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Properties

Label 37.37.345...561.1
Degree $37$
Signature $[37, 0]$
Discriminant $3.459\times 10^{84}$
Root discriminant $192.68$
Ramified prime $223$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{37}$ (as 37T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^37 - x^36 - 108*x^35 + 245*x^34 + 4913*x^33 - 17429*x^32 - 115073*x^31 + 605121*x^30 + 1258904*x^29 - 11895493*x^28 + 1401577*x^27 + 136786740*x^26 - 218698801*x^25 - 851925353*x^24 + 2810760540*x^23 + 1559768964*x^22 - 17408647490*x^21 + 14765496491*x^20 + 52555917850*x^19 - 112901823056*x^18 - 30825357400*x^17 + 320568348295*x^16 - 255996383256*x^15 - 343575615312*x^14 + 697695088122*x^13 - 135330997139*x^12 - 635742178755*x^11 + 559084397332*x^10 + 98854779029*x^9 - 379266961937*x^8 + 152281955055*x^7 + 65816775545*x^6 - 70480276872*x^5 + 12914339542*x^4 + 6483753572*x^3 - 3469705332*x^2 + 579396241*x - 30247313)
 
gp: K = bnfinit(x^37 - x^36 - 108*x^35 + 245*x^34 + 4913*x^33 - 17429*x^32 - 115073*x^31 + 605121*x^30 + 1258904*x^29 - 11895493*x^28 + 1401577*x^27 + 136786740*x^26 - 218698801*x^25 - 851925353*x^24 + 2810760540*x^23 + 1559768964*x^22 - 17408647490*x^21 + 14765496491*x^20 + 52555917850*x^19 - 112901823056*x^18 - 30825357400*x^17 + 320568348295*x^16 - 255996383256*x^15 - 343575615312*x^14 + 697695088122*x^13 - 135330997139*x^12 - 635742178755*x^11 + 559084397332*x^10 + 98854779029*x^9 - 379266961937*x^8 + 152281955055*x^7 + 65816775545*x^6 - 70480276872*x^5 + 12914339542*x^4 + 6483753572*x^3 - 3469705332*x^2 + 579396241*x - 30247313, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-30247313, 579396241, -3469705332, 6483753572, 12914339542, -70480276872, 65816775545, 152281955055, -379266961937, 98854779029, 559084397332, -635742178755, -135330997139, 697695088122, -343575615312, -255996383256, 320568348295, -30825357400, -112901823056, 52555917850, 14765496491, -17408647490, 1559768964, 2810760540, -851925353, -218698801, 136786740, 1401577, -11895493, 1258904, 605121, -115073, -17429, 4913, 245, -108, -1, 1]);
 

\( x^{37} - x^{36} - 108 x^{35} + 245 x^{34} + 4913 x^{33} - 17429 x^{32} - 115073 x^{31} + 605121 x^{30} + 1258904 x^{29} - 11895493 x^{28} + 1401577 x^{27} + 136786740 x^{26} - 218698801 x^{25} - 851925353 x^{24} + 2810760540 x^{23} + 1559768964 x^{22} - 17408647490 x^{21} + 14765496491 x^{20} + 52555917850 x^{19} - 112901823056 x^{18} - 30825357400 x^{17} + 320568348295 x^{16} - 255996383256 x^{15} - 343575615312 x^{14} + 697695088122 x^{13} - 135330997139 x^{12} - 635742178755 x^{11} + 559084397332 x^{10} + 98854779029 x^{9} - 379266961937 x^{8} + 152281955055 x^{7} + 65816775545 x^{6} - 70480276872 x^{5} + 12914339542 x^{4} + 6483753572 x^{3} - 3469705332 x^{2} + 579396241 x - 30247313 \)

