↑

Properties

Label 36.36.1255524126...0957.1
Degree $36$
Signature $[36, 0]$
Discriminant $7^{18}\cdot 37^{35}$
Root discriminant $88.55$
Ramified primes $7, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{36}$ (as 36T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![534059, 9165269, -9165269, -267265579, 267265579, 2082396629, -2082396629, -7316252203, 7316252203, 14222318037, -14222318037, -17302316587, 17302316587, 14222318037, -14222318037, -8295278123, 8295278123, 3543016917, -3543016917, -1129994283, 1129994283, 271909077, -271909077, -49475883, 49475883, 6766485, -6766485, -684427, 684427, 49653, -49653, -2443, 2443, 73, -73, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 73*x^34 + 73*x^33 + 2443*x^32 - 2443*x^31 - 49653*x^30 + 49653*x^29 + 684427*x^28 - 684427*x^27 - 6766485*x^26 + 6766485*x^25 + 49475883*x^24 - 49475883*x^23 - 271909077*x^22 + 271909077*x^21 + 1129994283*x^20 - 1129994283*x^19 - 3543016917*x^18 + 3543016917*x^17 + 8295278123*x^16 - 8295278123*x^15 - 14222318037*x^14 + 14222318037*x^13 + 17302316587*x^12 - 17302316587*x^11 - 14222318037*x^10 + 14222318037*x^9 + 7316252203*x^8 - 7316252203*x^7 - 2082396629*x^6 + 2082396629*x^5 + 267265579*x^4 - 267265579*x^3 - 9165269*x^2 + 9165269*x + 534059)
 
gp: K = bnfinit(x^36 - x^35 - 73*x^34 + 73*x^33 + 2443*x^32 - 2443*x^31 - 49653*x^30 + 49653*x^29 + 684427*x^28 - 684427*x^27 - 6766485*x^26 + 6766485*x^25 + 49475883*x^24 - 49475883*x^23 - 271909077*x^22 + 271909077*x^21 + 1129994283*x^20 - 1129994283*x^19 - 3543016917*x^18 + 3543016917*x^17 + 8295278123*x^16 - 8295278123*x^15 - 14222318037*x^14 + 14222318037*x^13 + 17302316587*x^12 - 17302316587*x^11 - 14222318037*x^10 + 14222318037*x^9 + 7316252203*x^8 - 7316252203*x^7 - 2082396629*x^6 + 2082396629*x^5 + 267265579*x^4 - 267265579*x^3 - 9165269*x^2 + 9165269*x + 534059, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 73 x^{34} + 73 x^{33} + 2443 x^{32} - 2443 x^{31} - 49653 x^{30} + 49653 x^{29} + 684427 x^{28} - 684427 x^{27} - 6766485 x^{26} + 6766485 x^{25} + 49475883 x^{24} - 49475883 x^{23} - 271909077 x^{22} + 271909077 x^{21} + 1129994283 x^{20} - 1129994283 x^{19} - 3543016917 x^{18} + 3543016917 x^{17} + 8295278123 x^{16} - 8295278123 x^{15} - 14222318037 x^{14} + 14222318037 x^{13} + 17302316587 x^{12} - 17302316587 x^{11} - 14222318037 x^{10} + 14222318037 x^{9} + 7316252203 x^{8} - 7316252203 x^{7} - 2082396629 x^{6} + 2082396629 x^{5} + 267265579 x^{4} - 267265579 x^{3} - 9165269 x^{2} + 9165269 x + 534059 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[36, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12555241263526989253001578573158506357399006856342819571588617542650957=7^{18}\cdot 