Properties

Label 37.37.171...601.1
Degree $37$
Signature $[37, 0]$
Discriminant $1.717\times 10^{78}$
Root discriminant $130.15$
Ramified prime $149$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{37}$ (as 37T1)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^37 - x^36 - 72*x^35 + 67*x^34 + 2278*x^33 - 1964*x^32 - 41849*x^31 + 33262*x^30 + 497142*x^29 - 362207*x^28 - 4027127*x^27 + 2672215*x^26 + 22871279*x^25 - 13719683*x^24 - 92273266*x^23 + 49606380*x^22 + 265291071*x^21 - 126402144*x^20 - 540999671*x^19 + 224635365*x^18 + 773506030*x^17 - 271832880*x^16 - 761221384*x^15 + 214848879*x^14 + 502240682*x^13 - 103923082*x^12 - 213804070*x^11 + 27920639*x^10 + 55105551*x^9 - 3603129*x^8 - 7708785*x^7 + 235974*x^6 + 497852*x^5 - 22640*x^4 - 12819*x^3 + 1020*x^2 + 28*x - 1)
 
gp: K = bnfinit(x^37 - x^36 - 72*x^35 + 67*x^34 + 2278*x^33 - 1964*x^32 - 41849*x^31 + 33262*x^30 + 497142*x^29 - 362207*x^28 - 4027127*x^27 + 2672215*x^26 + 22871279*x^25 - 13719683*x^24 - 92273266*x^23 + 49606380*x^22 + 265291071*x^21 - 126402144*x^20 - 540999671*x^19 + 224635365*x^18 + 773506030*x^17 - 271832880*x^16 - 761221384*x^15 + 214848879*x^14 + 502240682*x^13 - 103923082*x^12 - 213804070*x^11 + 27920639*x^10 + 55105551*x^9 - 3603129*x^8 - 7708785*x^7 + 235974*x^6 + 497852*x^5 - 22640*x^4 - 12819*x^3 + 1020*x^2 + 28*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 28, 1020, -12819, -22640, 497852, 235974, -7708785, -3603129, 55105551, 27920639, -213804070, -103923082, 502240682, 214848879, -761221384, -271832880, 773506030, 224635365, -540999671, -126402144, 265291071, 49606380, -92273266, -13719683, 22871279, 2672215, -4027127, -362207, 497142, 33262, -41849, -1964, 2278, 67, -72, -1, 1]);
 

\( x^{37} - x^{36} - 72 x^{35} + 67 x^{34} + 2278 x^{33} - 1964 x^{32} - 41849 x^{31} + 33262 x^{30} + 497142 x^{29} - 362207 x^{28} - 4027127 x^{27} + 2672215 x^{26} + 22871279 x^{25} - 13719683 x^{24} - 92273266 x^{23} + 49606380 x^{22} + 265291071 x^{21} - 126402144 x^{20} - 540999671 x^{19} + 224635365 x^{18} + 773506030 x^{17} - 271832880 x^{16} - 761221384 x^{15} + 214848879 x^{14} + 502240682 x^{13} - 103923082 x^{12} - 213804070 x^{11} + 27920639 x^{10} + 55105551 x^{9} - 3603129 x^{8} - 7708785 x^{7} + 235974 x^{6} + 497852 x^{5} - 22640 x^{4} - 12819 x^{3} + 1020 x^{2} + 28 x - 1 \)