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

Invariants

Degree:  $37$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[37, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(345\!\cdots\!561\)\(\medspace = 223^{36}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $192.68$
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:  $223$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $37$
This field is Galois and abelian over $\Q$.
Conductor:  \(223\)
Dirichlet character group:    $\lbrace$$\chi_{223}(128,·)$, $\chi_{223}(1,·)$, $\chi_{223}(2,·)$, $\chi_{223}(4,·)$, $\chi_{223}(68,·)$, $\chi_{223}(7,·)$, $\chi_{223}(8,·)$, $\chi_{223}(14,·)$, $\chi_{223}(15,·)$, $\chi_{223}(16,·)$, $\chi_{223}(17,·)$, $\chi_{223}(132,·)$, $\chi_{223}(28,·)$, $\chi_{223}(30,·)$, $\chi_{223}(32,·)$, $\chi_{223}(33,·)$, $\chi_{223}(34,·)$, $\chi_{223}(164,·)$, $\chi_{223}(41,·)$, $\chi_{223}(171,·)$, $\chi_{223}(49,·)$, $\chi_{223}(56,·)$, $\chi_{223}(60,·)$, $\chi_{223}(64,·)$, $\chi_{223}(66,·)$, $\chi_{223}(196,·)$, $\chi_{223}(197,·)$, $\chi_{223}(119,·)$, $\chi_{223}(82,·)$, $\chi_{223}(98,·)$, $\chi_{223}(105,·)$, $\chi_{223}(210,·)$, $\chi_{223}(112,·)$, $\chi_{223}(115,·)$, $\chi_{223}(169,·)$, $\chi_{223}(120,·)$, $\chi_{223}(136,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $\frac{1}{263} a^{34} + \frac{45}{263} a^{33} - \frac{100}{263} a^{32} + \frac{123}{263} a^{31} + \frac{10}{263} a^{30} + \frac{24}{263} a^{29} - \frac{102}{263} a^{28} + \frac{62}{263} a^{27} + \frac{113}{263} a^{26} - \frac{15}{263} a^{25} - \frac{76}{263} a^{24} - \frac{16}{263} a^{23} + \frac{22}{263} a^{22} - \frac{123}{263} a^{21} - \frac{3}{263} a^{20} - \frac{55}{263} a^{19} - \frac{122}{263} a^{18} - \frac{114}{263} a^{17} - \frac{99}{263} a^{16} - \frac{92}{263} a^{15} + \frac{46}{263} a^{14} + \frac{84}{263} a^{13} + \frac{50}{263} a^{12} - \frac{18}{263} a^{11} + \frac{28}{263} a^{10} - \frac{2}{263} a^{9} + \frac{90}{263} a^{8} - \frac{125}{263} a^{7} + \frac{92}{263} a^{6} - \frac{112}{263} a^{5} - \frac{111}{263} a^{4} + \frac{110}{263} a^{3} - \frac{58}{263} a^{2} - \frac{58}{263} a + \frac{114}{263}$, $\frac{1}{1759207} a^{35} + \frac{2169}{1759207} a^{34} + \frac{200943}{1759207} a^{33} + \frac{745043}{1759207} a^{32} - \frac{60124}{1759207} a^{31} - \frac{561807}{1759207} a^{30} - \frac{572962}{1759207} a^{29} + \frac{59827}{1759207} a^{28} + \frac{544711}{1759207} a^{27} - \frac{256021}{1759207} a^{26} + \frac{517471}{1759207} a^{25} - \frac{22050}{1759207} a^{24} + \frac{635110}{1759207} a^{23} - \frac{177734}{1759207} a^{22} - \frac{511631}{1759207} a^{21} + \frac{383865}{1759207} a^{20} + \frac{523463}{1759207} a^{19} - \frac{67778}{1759207} a^{18} + \frac{715085}{1759207} a^{17} - \frac{160924}{1759207} a^{16} + \frac{470554}{1759207} a^{15} - \frac{64220}{1759207} a^{14} - \frac{194731}{1759207} a^{13} + \frac{536713}{1759207} a^{12} + \frac{510151}{1759207} a^{11} + \frac{338513}{1759207} a^{10} + \frac{75531}{1759207} a^{9} + \frac{335948}{1759207} a^{8} - \frac{541032}{1759207} a^{7} + \frac{216336}{1759207} a^{6} + \frac{767713}{1759207} a^{5} - \frac{575187}{1759207} a^{4} - \frac{472573}{1759207} a^{3} - \frac{445951}{1759207} a^{2} - \frac{355570}{1759207} a - \frac{36907}{1759207}$, $\frac{1}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{36} - \frac{29833082494730708222744942083408371251345871203839213791845043162723521620357179639781537622653240075}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{35} + \frac{273029885338845929496541278749521488026340664613417337177215918230308221205568006386252156996634976197118}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{34} + \frac{71054967432456009669811560237059225672853108363363080531881915732482377886477987691233949776422831538586787}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{33} - \frac{94591250509320160352281651048108051160764713373548757302101950894832103869238602238441571237883370311313519}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{32} + \frac{6671043454854965461697434102149714667997502634927528367305802981057065741347430224983323987531102028616568}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{31} + \frac{96511541172348129442538480528932740831048197206581003467098763054473942374062997457703485901731510063175887}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{30} + \frac{31906795471129740497422631767063729852899695233971237444850453174467761102214719971623889911667219391320089}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{29} - \frac{1899628582993499564131875275112518248490296580014509193216031086837136416307275592379256154819403202624741}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{28} + \frac{23987140330323451013293182644415254457518306263375001454551556123520540544043809188940268558264983515460505}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{27} + \frac{28026905688385834139303218053173743799439070770450426941307389978146423738024561336740859807933533038535787}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{26} + \frac{103477895500718359998171074789709713548959673387579552288173422762148103067560372909265806609779986820322319}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