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Properties

Label 36.36.4999758568...0625.1
Degree $36$
Signature $[36, 0]$
Discriminant $3^{54}\cdot 5^{18}\cdot 7^{30}$
Root discriminant $58.81$
Ramified primes $3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_6^2$ (as 36T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -24, -360, 2408, 13686, -43479, -187354, 353091, 1304280, -1623250, -5391504, 4701951, 14426662, -9097842, -26324586, 12157547, 33816645, -11422746, -31151513, 7596969, 20757231, -3567239, -10013571, 1169019, 3476891, -261666, -858159, 38637, 147555, -3555, -17119, 183, 1269, -4, -54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 54*x^34 - 4*x^33 + 1269*x^32 + 183*x^31 - 17119*x^30 - 3555*x^29 + 147555*x^28 + 38637*x^27 - 858159*x^26 - 261666*x^25 + 3476891*x^24 + 1169019*x^23 - 10013571*x^22 - 3567239*x^21 + 20757231*x^20 + 7596969*x^19 - 31151513*x^18 - 11422746*x^17 + 33816645*x^16 + 12157547*x^15 - 26324586*x^14 - 9097842*x^13 + 14426662*x^12 + 4701951*x^11 - 5391504*x^10 - 1623250*x^9 + 1304280*x^8 + 353091*x^7 - 187354*x^6 - 43479*x^5 + 13686*x^4 + 2408*x^3 - 360*x^2 - 24*x + 1)
 
gp: K = bnfinit(x^36 - 54*x^34 - 4*x^33 + 1269*x^32 + 183*x^31 - 17119*x^30 - 3555*x^29 + 147555*x^28 + 38637*x^27 - 858159*x^26 - 261666*x^25 + 3476891*x^24 + 1169019*x^23 - 10013571*x^22 - 3567239*x^21 + 20757231*x^20 + 7596969*x^19 - 31151513*x^18 - 11422746*x^17 + 33816645*x^16 + 12157547*x^15 - 26324586*x^14 - 9097842*x^13 + 14426662*x^12 + 4701951*x^11 - 5391504*x^10 - 1623250*x^9 + 1304280*x^8 + 353091*x^7 - 187354*x^6 - 43479*x^5 + 13686*x^4 + 2408*x^3 - 360*x^2 - 24*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 54 x^{34} - 4 x^{33} + 1269 x^{32} + 183 x^{31} - 17119 x^{30} - 3555 x^{29} + 147555 x^{28} + 38637 x^{27} - 858159 x^{26} - 261666 x^{25} + 3476891 x^{24} + 1169019 x^{23} - 10013571 x^{22} - 3567239 x^{21} + 20757231 x^{20} + 7596969 x^{19} - 31151513 x^{18} - 11422746 x^{17} + 33816645 x^{16} + 12157547 x^{15} - 26324586 x^{14} - 9097842 x^{13} + 14426662 x^{12} + 4701951 x^{11} - 5391504 x^{10} - 1623250 x^{9} + 1304280 x^{8} + 353091 x^{7} - 187354 x^{6} - 43479 x^{5} + 13686 x^{4} + 2408 x^{3} - 360 x^{2} - 24 x + 1 \)

