Properties

Label 42.42.805...125.1
Degree $42$
Signature $[42, 0]$
Discriminant $8.050\times 10^{78}$
Root discriminant $75.63$
Ramified primes $5, 7$
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 - 63*x^40 + 1764*x^38 - 29043*x^36 - 29*x^35 + 313845*x^34 + 1295*x^33 - 2356263*x^32 - 24605*x^31 + 12710649*x^30 + 262360*x^29 - 50338309*x^28 - 1751015*x^27 + 148468453*x^26 + 7757582*x^25 - 329114310*x^24 - 23675820*x^23 + 550880022*x^22 + 50961633*x^21 - 696364921*x^20 - 78352652*x^19 + 661675518*x^18 + 86284422*x^17 - 467756891*x^16 - 67555180*x^15 + 241883075*x^14 + 36893815*x^13 - 89215119*x^12 - 13584956*x^11 + 22638112*x^10 + 3191321*x^9 - 3752308*x^8 - 437383*x^7 + 374801*x^6 + 29995*x^5 - 19355*x^4 - 798*x^3 + 343*x^2 + 14*x - 1)
 
gp: K = bnfinit(x^42 - 63*x^40 + 1764*x^38 - 29043*x^36 - 29*x^35 + 313845*x^34 + 1295*x^33 - 2356263*x^32 - 24605*x^31 + 12710649*x^30 + 262360*x^29 - 50338309*x^28 - 1751015*x^27 + 148468453*x^26 + 7757582*x^25 - 329114310*x^24 - 23675820*x^23 + 550880022*x^22 + 50961633*x^21 - 696364921*x^20 - 78352652*x^19 + 661675518*x^18 + 86284422*x^17 - 467756891*x^16 - 67555180*x^15 + 241883075*x^14 + 36893815*x^13 - 89215119*x^12 - 13584956*x^11 + 22638112*x^10 + 3191321*x^9 - 3752308*x^8 - 437383*x^7 + 374801*x^6 + 29995*x^5 - 19355*x^4 - 798*x^3 + 343*x^2 + 14*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 14, 343, -798, -19355, 29995, 374801, -437383, -3752308, 3191321, 22638112, -13584956, -89215119, 36893815, 241883075, -67555180, -467756891, 86284422, 661675518, -78352652, -696364921, 50961633, 550880022, -23675820, -329114310, 7757582, 148468453, -1751015, -50338309, 262360, 12710649, -24605, -2356263, 1295, 313845, -29, -29043, 0, 1764, 0, -63, 0, 1]);
 

\( x^{42} - 63 x^{40} + 1764 x^{38} - 29043 x^{36} - 29 x^{35} + 313845 x^{34} + 1295 x^{33} - 2356263 x^{32} - 24605 x^{31} + 12710649 x^{30} + 262360 x^{29} - 50338309 x^{28} - 1751015 x^{27} + 148468453 x^{26} + 7757582 x^{25} - 329114310 x^{24} - 23675820 x^{23} + 550880022 x^{22} + 50961633 x^{21} - 696364921 x^{20} - 78352652 x^{19} + 661675518 x^{18} + 86284422 x^{17} - 467756891 x^{16} - 67555180 x^{15} + 241883075 x^{14} + 36893815 x^{13} - 89215119 x^{12} - 13584956 x^{11} + 22638112 x^{10} + 3191321 x^{9} - 3752308 x^{8} - 437383 x^{7} + 374801 x^{6} + 29995 x^{5} - 19355 x^{4} - 798 x^{3} + 343 x^{2} + 14 x - 1 \)