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:  \(171\!\cdots\!601\)\(\medspace = 149^{36}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $130.15$
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:  $149$
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:  \(149\)
Dirichlet character group:    $\lbrace$$\chi_{149}(1,·)$, $\chi_{149}(5,·)$, $\chi_{149}(6,·)$, $\chi_{149}(129,·)$, $\chi_{149}(140,·)$, $\chi_{149}(142,·)$, $\chi_{149}(16,·)$, $\chi_{149}(17,·)$, $\chi_{149}(19,·)$, $\chi_{149}(25,·)$, $\chi_{149}(28,·)$, $\chi_{149}(29,·)$, $\chi_{149}(30,·)$, $\chi_{149}(31,·)$, $\chi_{149}(33,·)$, $\chi_{149}(36,·)$, $\chi_{149}(37,·)$, $\chi_{149}(39,·)$, $\chi_{149}(46,·)$, $\chi_{149}(49,·)$, $\chi_{149}(63,·)$, $\chi_{149}(67,·)$, $\chi_{149}(73,·)$, $\chi_{149}(80,·)$, $\chi_{149}(81,·)$, $\chi_{149}(85,·)$, $\chi_{149}(88,·)$, $\chi_{149}(95,·)$, $\chi_{149}(96,·)$, $\chi_{149}(102,·)$, $\chi_{149}(145,·)$, $\chi_{149}(104,·)$, $\chi_{149}(107,·)$, $\chi_{149}(114,·)$, $\chi_{149}(123,·)$, $\chi_{149}(125,·)$, $\chi_{149}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $\frac{1}{94763} a^{34} + \frac{3403}{94763} a^{33} - \frac{35915}{94763} a^{32} + \frac{17439}{94763} a^{31} - \frac{30790}{94763} a^{30} - \frac{15730}{94763} a^{29} + \frac{23767}{94763} a^{28} + \frac{44513}{94763} a^{27} - \frac{46739}{94763} a^{26} + \frac{7429}{94763} a^{25} + \frac{42361}{94763} a^{24} + \frac{169}{491} a^{23} + \frac{24337}{94763} a^{22} - \frac{13157}{94763} a^{21} + \frac{16460}{94763} a^{20} - \frac{37771}{94763} a^{19} - \frac{2599}{94763} a^{18} + \frac{29942}{94763} a^{17} - \frac{17754}{94763} a^{16} - \frac{3614}{94763} a^{15} - \frac{28376}{94763} a^{14} - \frac{5201}{94763} a^{13} - \frac{2229}{94763} a^{12} + \frac{38934}{94763} a^{11} - \frac{25674}{94763} a^{10} - \frac{25185}{94763} a^{9} + \frac{3362}{94763} a^{8} - \frac{20310}{94763} a^{7} + \frac{27337}{94763} a^{6} + \frac{10720}{94763} a^{5} + \frac{45180}{94763} a^{4} - \frac{32539}{94763} a^{3} + \frac{19708}{94763} a^{2} - \frac{27486}{94763} a + \frac{4000}{94763}$, $\frac{1}{94763} a^{35} + \frac{39525}{94763} a^{33} - \frac{8086}{94763} a^{32} + \frac{40694}{94763} a^{31} - \frac{45238}{94763} a^{30} + \frac{11862}{94763} a^{29} - \frac{1749}{94763} a^{28} + \frac{1559}{94763} a^{27} - \frac{46831}{94763} a^{26} - \frac{31568}{94763} a^{25} + \frac{12657}{94763} a^{24} - \frac{3841}{94763} a^{23} - \frac{9106}{94763} a^{22} - \frac{33168}{94763} a^{21} - \frac{46218}{94763} a^{20} + \frac{33486}{94763} a^{19} - \frac{33383}{94763} a^{18} - \frac{40155}{94763} a^{17} - \frac{45546}{94763} a^{16} + \frac{45639}{94763} a^{15} - \frac{5170}{94763} a^{14} - \frac{23907}{94763} a^{13} + \frac{43181}{94763} a^{12} - \frac{39402}{94763} a^{11} - \frac{28049}{94763} a^{10} + \frac{42165}{94763} a^{9} + \frac{5127}{94763} a^{8} - \frac{34723}{94763} a^{7} + \frac{40175}{94763} a^{6} - \frac{45988}{94763} a^{5} + \frac{20270}{94763} a^{4} - \frac{28022}{94763} a^{3} - \frac{1606}{94763} a^{2} + \frac{7777}{94763} a + \frac{33872}{94763}$, $\frac{1}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{36} - \frac{595037869703802503909083166917321630929675754811856509180232363432022144640964611485292}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{35} - \frac{273244333771102060152261827662237473200129304841526420532041958112097915848818480645118}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{34} + \frac{50438541151483921140107477972166894089059157581586968666647851885836674138713656459900401850}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{33} - \frac{10343282928232326081605415895787308945593066549692738311515531778921582122479812768682695972}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{32} - \frac{27867425467145712227082857497026714084444364437192860031356308466046269991025090490229740099}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{31} - \frac{67220422999077937614061868520978202668370313516858127577624722188789181362014290002860860666}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{30} - \frac{24133258144847337760648130825712087985732727607971418457222160622801786054221398206665395920}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{29} + \frac{32734160719262877362153681655571634001228421880554283932277632631496695917281071634638126901}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{28} - \frac{68303145258082708283949865312592963170794925617481938356742686007795505802086359607479127624}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{27} + \frac{17729618267654475784061046496868122387373204136459028466394215446