x^2 - 11*x - 1)
 
gp: K = bnfinit(x^42 - x^41 - 61*x^40 + 56*x^39 + 1654*x^38 - 1379*x^37 - 26380*x^36 + 19735*x^35 + 276302*x^34 - 183047*x^33 - 2012061*x^32 + 1164081*x^31 + 10536175*x^30 - 5248100*x^29 - 40536685*x^28 + 17139885*x^27 + 116229751*x^26 - 41141876*x^25 - 250648402*x^24 + 73243022*x^23 + 408521488*x^22 - 97102633*x^21 - 503598537*x^20 + 95768202*x^19 + 467783098*x^18 - 69790048*x^17 - 324502952*x^16 + 37071837*x^15 + 165565463*x^14 - 14037193*x^13 - 60679502*x^12 + 3661307*x^11 + 15419833*x^10 - 625768*x^9 - 2575782*x^8 + 66132*x^7 + 260062*x^6 - 4477*x^5 - 13750*x^4 + 275*x^3 + 286*x^2 - 11*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -11, 286, 275, -13750, -4477, 260062, 66132, -2575782, -625768, 15419833, 3661307, -60679502, -14037193, 165565463, 37071837, -324502952, -69790048, 467783098, 95768202, -503598537, -97102633, 408521488, 73243022, -250648402, -41141876, 116229751, 17139885, -40536685, -5248100, 10536175, 1164081, -2012061, -183047, 276302, 19735, -26380, -1379, 1654, 56, -61, -1, 1]);
 

\( x^{42} - x^{41} - 61 x^{40} + 56 x^{39} + 1654 x^{38} - 1379 x^{37} - 26380 x^{36} + 19735 x^{35} + 276302 x^{34} - 183047 x^{33} - 2012061 x^{32} + 1164081 x^{31} + 10536175 x^{30} - 5248100 x^{29} - 40536685 x^{28} + 17139885 x^{27} + 116229751 x^{26} - 41141876 x^{25} - 250648402 x^{24} + 73243022 x^{23} + 408521488 x^{22} - 97102633 x^{21} - 503598537 x^{20} + 95768202 x^{19} + 467783098 x^{18} - 69790048 x^{17} - 324502952 x^{16} + 37071837 x^{15} + 165565463 x^{14} - 14037193 x^{13} - 60679502 x^{12} + 3661307 x^{11} + 15419833 x^{10} - 625768 x^{9} - 2575782 x^{8} + 66132 x^{7} + 260062 x^{6} - 4477 x^{5} - 13750 x^{4} + 275 x^{3} + 286 x^{2} - 11 x - 1 \)