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:  \(4999758568289868528789868885747458073284974388119052886962890625=3^{54}\cdot 5^{18}\cdot 7^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.81$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(315=3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{315}(256,·)$, $\chi_{315}(1,·)$, $\chi_{315}(131,·)$, $\chi_{315}(4,·)$, $\chi_{315}(269,·)$, $\chi_{315}(109,·)$, $\chi_{315}(16,·)$, $\chi_{315}(274,·)$, $\chi_{315}(151,·)$, $\chi_{315}(26,·)$, $\chi_{315}(289,·)$, $\chi_{315}(164,·)$, $\chi_{315}(41,·)$, $\chi_{315}(299,·)$, $\chi_{315}(46,·)$, $\chi_{315}(311,·)$, $\chi_{315}(184,·)$, $\chi_{315}(314,·)$, $\chi_{315}(59,·)$, $\chi_{315}(64,·)$, $\chi_{315}(194,·)$, $\chi_{315}(206,·)$, $\chi_{315}(79,·)$, $\chi_{315}(209,·)$, $\chi_{315}(211,·)$, $\chi_{315}(214,·)$, $\chi_{315}(89,·)$, $\chi_{315}(226,·)$, $\chi_{315}(101,·)$, $\chi_{315}(104,·)$, $\chi_{315}(106,·)$, $\chi_{315}(236,·)$, $\chi_{315}(146,·)$, $\chi_{315}(169,·)$, $\chi_{315}(121,·)$, $\chi_{315}(251,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{26} - \frac{1}{2} a^{25} - \frac{1}{2} a^{24} - \frac{1}{2} a^{23} - \frac{1}{2} a^{22} - \frac{1}{2} a^{21} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{28} - \frac{1}{2} a^{21} - \frac{1}{2} a^{20} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{29} - \frac{1}{2} a^{22} - \frac{1}{2} a^{21} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{30} - \frac{1}{2} a^{23} - \frac{1}{2} a^{22} - \frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{31} - \frac{1}{2} a^{24} - \frac{1}{2} a^{23} - \frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{32} - \frac{1}{2} a^{25} - \frac{1}{2} a^{24} - \frac{1}{2} a^{18} - \frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{33} - \frac{1}{2} a^{26} - \frac{1}{2} a^{25} - \frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{34} - \frac{1}{2} a^{25} - \frac{1}{2} a^{24} - \frac{1}{2} a^{23} - \frac{1}{2} a^{22} - \frac{1}{2} a^{21} - \frac{1}{2} a^{20} - \frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{65390383628944674791806320901162606644591953729538346212992554} a^{35} - \frac{412350709804256477553150509654194049842723765852697131935494}{32695191814472337395903160450581303322295976864769173106496277} a^{34} + \frac{7986703662774878339049633997096657639694275308420948169764354}{32695191814472337395903160450581303322295976864769173106496277} a^{33} + \frac{827590782179691723343598480214469217403785337015890412173985}{32695191814472337395903160450581303322295976864769173106496277} a^{32} - \frac{7066793753674735812718561884440421192487885921028327254900656}{32695191814472337395903160450581303322295976864769173106496277} a^{31} + \frac{4709781387427034152049054864461309840288049521096234143386687}{32695191814472337395903160450581303322295976864769173106496277} a^{30} + \frac{4034553437470880676407576281108882274145331300947143336129999}{32695191814472337395903160450581303322295976864769173106496277} a^{29} + \frac{13734337423020128696530032131784265612218582884811151198160767}{65390383628944674791806320901162606644591953729538346212992554} a^{28} - \frac{7444355932652210865508457408064183260320650226413292720740339}{65390383628944674791806320901162606644591953729538346212992554} a^{27} - \frac{3787947303184083675976905931387207214039258235823133923581573}{32695191814472337395903160450581303322295976864769173106496277} a^{26} + \frac{3684111338518417035726425263549710638531755071049588376530823}{32695191814472337395903160450581303322295976864769173106496277} a^{25} - \frac{7611854921837380965495773244698176761320014694971671224251396}{32695191814472337395903160450581303322295976864769173106496277} a^{24} - \frac{1632500470238413678864690009952314824363226110165805992097122}{32695191814472337395903160450581303322295976864769173106496277} a^{23} - \frac{4110644591874765949323886247702329480618791679814104024346533}{32695191814472337395903160450581303322295976864769173106496277} a^{22} - \frac{2792733448731905082170741425383252093490334970423058951289485}{65390383628944674791806320901162606644591953729538346212992554} a^{21} - \frac{11906930987959871134650150679763541093557726108551446953022993}{32695191814472337395903160450581303322295976864769173106496277} a^{20} + \frac{24870277633506483665372000388098560110504461162624972390675207}{65390383628944674791806320901162606644591953729538346212992554} a^{19} - \frac{4912135710787789193033266738068202484862617581718523940396206}{32695191814472337395903160450581303322295976864769173106496277} a^{18} - \frac{4305412191178385105672947744080694603875212157248419650955199}{65390383628944674791806320901162606644591953729538346212992554} a^{17} - \frac{7776389511339768558555686899802522897364711175799331476322280}{32695191814472337395903160450581303322295976864769173106496277} a^{16} - \frac{2177189127602133478472167349771271217436308534496779135437143}{32695191814472337395903160450581303322295976864769173106496277} a^{15} + \frac{11230687264802538342994700513325687640517302751040630590495439}{65390383628944674791806320901162606644591953729538346212992554} a^{14} + \frac{6748952092061948403612470727495160218679686946913141179009205}{65390383628944674791806320901162606644591953729538346212992554} a^{13} + \frac{19472453521380252705718502312406616771101438115619177848812593}{65390383628944674791806320901162606644591953729538346212992554} a^{12} + \frac{19451820420788126450772170728252672501700311251339265506520413}{65390383628944674791806320901162606644591953729538346212992554} a^{11} + \frac{8749859376077797986787412301720493650395858014976549468117387}{65390383628944674791806320901162606644591953729538346212992554} a^{10} + \frac{31885710005161663887470489742522150682027778417336096323581263}{65390383628944674791806320901162606644591953729538346212992554} a^{9} - \frac{2015785172998845686743262773882431911569714097450734367064423}{65390383628944674791806320901162606644591953729538346212992554} a^{8} + \frac{13107581863988380088723573039184175625729089302817875724023507}{65390383628944674791806320901162606644591953729538346212992554} a^{7} - \frac{5015025634568345202155268141470790056008431271259146223785700}{32695191814472337395903160450581303322295976864769173106496277} a^{6} - \frac{7243340778092799369414575354672125279023404345578834879833290}{32695191814472337395903160450581303322295976864769173106496277} a^{5} + \frac{15323284517745873531915712284646036967576583286833092179077834}{32695191814472337395903160450581303322295976864769173106496277} a^{4} - \frac{7239473431976332671059675290468459406337689283001592181372341}{32695191814472337395903160450581303322295976864769173106496277} a^{3} - \frac{2604017882743504561659304755387960247939181767139819540869584}{32695191814472337395903160450581303322295976864769173106496277} a^{2} - \frac{1576446155227895715539831557839052378128479991677463506097864}{32695191814472337395903160450581303322295976864769173106496277} a + \frac{6104807135642630302213633635772862520053683532991178500735746}{32695191814472337395903160450581303322295976864769173106496277}$

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:  \( 268381905239177630000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6^2$ (as 36T4):

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 $C_6^2$
Character table for $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, \(\Q(\sqrt{5}, \sqrt{21})\), 6.6.843908625.1, 6.6.41351522625.2, 6.6.56723625.1, 6.6.41351522625.1, 6.6.820125.1, 6.6.6751269.1, 6.6.1969120125.2, 6.6.330812181.1, 6.6.300125.1, \(\Q(\zeta_{21})^+\), 6.6.1969120125.1, 6.6.330812181.2, 9.9.62523502209.1, 12.12.712181767349390625.1, 12.12.1709948423405886890625.2, 12.12.3217569633140625.1, 12.12.1709948423405886890625.1, 18.18.70708970918051611315870587890625.1, 18.18.7635133454060210702501953125.1, \(\Q(\zeta_{63})^+\)

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{6}$ R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{12}$

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
3Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed