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Properties

Label 36.0.58065421278...5936.2
Degree $36$
Signature $[0, 18]$
Discriminant $2^{54}\cdot 3^{18}\cdot 19^{32}$
Root discriminant $67.11$
Ramified primes $2, 3, 19$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{18}$ (as 36T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![262144, 0, 5898240, 0, 111083520, 0, 426049536, 0, 1081884672, 0, 1557225472, 0, 1576677376, 0, 1124603904, 0, 606287872, 0, 249758720, 0, 81078016, 0, 20814464, 0, 4279936, 0, 699296, 0, 90624, 0, 9040, 0, 676, 0, 34, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 34*x^34 + 676*x^32 + 9040*x^30 + 90624*x^28 + 699296*x^26 + 4279936*x^24 + 20814464*x^22 + 81078016*x^20 + 249758720*x^18 + 606287872*x^16 + 1124603904*x^14 + 1576677376*x^12 + 1557225472*x^10 + 1081884672*x^8 + 426049536*x^6 + 111083520*x^4 + 5898240*x^2 + 262144)
 
gp: K = bnfinit(x^36 + 34*x^34 + 676*x^32 + 9040*x^30 + 90624*x^28 + 699296*x^26 + 4279936*x^24 + 20814464*x^22 + 81078016*x^20 + 249758720*x^18 + 606287872*x^16 + 1124603904*x^14 + 1576677376*x^12 + 1557225472*x^10 + 1081884672*x^8 + 426049536*x^6 + 111083520*x^4 + 5898240*x^2 + 262144, 1)
 

Normalized defining polynomial

\( x^{36} + 34 x^{34} + 676 x^{32} + 9040 x^{30} + 90624 x^{28} + 699296 x^{26} + 4279936 x^{24} + 20814464 x^{22} + 81078016 x^{20} + 249758720 x^{18} + 606287872 x^{16} + 1124603904 x^{14} + 1576677376 x^{12} + 1557225472 x^{10} + 1081884672 x^{8} + 426049536 x^{6} + 111083520 x^{4} + 5898240 x^{2} + 262144 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(580654212789242722070512622423229170767369878981703159664854695936=2^{54}\cdot 3^{18}\cdot 19^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.11$
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:  $2, 3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(456=2^{3}\cdot 3\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{456}(1,·)$, $\chi_{456}(5,·)$, $\chi_{456}(385,·)$, $\chi_{456}(137,·)$, $\chi_{456}(397,·)$, $\chi_{456}(17,·)$, $\chi_{456}(277,·)$, $\chi_{456}(25,·)$, $\chi_{456}(157,·)$, $\chi_{456}(389,·)$, $\chi_{456}(289,·)$, $\chi_{456}(305,·)$, $\chi_{456}(169,·)$, $\chi_{456}(301,·)$, $\chi_{456}(49,·)$, $\chi_{456}(329,·)$, $\chi_{456}(313,·)$, $\chi_{456}(61,·)$, $\chi_{456}(197,·)$, $\chi_{456}(161,·)$, $\chi_{456}(73,·)$, $\chi_{456}(77,·)$, $\chi_{456}(85,·)$, $\chi_{456}(377,·)$, $\chi_{456}(349,·)$, $\chi_{456}(101,·)$, $\chi_{456}(353,·)$, $\chi_{456}(229,·)$, $\chi_{456}(233,·)$, $\chi_{456}(365,·)$, $\chi_{456}(125,·)$, $\chi_{456}(245,·)$, $\chi_{456}(425,·)$, $\chi_{456}(121,·)$, $\chi_{456}(253,·)$, $\chi_{456}(149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{4096} a^{24}$, $\frac{1}{4096} a^{25}$, $\frac{1}{8192} a^{26}$, $\frac{1}{8192} a^{27}$, $\frac{1}{606208} a^{28} - \frac{1}{75776} a^{26} + \frac{1}{9472} a^{24} - \frac{11}{9472} a^{16} + \frac{7}{4736} a^{14} + \frac{9}{2368} a^{12} + \frac{5}{74} a^{4} - \frac{3}{74} a^{2} + \frac{12}{37}$, $\frac{1}{606208} a^{29} - \frac{1}{75776} a^{27} + \frac{1}{9472} a^{25} - \frac{11}{9472} a^{17} + \frac{7}{4736} a^{15} + \frac{9}{2368} a^{13} + \frac{5}{74} a^{5} - \frac{3}{74} a^{3} + \frac{12}{37} a$, $\frac{1}{1212416} a^{30} - \frac{5}{75776} a^{24} - \frac{11}{18944} a^{18} - \frac{1}{2368} a^{12} + \frac{5}{148} a^{6} + \frac{11}{37}$, $\frac{1}{1212416} a^{31} - \frac{5}{75776} a^{25} - \frac{11}{18944} a^{19} - \frac{1}{2368} a^{13} + \frac{5}{148} a^{7} + \frac{11}{37} a$, $\frac{1}{2424832} a^{32} - \frac{5}{151552} a^{26} - \frac{11}{37888} a^{20} - \frac{1}{4736} a^{14} + \frac{5}{296} a^{8} + \frac{11}{74} a^{2}$, $\frac{1}{2424832} a^{33} - \frac{5}{151552} a^{27} - \frac{11}{37888} a^{21} - \frac{1}{4736} a^{15} + \frac{5}{296} a^{9} + \frac{11}{74} a^{3}$, $\frac{1}{7486578500729254609948939165761536} a^{34} + \frac{281300412479481997906040743}{1871644625182313652487234791440384} a^{32} - \frac{112201860836853889114726237}{467911156295578413121808697860096} a^{30} + \frac{266908974984267448615875399}{467911156295578413121808697860096} a^{28} + \frac{16981243015673558372280861}{1021640079247987801576001523712} a^{26} + \frac{1241362507561130718870617621}{29244447268473650820113043616256} a^{24} + \frac{15691695729946070417197777447}{116977789073894603280452174465024} a^{22} - \frac{6443514245908241039294162037}{14622223634236825410056521808128} a^{20} - \frac{101001686791918555097976543}{1827777954279603176257065226016} a^{18} + \frac{22156432587936718189242127121}{14622223634236825410056521808128} a^{16} - \frac{6600734249705680665965007911}{3655555908559206352514130452032} a^{14} + \frac{15882856580042423864710821167}{3655555908559206352514130452032} a^{12} + \frac{4471241508449024007978869677}{1827777954279603176257065226016} a^{10} - \frac{3407986087532580637108559025}{114236122142475198516066576626} a^{8} - \frac{10280425897839528682011277851}{228472244284950397032133153252} a^{6} - \frac{20976725452887937630990242929}{228472244284950397032133153252} a^{4} + \frac{23706273027634887149215856239}{114236122142475198516066576626} a^{2} + \frac{9961109638644629283251629416}{57118061071237599258033288313}$, $\frac{1}{7486578500729254609948939165761536} a^{35} + \frac{281300412479481997906040743}{1871644625182313652487234791440384} a^{33} - \frac{112201860836853889114726237}{467911156295578413121808697860096} a^{31} + \frac{266908974984267448615875399}{467911156295578413121808697860096} a^{29} + \frac{16981243015673558372280861}{1021640079247987801576001523712} a^{27} + \frac{1241362507561130718870617621}{29244447268473650820113043616256} a^{25} + \frac{15691695729946070417197777447}{116977789073894603280452174465024} a^{23} - \frac{6443514245908241039294162037}{14622223634236825410056521808128} a^{21} - \frac{101001686791918555097976543}{1827777954279603176257065226016} a^{19} + \frac{22156432587936718189242127121}{14622223634236825410056521808128} a^{17} - \frac{6600734249705680665965007911}{3655555908559206352514130452032} a^{15} + \frac{15882856580042423864710821167}{3655555908559206352514130452032} a^{13} + \frac{4471241508449024007978869677}{1827777954279603176257065226016} a^{11} - \frac{3407986087532580637108559025}{114236122142475198516066576626} a^{9} - \frac{10280425897839528682011277851}{228472244284950397032133153252} a^{7} - \frac{20976725452887937630990242929}{228472244284950397032133153252} a^{5} + \frac{23706273027634887149215856239}{114236122142475198516066576626} a^{3} + \frac{9961109638644629283251629416}{57118061071237599258033288313} a$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1742784048321520613392253361}{7486578500729254609948939165761536} a^{34} - \frac{3693825510350631872540163025}{467911156295578413121808697860096} a^{32} - \frac{146621167645672965906869296181}{935822312591156826243617395720192} a^{30} - \frac{1956648776191828306544726157119}{935822312591156826243617395720192} a^{28} - \frac{42738307960775112954483221721}{2043280158495975603152003047424} a^{26} - \frac{37667896439467774774793265896323}{233955578147789206560904348930048} a^{24} - \frac{114956248500605056936346196574849}{116977789073894603280452174465024} a^{22} - \frac{278594490082387740175473699925741}{58488894536947301640226087232512} a^{20} - \frac{270230873428136620688433097061729}{14622223634236825410056521808128} a^{18} - \frac{828175646028262090490021672275013}{14622223634236825410056521808128} a^{16} - \frac{499427116839174479609429317685573}{3655555908559206352514130452032} a^{14} - \frac{918155071348781884221179527924461}{3655555908559206352514130452032} a^{12} - \frac{159099282680069082953770833151219}{456944488569900794064266306504} a^{10} - \frac{154244491858455031945920065198521}{456944488569900794064266306504} a^{8} - \frac{13116483585798664839994874126889}{57118061071237599258033288313} a^{6} - \frac{19557018116840705443989314105499}{228472244284950397032133153252} a^{4} - \frac{2621525988583112397929480907399}{114236122142475198516066576626} a^{2} - \frac{12431064570713107002485864270}{57118061071237599258033288313} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{18}$ (as 36T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_2\times C_{18}$
Character table for $C_2\times C_{18}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), 3.3.361.1, \(\Q(\sqrt{2}, \sqrt{-3})\), 6.0.3518667.1, 6.0.1801557504.1, 6.6.66724352.1, \(\Q(\zeta_{19})^+\), 12.0.3245609440218710016.1, 18.0.5677392343251487443465123.1, 18.0.762006701275810777282417256300544.1, 18.18.38713951190154487490850848768.1

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 R R $18^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ R $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$ $18^{2}$ $18^{2}$ $18^{2}$ $18^{2}$ $18^{2}$

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
2Data not computed
3Data not computed
19Data not computed