37^{35}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.55$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $7, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(259=7\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{259}(1,·)$, $\chi_{259}(6,·)$, $\chi_{259}(13,·)$, $\chi_{259}(146,·)$, $\chi_{259}(20,·)$, $\chi_{259}(153,·)$, $\chi_{259}(155,·)$, $\chi_{259}(69,·)$, $\chi_{259}(36,·)$, $\chi_{259}(167,·)$, $\chi_{259}(169,·)$, $\chi_{259}(176,·)$, $\chi_{259}(55,·)$, $\chi_{259}(64,·)$, $\chi_{259}(197,·)$, $\chi_{259}(71,·)$, $\chi_{259}(202,·)$, $\chi_{259}(76,·)$, $\chi_{259}(78,·)$, $\chi_{259}(141,·)$, $\chi_{259}(209,·)$, $\chi_{259}(211,·)$, $\chi_{259}(85,·)$, $\chi_{259}(216,·)$, $\chi_{259}(218,·)$, $\chi_{259}(97,·)$, $\chi_{259}(99,·)$, $\chi_{259}(230,·)$, $\chi_{259}(232,·)$, $\chi_{259}(225,·)$, $\chi_{259}(237,·)$, $\chi_{259}(244,·)$, $\chi_{259}(120,·)$, $\chi_{259}(251,·)$, $\chi_{259}(125,·)$, $\chi_{259}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{194399} a^{19} - \frac{18291}{194399} a^{18} - \frac{38}{194399} a^{17} + \frac{75279}{194399} a^{16} + \frac{608}{194399} a^{15} + \frac{37209}{194399} a^{14} - \frac{5320}{194399} a^{13} - \frac{2901}{194399} a^{12} + \frac{27664}{194399} a^{11} + \frac{96990}{194399} a^{10} - \frac{86944}{194399} a^{9} + \frac{75349}{194399} a^{8} - \frac{33887}{194399} a^{7} - \frac{30810}{194399} a^{6} + \frac{33887}{194399} a^{5} - \frac{97176}{194399} a^{4} + \frac{72960}{194399} a^{3} - \frac{9727}{194399} a^{2} - \frac{9728}{194399} a + \frac{67680}{194399}$, $\frac{1}{194399} a^{20} - \frac{40}{194399} a^{18} - \frac{36582}{194399} a^{17} + \frac{680}{194399} a^{16} + \frac{77394}{194399} a^{15} - \frac{6400}{194399} a^{14} + \frac{82878}{194399} a^{13} + \frac{36400}{194399} a^{12} + \frac{78617}{194399} a^{11} + \frac{66271}{194399} a^{10} - \frac{33535}{194399} a^{9} + \frac{80161}{194399} a^{8} + \frac{80484}{194399} a^{7} + \frac{50878}{194399} a^{6} - \frac{14071}{194399} a^{5} + \frac{16801}{194399} a^{4} - \frac{47502}{194399} a^{3} - \frac{51200}{194399} a^{2} + \frac{7917}{194399} a + \frac{2048}{194399}$, $\frac{1}{194399} a^{21} + \frac{9374}{194399} a^{18} - \frac{840}{194399} a^{17} - \frac{21830}{194399} a^{16} + \frac{17920}{194399} a^{15} + \frac{16046}{194399} a^{14} + \frac{17999}{194399} a^{13} - \frac{37423}{194399} a^{12} + \frac{6437}{194399} a^{11} - \frac{41915}{194399} a^{10} - \frac{92816}{194399} a^{9} - \frac{15940}{194399} a^{8} + \frac{56191}{194399} a^{7} - \frac{80077}{194399} a^{6} + \frac{11488}{194399} a^{5} - \frac{46562}{194399} a^{4} - \frac{48785}{194399} a^{3} + \frac{7635}{194399} a^{2} + \frac{1726}{194399} a - \frac{14386}{194399}$, $\frac{1}{194399} a^{22} - \frac{924}{194399} a^{18} - \frac{54416}{194399} a^{17} + \frac{20944}{194399} a^{16} - \frac{45775}{194399} a^{15} - \frac{27361}{194399} a^{14} + \frac{66113}{194399} a^{13} - \frac{15449}{194399} a^{12} - \frac{35985}{194399} a^{11} - \frac{72953}{194399} a^{10} + \frac{76508}{194399} a^{9} - \frac{13768}{194399} a^{8} - \frac{71305}{194399} a^{7} - \frac{52486}{194399} a^{6} - \frac{758}{2663} a^{5} - \frac{74675}{194399} a^{4} - \frac{23723}{194399} a^{3} + \frac{9493}{194399} a^{2} + \frac{2755}{194399} a + \frac{86016}{194399}$, $\frac{1}{194399} a^{23} - \frac{42587}{194399} a^{18} - \frac{14168}{194399} a^{17} - \frac{82821}{194399} a^{16} - \frac{48766}{194399} a^{15} + \frac{38606}{194399} a^{14} - \frac{71154}{194399} a^{13} + \frac{5077}{194399} a^{12} + \frac{22314}{194399} a^{11} + \frac{77329}{194399} a^{10} - \frac{63237}{194399} a^{9} - \frac{43671}{194399} a^{8} - \frac{65835}{194399} a^{7} + \frac{52879}{194399} a^{6} - \frac{61326}{194399} a^{5} - \frac{2009}{194399} a^{4} - \frac{31920}{194399} a^{3} - \frac{42639}{194399} a^{2} + \frac{39698}{194399} a - \frac{60158}{194399}$, $\frac{1}{194399} a^{24} - \frac{16192}{194399} a^{18} + \frac{48464}{194399} a^{17} + \frac{24098}{194399} a^{16} + \frac{76435}{194399} a^{15} + \frac{2280}{194399} a^{14} - \frac{1136}{2663} a^{13} - \frac{79208}{194399} a^{12} - \frac{48242}{194399} a^{11} + \frac{54340}{194399} a^{10} - \frac{10046}{194399} a^{9} + \frac{72134}{194399} a^{8} - \frac{69013}{194399} a^{7} + \frac{26454}{194399} a^{6} - \frac{74516}{194399} a^{5} + \frac{94079}{194399} a^{4} + \frac{25664}{194399} a^{3} + \frac{60218}{194399} a^{2} - \frac{82225}{194399} a - \frac{65813}{194399}$, $\frac{1}{194399} a^{25} - \frac{49731}{194399} a^{18} - \frac{8001}{194399} a^{17} - \frac{82126}{194399} a^{16} - \frac{67333}{194399} a^{15} - \frac{37301}{194399} a^{14} + \frac{92508}{194399} a^{13} + \frac{23324}{194399} a^{12} + \frac{94532}{194399} a^{11} + \frac{96912}{194399} a^{10} - \frac{81955}{194399} a^{9} - \frac{66129}{194399} a^{8} - \frac{77872}{194399} a^{7} + \frac{989}{2663} a^{6} + \frac{4006}{194399} a^{5} + \frac{17378}{194399} a^{4} + \frac{65815}{194399} a^{3} + \frac{75780}{194399} a^{2} + \frac{76000}{194399} a + \frac{47397}{194399}$, $\frac{1}{194399} a^{26} - \frac{44801}{194399} a^{18} - \frac{27914}{194399} a^{17} + \frac{91073}{194399} a^{16} + \frac{67302}{194399} a^{15} + \frac{49206}{194399} a^{14} + \frac{31443}{194399} a^{13} + \frac{68959}{194399} a^{12} + \frac{93573}{194399} a^{11} + \frac{94146}{194399} a^{10} - \frac{55635}{194399} a^{9} + \frac{62522}{194399} a^{8} + \frac{82731}{194399} a^{7} + \frac{44814}{194399} a^{6} + \frac{6844}{194399} a^{5} - \frac{29100}{194399} a^{4} - \frac{7795}{194399} a^{3} + \frac{7275}{194399} a^{2} - \frac{71059}{194399} a - \frac{30206}{194399}$, $\frac{1}{194399} a^{27} - \frac{91220}{194399} a^{18} - \frac{56173}{194399} a^{17} + \frac{13530}{194399} a^{16} + \frac{72354}{194399} a^{15} + \frac{60427}{194399} a^{14} + \frac{60813}{194399} a^{13} - \frac{15596}{194399} a^{12} - \frac{19014}{194399} a^{11} - \frac{13093}{194399} a^{10} + \frac{57141}{194399} a^{9} + \frac{54645}{194399} a^{8} - \frac{64882}{194399} a^{7} - \frac{79066}{194399} a^{6} + \frac{80596}{194399} a^{5} - \frac{24166}{194399} a^{4} + \frac{63449}{194399} a^{3} - \frac{7828}{194399} a^{2} - \frac{11776}{194399} a + \frac{90477}{194399}$, $\frac{1}{194399} a^{28} - \frac{34576}{194399} a^{18} + \frac{46352}{194399} a^{17} + \frac{72458}{194399} a^{16} - \frac{75927}{194399} a^{15} + \frac{59253}{194399} a^{14} - \frac{86092}{194399} a^{13} - \frac{71195}{194399} a^{12} + \frac{3568}{194399} a^{11} - \frac{2347}{194399} a^{10} - \frac{81032}{194399} a^{9} - \frac{94545}{194399} a^{8} + \frac{81692}{194399} a^{7} + \frac{18739}{194399} a^{6} + \frac{9475}{194399} a^{5} + \frac{68730}{194399} a^{4} - \frac{40792}{194399} a^{3} - \frac{71680}{194399} a^{2} - \frac{60647}{194399} a + \frac{46158}{194399}$, $\frac{1}{194399} a^{29} - \frac{3317}{194399} a^{18} - \frac{75036}{194399} a^{17} - \frac{37434}{194399} a^{16} + \frac{86369}{194399} a^{15} - \frac{80290}{194399} a^{14} + \frac{80338}{194399} a^{13} + \frac{8476}{194399} a^{12} + \frac{65037}{194399} a^{11} + \frac{62458}{194399} a^{10} - \frac{84153}{194399} a^{9} + \frac{13318}{194399} a^{8} - \frac{15400}{194399} a^{7} + \frac{29435}{194399} a^{6} - \frac{91530}{194399} a^{5} - \frac{5852}{194399} a^{4} + \frac{71856}{194399} a^{3} - \frac{71129}{194399} a^{2} + \frac{1100}{194399} a - \frac{71482}{194399}$, $\frac{1}{194399} a^{30} - \frac{93795}{194399} a^{18} + \frac{30919}{194399} a^{17} - \frac{15903}{194399} a^{16} - \frac{7544}{194399} a^{15} + \frac{59226}{194399} a^{14} + \frac{52345}{194399} a^{13} - \frac{32029}{194399} a^{12} + \frac{67618}{194399} a^{11} + \frac{95731}{194399} a^{10} - \frac{1181}{2663} a^{9} - \frac{79881}{194399} a^{8} - \frac{11122}{194399} a^{7} - \frac{34426}{194399} a^{6} + \frac{34705}{194399} a^{5} + \frac{52606}{194399} a^{4} - \frac{89564}{194399} a^{3} + \frac{6875}{194399} a^{2} - \frac{69024}{194399} a - \frac{36285}{194399}$, $\frac{1}{194399} a^{31} - \frac{2251}{194399} a^{18} - \frac{80931}{194399} a^{17} + \frac{20182}{194399} a^{16} - \frac{66720}{194399} a^{15} + \frac{25253}{194399} a^{14} + \frac{804}{194399} a^{13} - \frac{67476}{194399} a^{12} + \frac{2759}{194399} a^{11} - \frac{4767}{194399} a^{10} + \frac{45689}{194399} a^{9} - \frac{27312}{194399} a^{8} - \frac{41941}{194399} a^{7} - \frac{48110}{194399} a^{6} + \frac{60121}{194399} a^{5} + \frac{73429}{194399} a^{4} + \frac{56477}{194399} a^{3} + \frac{95917}{194399} a^{2} + \frac{34861}{194399} a - \frac{53745}{194399}$, $\frac{1}{194399} a^{32} - \frac{41384}{194399} a^{18} - \frac{65356}{194399} a^{17} + \frac{64780}{194399} a^{16} + \frac{33068}{194399} a^{15} - \frac{27706}{194399} a^{14} + \frac{9942}{194399} a^{13} + \frac{82174}{194399} a^{12} + \frac{59217}{194399} a^{11} + \frac{60102}{194399} a^{10} + \frac{21537}{194399} a^{9} + \frac{52730}{194399} a^{8} + \frac{71060}{194399} a^{7} - \frac{87145}{194399} a^{6} - \frac{45741}{194399} a^{5} + \frac{12176}{194399} a^{4} + \frac{61722}{194399} a^{3} - \frac{87928}{194399} a^{2} + \frac{15614}{194399} a - \frac{61136}{194399}$, $\frac{1}{194399} a^{33} - \frac{30394}{194399} a^{18} + \frac{47380}{194399} a^{17} - \frac{59170}{194399} a^{16} + \frac{56295}{194399} a^{15} + \frac{32719}{194399} a^{14} - \frac{21038}{194399} a^{13} - \frac{51584}{194399} a^{12} + \frac{91367}{194399} a^{11} - \frac{94855}{194399} a^{10} + \frac{93325}{194399} a^{9} - \frac{40283}{194399} a^{8} - \frac{72367}{194399} a^{7} - \frac{23740}{194399} a^{6} - \frac{2602}{194399} a^{5} + \frac{62251}{194399} a^{4} + \frac{77843}{194399} a^{3} + \frac{73775}{194399} a^{2} - \frac{44359}{194399} a - \frac{31672}{194399}$, $\frac{1}{194399} a^{34} + \frac{91866}{194399} a^{18} - \frac{47748}{194399} a^{17} + \frac{9991}{194399} a^{16} + \frac{44366}{194399} a^{15} + \frac{90325}{194399} a^{14} - \frac{7696}{194399} a^{13} - \frac{18880}{194399} a^{12} - \frac{50914}{194399} a^{11} - \frac{53450}{194399} a^{10} + \frac{43787}{194399} a^{9} + \frac{64919}{194399} a^{8} - \frac{59316}{194399} a^{7} - \frac{21759}{194399} a^{6} - \frac{96572}{194399} a^{5} + \frac{14506}{194399} a^{4} - \frac{83777}{194399} a^{3} - \frac{5918}{194399} a^{2} - \frac{23625}{194399} a - \frac{64298}{194399}$, $\frac{1}{194399} a^{35} + \frac{82701}{194399} a^{18} + \frac{1717}{194399} a^{17} + \frac{13778}{194399} a^{16} + \frac{28310}{194399} a^{15} + \frac{62326}{194399} a^{14} - \frac{10846}{194399} a^{13} - \frac{68677}{194399} a^{12} - \frac{56347}{194399} a^{11} + \frac{44213}{194399} a^{10} - \frac{9290}{194399} a^{9} + \frac{89042}{194399} a^{8} - \frac{64203}{194399} a^{7} + \frac{39847}{194399} a^{6} + \frac{56950}{194399} a^{5} + \frac{90160}{194399} a^{4} - \frac{60556}{194399} a^{3} - \frac{95246}{194399} a^{2} - \frac{44053}{194399} a - \frac{27663}{194399}$

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

Class group and class number

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

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $35$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 612451006629215400000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{36}$ (as 36T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 36
The 36 conjugacy class representatives for $C_{36}$
Character table for $C_{36}$ is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.1369.1, 4.4.2481997.1, 6.6.69343957.1, 9.9.3512479453921.1, 12.12.20931830284731738129037.1, \(\Q(\zeta_{37})^+\)

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 $36$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{4}$ $36$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ $36$ $36$ $36$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/29.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{9}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{4}$ $36$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
37Data not computed