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:  \(805\!\cdots\!125\)\(\medspace = 5^{21}\cdot 7^{76}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $75.63$
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:  $5, 7$
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:  \(245=5\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{245}(1,·)$, $\chi_{245}(4,·)$, $\chi_{245}(86,·)$, $\chi_{245}(134,·)$, $\chi_{245}(9,·)$, $\chi_{245}(11,·)$, $\chi_{245}(141,·)$, $\chi_{245}(16,·)$, $\chi_{245}(149,·)$, $\chi_{245}(151,·)$, $\chi_{245}(156,·)$, $\chi_{245}(29,·)$, $\chi_{245}(51,·)$, $\chi_{245}(36,·)$, $\chi_{245}(39,·)$, $\chi_{245}(169,·)$, $\chi_{245}(44,·)$, $\chi_{245}(46,·)$, $\chi_{245}(176,·)$, $\chi_{245}(179,·)$, $\chi_{245}(184,·)$, $\chi_{245}(186,·)$, $\chi_{245}(191,·)$, $\chi_{245}(64,·)$, $\chi_{245}(71,·)$, $\chi_{245}(74,·)$, $\chi_{245}(204,·)$, $\chi_{245}(79,·)$, $\chi_{245}(81,·)$, $\chi_{245}(211,·)$, $\chi_{245}(214,·)$, $\chi_{245}(219,·)$, $\chi_{245}(221,·)$, $\chi_{245}(144,·)$, $\chi_{245}(226,·)$, $\chi_{245}(99,·)$, $\chi_{245}(106,·)$, $\chi_{245}(109,·)$, $\chi_{245}(239,·)$, $\chi_{245}(114,·)$, $\chi_{245}(116,·)$$\chi_{245}(121,·)$$\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}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{41} - \frac{419802150277063304833901008937743260546954026958290224983522378842242303613939713216}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{40} - \frac{1471035029198610395087072600850415197412469763046774956790015354107579485537980375361}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{39} - \frac{178147581111743355302967261314472484091820476938649492080141873628586554088083773123}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{38} - \frac{2033745488011368667493044959379858345622109294035927615541487542367159217906381473366}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{37} + \frac{1956479879462696131703877233028640640584054876680896884567804058439076805334957638871}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{36} - \frac{302692209309895326027646344271271289064045904373083011726998052683764098079084543724}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{35} - \frac{1459218121793433358566944677481669982514926857665708131694161250625851910907150585227}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{34} + \frac{883132117750862109637612744234354110799083668897463659347308041029477060680080667803}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{33} - \frac{963196252265693766079132875988255455184594125297327025570436590031661371198774810750}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{32} + \frac{1585981878478714790468355407750516485980092848567740426176041649234220692499423640410}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{31} - \frac{172205685844614270072687599163491225191233233236051814503221736672963960273855804480}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{30} + \frac{2295084418229037799870616468362213830688628644038008254837490828217796025542437564148}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{29} + \frac{2359923129067470633289664723981637638993145420096597237957916053292154411372996271727}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{28} + \frac{1050341687307242454434453329470997079665212809395735685967006230880342970846475733253}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{27} + \frac{779867946317405466994564930594820324541730351888670934220500884164354531690014105532}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{26} - \frac{85161079767038977163587247072171138045423080252353040411069705408091439828262036975}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{25} - \frac{2723665688371616923047095897623301246136278767000641872639762790864181414038037860937}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{24} - \frac{435884415956051212485598701529875926416006927132492134171646979298624991256595090482}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{23} - \frac{2588581461699903491790706615733081610508047431136366923130387470961572341390119344847}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{22} - \frac{986478990107979802525418922727005129346423007385760456631025089129963850232404532736}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{21} - \frac{1250002761527275805165339271112278904871953648116605618260716490316225630197856720596}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{20} + \frac{295310253413550824054184485098652102602983207211538319564139125534243514771290601563}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{19} - \frac{2060398908873633969098770282811563983435703582477037052833490549150205079480606977022}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{18} - \frac{652842495243417219035055837368960909741032235715866461293074450374673048682856795670}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{17} + \frac{50571618761604495796151968176176217397965843596763118960528960953320943435149046371}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{16} + \frac{972977242844462932328726923451344522805303532881476071901895780010531075623469845148}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{15} + \frac{545687630573303154437653721368091813885618020738267542858166481238886001475583941608}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{14} + \frac{1008595933992687179061641917838711336238414921906197038999410565347679244580324186111}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{13} + \frac{1056564887298088564839137680993768116619585239489066103268550253348736290366415147824}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{12} - \frac{1092870283051047101306468128116776631942006299124019923218209722501262490850362846068}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{11} - \frac{740921059837909116871765880879744139553918064713782677705906448569571740633507477774}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{10} - \frac{1970554289953415088855113644709633218854424413809627819082097714373336799633099133060}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{9} + \frac{727130019561874876197586503500920509101777648177024265659448311786943676902566839292}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{8} - \frac{611399016462095752712406622724226090203200547472004368283004664252166816980315231602}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{7} - \frac{2350573257509076055637834131915216817181130497988737496729832933712997751625342320756}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{6} + \frac{1112149493467046316292456581995747040525525163663972361995274092849424368486042573762}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{5} + \frac{244776757177600817265560335730665610447211470778264018752965925004701598697765693654}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{4} - \frac{2097615419869473403243653944333261571457138542586660657416036935090093370132739334540}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{3} + \frac{242152958200700588180057082297881060375942942590059855880681668980487485954436712717}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a^{2} + \frac{1014130534675329717133069556067664652418308380085192988450642291321991301524536835825}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501} a - \frac{2047966679726240753026002689119110138394710432401388077544981730120563537240947924747}{5574417667875906684571331862907077791420203092569040939871906901501749358719930594501}$

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:  \( 119589768703493500000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{42}\cdot(2\pi)^{0}\cdot 119589768703493500000000000 \cdot 1}{2\sqrt{8050468075656610214837511220114705524038488445061950919170859146595001220703125}}\approx 0.0926858169158281$ (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 6.6.300125.1, 7.7.13841287201.1, 14.14.14967283701606751125078125.1, \(\Q(\zeta_{49})^+\)

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 $42$ $42$ R R $21^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{3}$ $42$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{14}$ $42$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{14}$ $42$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{6}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{3}$ $42$ $42$ $21^{2}$

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
7Data not computed