↑

Properties

Label 36.0.12527980153...3201.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{54}\cdot 7^{24}\cdot 13^{18}$
Root discriminant $68.56$
Ramified primes $3, 7, 13$
Class number Not computed
Class group Not computed
Galois group $C_6^2$ (as 36T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 5749, 0, 0, 33096582, 0, 0, -262040945, 0, 0, 2089768094, 0, 0, 80687298, 0, 0, 66130813, 0, 0, -1408917, 0, 0, 1824236, 0, 0, 17456, 0, 0, 1611, 0, 0, -16, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 16*x^33 + 1611*x^30 + 17456*x^27 + 1824236*x^24 - 1408917*x^21 + 66130813*x^18 + 80687298*x^15 + 2089768094*x^12 - 262040945*x^9 + 33096582*x^6 + 5749*x^3 + 1)
 
gp: K = bnfinit(x^36 - 16*x^33 + 1611*x^30 + 17456*x^27 + 1824236*x^24 - 1408917*x^21 + 66130813*x^18 + 80687298*x^15 + 2089768094*x^12 - 262040945*x^9 + 33096582*x^6 + 5749*x^3 + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 16 x^{33} + 1611 x^{30} + 17456 x^{27} + 1824236 x^{24} - 1408917 x^{21} + 66130813 x^{18} + 80687298 x^{15} + 2089768094 x^{12} - 262040945 x^{9} + 33096582 x^{6} + 5749 x^{3} + 1 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1252798015380565638574977910764073918937637802772869190703398883201=3^{54}\cdot 7^{24}\cdot 13^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.56$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(819=3^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{819}(1,·)$, $\chi_{819}(389,·)$, $\chi_{819}(779,·)$, $\chi_{819}(781,·)$, $\chi_{819}(142,·)$, $\chi_{819}(274,·)$, $\chi_{819}(662,·)$, $\chi_{819}(25,·)$, $\chi_{819}(155,·)$, $\chi_{819}(415,·)$, $\chi_{819}(547,·)$, $\chi_{819}(298,·)$, $\chi_{819}(428,·)$, $\chi_{819}(688,·)$, $\chi_{819}(53,·)$, $\chi_{819}(571,·)$, $\chi_{819}(701,·)$, $\chi_{819}(64,·)$, $\chi_{819}(326,·)$, $\chi_{819}(716,·)$, $\chi_{819}(79,·)$, $\chi_{819}(337,·)$, $\chi_{819}(599,·)$, $\chi_{819}(92,·)$, $\chi_{819}(352,·)$, $\chi_{819}(610,·)$, $\chi_{819}(443,·)$, $\chi_{819}(233,·)$, $\chi_{819}(235,·)$, $\chi_{819}(365,·)$, $\chi_{819}(625,·)$, $\chi_{819}(116,·)$, $\chi_{819}(506,·)$, $\chi_{819}(508,·)$, $\chi_{819}(170,·)$, $\chi_{819}(638,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{18} - \frac{1}{2}$, $\frac{1}{2} a^{28} - \frac{1}{2} a^{19} - \frac{1}{2} a$, $\frac{1}{2} a^{29} - \frac{1}{2} a^{20} - \frac{1}{2} a^{2}$, $\frac{1}{44147300117967142} a^{30} - \frac{9363029901186395}{44147300117967142} a^{27} + \frac{427396293944907}{22073650058983571} a^{24} + \frac{8316871360247821}{44147300117967142} a^{21} - \frac{8634908325083647}{44147300117967142} a^{18} - \frac{4565463109188187}{22073650058983571} a^{15} + \frac{9777836872082155}{22073650058983571} a^{12} + \frac{877819502609860}{22073650058983571} a^{9} + \frac{249545921815734}{22073650058983571} a^{6} + \frac{3969414815587777}{44147300117967142} a^{3} + \frac{3471855785505801}{44147300117967142}$, $\frac{1}{44147300117967142} a^{31} - \frac{9363029901186395}{44147300117967142} a^{28} + \frac{427396293944907}{22073650058983571} a^{25} + \frac{8316871360247821}{44147300117967142} a^{22} - \frac{8634908325083647}{44147300117967142} a^{19} - \frac{4565463109188187}{22073650058983571} a^{16} + \frac{9777836872082155}{22073650058983571} a^{13} + \frac{877819502609860}{22073650058983571} a^{10} + \frac{249545921815734}{22073650058983571} a^{7} + \frac{3969414815587777}{44147300117967142} a^{4} + \frac{3471855785505801}{44147300117967142} a$, $\frac{1}{44147300117967142} a^{32} - \frac{9363029901186395}{44147300117967142} a^{29} + \frac{427396293944907}{22073650058983571} a^{26} + \frac{8316871360247821}{44147300117967142} a^{23} - \frac{8634908325083647}{44147300117967142} a^{20} - \frac{4565463109188187}{22073650058983571} a^{17} + \frac{9777836872082155}{22073650058983571} a^{14} + \frac{877819502609860}{22073650058983571} a^{11} + \frac{249545921815734}{22073650058983571} a^{8} + \frac{3969414815587777}{44147300117967142} a^{5} + \frac{3471855785505801}{44147300117967142} a^{2}$, $\frac{1}{4390281517425555606250289780598925147270305986} a^{33} + \frac{6149397509176430644625779389}{2195140758712777803125144890299462573635152993} a^{30} - \frac{22400052802067584046436452976354645630050332}{2195140758712777803125144890299462573635152993} a^{27} + \frac{416066728899615549660685039696242418801117567}{4390281517425555606250289780598925147270305986} a^{24} - \frac{80994660057181684097410331940768771565137922}{2195140758712777803125144890299462573635152993} a^{21} + \frac{709881917714138706000463285191654916458080927}{2195140758712777803125144890299462573635152993} a^{18} - \frac{1051809413987195039258168368087415825281017390}{2195140758712777803125144890299462573635152993} a^{15} + \frac{1014584599835562728160690877644684248870982849}{2195140758712777803125144890299462573635152993} a^{12} + \frac{574347796546557664588978272974740605258842040}{2195140758712777803125144890299462573635152993} a^{9} - \frac{930509301846344659220123748274030209893469961}{4390281517425555606250289780598925147270305986} a^{6} - \frac{48830985808419896089600036521658457475820376}{2195140758712777803125144890299462573635152993} a^{3} + \frac{393784886398436695286111505174918577463404979}{2195140758712777803125144890299462573635152993}$, $\frac{1}{4390281517425555606250289780598925147270305986} a^{34} + \frac{6149397509176430644625779389}{2195140758712777803125144890299462573635152993} a^{31} - \frac{22400052802067584046436452976354645630050332}{2195140758712777803125144890299462573635152993} a^{28} + \frac{416066728899615549660685039696242418801117567}{4390281517425555606250289780598925147270305986} a^{25} - \frac{80994660057181684097410331940768771565137922}{2195140758712777803125144890299462573635152993} a^{22} + \frac{709881917714138706000463285191654916458080927}{2195140758712777803125144890299462573635152993} a^{19} - \frac{1051809413987195039258168368087415825281017390}{2195140758712777803125144890299462573635152993} a^{16} + \frac{1014584599835562728160690877644684248870982849}{2195140758712777803125144890299462573635152993} a^{13} + \frac{574347796546557664588978272974740605258842040}{2195140758712777803125144890299462573635152993} a^{10} - \frac{930509301846344659220123748274030209893469961}{4390281517425555606250289780598925147270305986} a^{7} - \frac{48830985808419896089600036521658457475820376}{2195140758712777803125144890299462573635152993} a^{4} + \frac{393784886398436695286111505174918577463404979}{2195140758712777803125144890299462573635152993} a$, $\frac{1}{4390281517425555606250289780598925147270305986} a^{35} + \frac{6149397509176430644625779389}{2195140758712777803125144890299462573635152993} a^{32} - \frac{22400052802067584046436452976354645630050332}{2195140758712777803125144890299462573635152993} a^{29} + \frac{416066728899615549660685039696242418801117567}{4390281517425555606250289780598925147270305986} a^{26} - \frac{80994660057181684097410331940768771565137922}{2195140758712777803125144890299462573635152993} a^{23} + \frac{709881917714138706000463285191654916458080927}{2195140758712777803125144890299462573635152993} a^{20} - \frac{1051809413987195039258168368087415825281017390}{2195140758712777803125144890299462573635152993} a^{17} + \frac{1014584599835562728160690877644684248870982849}{2195140758712777803125144890299462573635152993} a^{14} + \frac{574347796546557664588978272974740605258842040}{2195140758712777803125144890299462573635152993} a^{11} - \frac{930509301846344659220123748274030209893469961}{4390281517425555606250289780598925147270305986} a^{8} - \frac{48830985808419896089600036521658457475820376}{2195140758712777803125144890299462573635152993} a^{5} + \frac{393784886398436695286111505174918577463404979}{2195140758712777803125144890299462573635152993} a^{2}$

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:  \( -\frac{49860992545937271898506957679336421587}{10477998848271015766707135514555907272721494} a^{34} + \frac{395743368857565118104361967031924941178}{5238999424135507883353567757277953636360747} a^{31} - \frac{40112716047786975483533928283730596542225}{5238999424135507883353567757277953636360747} a^{28} - \frac{2612775353345573035538039206641547837213}{31091984712970373194976663247940377663862} a^{25} - \frac{45534000044497145255408208318403326460889496}{5238999424135507883353567757277953636360747} a^{22} + \frac{29388573674842919300872452126152226399333349}{5238999424135507883353567757277953636360747} a^{19} - \frac{1644242908261703855566475990496841580847736281}{5238999424135507883353567757277953636360747} a^{16} - \frac{2219518059418090038684465689440511523460735973}{5238999424135507883353567757277953636360747} a^{13} - \frac{52352683235575937172342275846879158415173186659}{5238999424135507883353567757277953636360747} a^{10} - \frac{76472027186800349616986364231589579568141451}{10477998848271015766707135514555907272721494} a^{7} - \frac{6641739499855096789313710055052427609090}{5238999424135507883353567757277953636360747} a^{4} - \frac{93895553075212754887595398225008769921115849}{5238999424135507883353567757277953636360747} a \) (order $18$)
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

$C_6^2$ (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{13}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\zeta_{9})\), 6.0.47258883.2, 6.0.47258883.1, 6.0.64827.1, 6.0.43243551.1, 6.6.14414517.1, 6.0.103827765951.7, 6.6.34609255317.1, 6.0.103827765951.8, 6.6.34609255317.2, 6.0.142424919.1, 6.6.5274997.1, 9.9.62523502209.1, 12.0.1870004703089601.1, 12.0.10780204982375634934401.1, 12.0.10780204982375634934401.2, 12.0.20284857552156561.1, 18.0.105548084868928352751387.1, 18.0.1119284599813901503934006266380351.1, 18.18.41454985178292648293852083940013.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
7Data not computed
$13$13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$