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:  \(104\!\cdots\!125\)\(\medspace = 5^{21}\cdot 43^{40}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $80.38$
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:  $5, 43$
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:  \(215=5\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{215}(1,·)$, $\chi_{215}(4,·)$, $\chi_{215}(6,·)$, $\chi_{215}(9,·)$, $\chi_{215}(11,·)$, $\chi_{215}(14,·)$, $\chi_{215}(16,·)$, $\chi_{215}(146,·)$, $\chi_{215}(21,·)$, $\chi_{215}(24,·)$, $\chi_{215}(154,·)$, $\chi_{215}(31,·)$, $\chi_{215}(36,·)$, $\chi_{215}(41,·)$, $\chi_{215}(44,·)$, $\chi_{215}(176,·)$, $\chi_{215}(49,·)$, $\chi_{215}(181,·)$, $\chi_{215}(54,·)$, $\chi_{215}(56,·)$, $\chi_{215}(186,·)$, $\chi_{215}(59,·)$, $\chi_{215}(189,·)$, $\chi_{215}(64,·)$, $\chi_{215}(66,·)$, $\chi_{215}(139,·)$, $\chi_{215}(196,·)$, $\chi_{215}(74,·)$, $\chi_{215}(79,·)$, $\chi_{215}(81,·)$, $\chi_{215}(84,·)$, $\chi_{215}(164,·)$, $\chi_{215}(96,·)$, $\chi_{215}(144,·)$, $\chi_{215}(99,·)$, $\chi_{215}(101,·)$, $\chi_{215}(109,·)$, $\chi_{215}(111,·)$, $\chi_{215}(169,·)$, $\chi_{215}(121,·)$, $\chi_{215}(124,·)$$\chi_{215}(126,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $\frac{1}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{41} - \frac{391770097095061334001925904115168815491038202071746550490926699126083157813940465}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{40} - \frac{3128945255897116428079948372984864441738977892499292861645305796975485996330223893}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{39} + \frac{6356614100105624880161561027763412049275792423781552831197464550897165389384448101}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{38} + \frac{6756643745296521367932435402910215035448825534085209375341346538653252881556640365}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{37} - \frac{2558792168307923404044195429488012195611926270598261831613797101778147239849507510}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{36} - \frac{80468552792210009865557673446151319237728987318806064901937391550187398433367317}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{35} + \frac{3111100005542695317648106931327051671194548554202026688599497786776616768140819795}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{34} - \frac{1032808141115945058414579978999054931260454229959903515132911371380990155500027389}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{33} + \frac{9274555261247020915843621597252734490889245749065488995478446013957217875949308095}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{32} - \frac{620095183314661628594004298084267552735560655596330360278496866410107307793835519}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{31} + \frac{11261774264185287800162411434467418476008514977884227027256099311043887729798585714}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{30} - \frac{8482406484375717049824872877657896258164012783749437205087134687713854620719771499}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{29} + \frac{8247337386666127281917943545869988324158142750946858320010786073239574380242269969}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{28} - \frac{786697559571025349016581641308666862878411609510649616833112059930937309167215298}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{27} + \frac{4658925853692242399359022424780177760419342789603915563840563885777432592408758021}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{26} + \frac{10328609064079975364772154099442960956079656044171004351821822951197194746281482621}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{25} - \frac{10845162218980335247332143620186732164335506922143476672541496133255755438097760603}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{24} + \frac{3573435638909451525748511385407280234973601321286354210387694347834137848712536389}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{23} - \frac{8407834815773379693813062580068157351171627212760866181487853697366963419063926702}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{22} + \frac{6776985496375095679717433834131347842510398522354935427289305985226636312565869397}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{21} - \frac{10854815518371359649838969497819169659829520203120603772048631192117548900912524200}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{20} + \frac{3469715472051693751300590354532461446084216993167513338308039138823611916646158380}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{19} - \frac{730437470864976407656958122992092056330587730375042776550396566161376730838345217}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{18} - \frac{7560174817909395814182544313019816894861066188025535120279556730864561815025962885}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{17} + \frac{4331218433059007241422730278187924846073174559857058041172532027894236168557510594}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{16} + \frac{3382892903508950224175976373925141683531815955542462917891675128633059162445649825}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{15} - \frac{10543101717811755762049744373406127289784322900525366101818480474008233635111493542}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{14} - \frac{1799683743636941344083884848011948611993631184765029495565479636316332554682463396}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{13} + \frac{2074199844622163326271459071789795601417885144349737969730542814753442242202507519}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{12} + \frac{5357662192923225173633935486260037601848351488106824715695887071604514115536933898}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{11} + \frac{8275872252391908086143406006955553521287369031732034307581928605941340502475248042}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{10} + \frac{2424651070413887948146684338570405227260318904096559300702726454717320877857695887}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{9} + \frac{6414543266918718974308186356290432485658585705613981276318293260045589686032926859}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{8} + \frac{1851023631508889479219361805329696033715357407295115628779149391742115939290099603}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{7} - \frac{7501864892068266761453739948748745549603981185007378558549339324247611497422345439}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{6} + \frac{4694057781844476934753396562785264235879591539711366296514447096856132808475731757}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{5} - \frac{5735740888470205274577281362909650363678858918554442042450857892221622342821288599}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{4} - \frac{11233749016921078373911841811586522853412611464657566420495191756184007637988191669}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{3} + \frac{6400621399068060385163391446738117284244551499700461390254724707395767926745960528}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a^{2} + \frac{9229185632214565927967214961967383623440065326513906296718858004950522830104272857}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229} a + \frac{6266612218079721778645019467871683937293947928711897681662199014989613163571972441}{24020006024593681350498127947751686309897738477817714828669095279335555750795667229}$

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:  $41$
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:  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

\(\Q(\sqrt{5}) \), 3.3.1849.1, 6.6.427350125.1, 7.7.6321363049.1, 14.14.3121846156036138781328125.1, \(\Q(\zeta_{43})^+\)

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 ${\href{/LocalNumberField/2.14.0.1}{14} }^{3}$ $42$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{7}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{6}$ $42$ $42$ $21^{2}$ $42$ $21^{2}$ $21^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{6}$ R ${\href{/LocalNumberField/47.14.0.1}{14} }^{3}$ $42$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{6}$

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
5Data not computed
43Data not computed