{25} + \frac{14156321327292003310604891475429934155995214548360086467320857229136879381753086745224222505746764525697290}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{24} + \frac{104739925069587464270959127326703390672278607345843117093918176901515069260628074161885220616505917442568345}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{23} - \frac{67078267917524242542179841375376297624854567311353856392604022421574633571950233187423621210739209629662435}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{22} + \frac{81335023540693953343871931929246197112050348848185135727580592319735350937524114825266598920834972594054522}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{21} - \frac{30263597044054591139924060592384723341639050540041050274119023630492428496010459463062927345660687620679867}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{20} + \frac{89295341043423224433157733207839295269008706836062991419401897672691776448247814987437616674925171581760024}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{19} + \frac{56531872577834651168482251738980483561732491553696241310291865472271052686327648664818221862769958989568247}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{18} + \frac{20833686475213896250359652323370422098708887780518052792062240761915459291770757313452379988149487186038367}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{17} + \frac{14011913878210449449075267809927447549536965249830626613242378810538261421811646565136098913721425776976826}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{16} + \frac{87800659164036907918644344130030406106863263311059020098195587191237537516945973513256288391718932547110546}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{15} + \frac{30832271090942909566058473243317432391446773232082736781273575189842791115035156984261488631949489650515553}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{14} - \frac{107411081783795360273460237772386352961191594897861646358740807503283218905171736139678074639509528774416895}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{13} + \frac{60601557963161595962346649447760352842264819987990387824090036669014108242243361301033599899169813354170658}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{12} + \frac{29622339208121130819037793074861902256561520214666283819193269261977498491593584075060715528594019622516873}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{11} - \frac{100465719766846687794814591315567265049831573312363012559034951824855112345535822732345888651848668242891368}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{10} + \frac{54540402452727616499502948777780228412905031472614199079775075513573387138015051584903075437520101060481302}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{9} + \frac{14725879211894453866144280750842059433472753472031012978693669896118523521051296765059054560116329741393242}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{8} + \frac{63813357185986793505570048218650670735179067744745710566313812104741048646252592180696038064759725811459264}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{7} + \frac{70664837686975546073484048776602932690145405224146879040233556152380620574359318069726328876352526976321309}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{6} + \frac{106339673587495917465259618419370302748650527708218352014770747747491419688781887066231772465614652472922560}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{5} - \frac{73414677482286749643841510696713650844181899045085732810290594796676056511371263111787814936567300078218230}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{4} - \frac{54987619681677642801021162503545429067233652280811338208683123413737887310062435304748814101829414333845597}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{3} - \frac{36418894478797653395268225325221675131648415295394174312622886049279213213519589653915259939077147358495220}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a^{2} - \frac{67075407826719266625640005914271324316862789497821734239944030208489966981682479887556519889487772584006708}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249} a - \frac{28495415372288937930368371370142311850471360031879890649397478781726175986656265883000043282372204818576601}{216744860473247717200728966213046476132270967233619462527804749365710779178051356311178084523203340091486249}$

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:  $36$
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:  \( 2669696754256726600000000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{37}\cdot(2\pi)^{0}\cdot 2669696754256726600000000000000 \cdot 1}{2\sqrt{3459195200715080658084612917441952732107425985306852567952080652975182679421408786561}}\approx 0.0986402677685641$ (assuming GRH)

Galois group

$C_{37}$ (as 37T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 37
The 37 conjugacy class representatives for $C_{37}$
Character table for $C_{37}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $37$ $37$ $37$ $37$ $37$ $37$ $37$ $37$ $37$ $37$ $37$ $37$ $37$ $37$ $37$ $37$ $37$

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
223Data not computed