207367915094184489543117896}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{26} + \frac{23132706987617346016839018528943810253557998284453289228519305367326886652545730825727003815}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{25} - \frac{12136244208220765295353262672772795499706627057446527354985467762510507438752006374749292626}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{24} + \frac{32269976974797485938857134679009204268019574471795967814383619473887327259822232869499086110}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{23} + \frac{50369043092369423309071474137983880592553740664748983760098758308101855754563616688972602787}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{22} - \frac{74254934019370949497659485999065026252054777630585081200424318221392844772189813615089392312}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{21} - \frac{2497627595424291177555710863887118955841248702794865446833632614018867731321149770016086566}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{20} - \frac{20384569464172606181153598311739920149969910156511512310569626121715531139226675971464456080}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{19} - \frac{9156087147768696118679615584627421684594227555769181672208045614999054727244425525475545219}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{18} - \frac{41867662355196305895813889669113261538264774991699008797432209239465215214192744180520230286}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{17} - \frac{47972869891501263266089682367311576225262149579620579017686815095390409210158859005188644553}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{16} + \frac{38154139349793943531252762602913735037941477274338481717293486423119457608705356703391009198}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{15} + \frac{58609837009586179052268453595125136790115386463687067481296106316563148965144576411082964914}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{14} + \frac{24475981291059232839261529915329468518820358902305340503035322269233031157486504119609707516}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{13} - \frac{58496874114877758965067907009379905224713333919836235970203560688063331959678586172646476150}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{12} - \frac{29365725474664805683153957051792697714295562341149249762273536468365898913189486118389626842}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{11} - \frac{50111516218787650509899514970034380939207907998144638758300400354188839492218848679783097749}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{10} - \frac{31145890406876103073713108327901928407304887737534904769198372836856017604413564870802855501}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{9} + \frac{58389121242056520339642557001642343204938748520444107946154700022763320261740304981868840119}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{8} - \frac{64794451124594564779394921489260129312226160520620589581859252331200126861738745963767107582}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{7} - \frac{36608408628855876227816960528263363777325374035829681560494572528250874783999944558769854858}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{6} + \frac{28613551633336581918298020457490718522942350392180699532103848750026424602498118394970784978}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{5} - \frac{40932148902876501924605841207059938141736533603772677894301179330822764130331361403446751670}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{4} + \frac{73405781875102555400906721048788846408417893137973597446468532940665770106757275645061485692}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{3} + \frac{27852771979113811716904466332527348183528706111333012878933776940391669499974147877130426408}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a^{2} - \frac{12940548230582574691866149961193062651971832065968713449511574256154398939511692857122416042}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927} a + \frac{51351502205591418643964523511064508742584427514324775092821509686469797338026750049861722387}{151825131072484496340769137982488217780564502277603765267000993645782154241330373914336442927}$

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:  \( 2217351755002491400000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{37}\cdot(2\pi)^{0}\cdot 2217351755002491400000000000 \cdot 1}{2\sqrt{1716744492644476569905693744668682750842648214113467507247085096126466572919601}}\approx 0.116295150867153$ (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
149Data not computed