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Properties

Label 36.36.2287833182...2289.1
Degree $36$
Signature $[36, 0]$
Discriminant $3^{18}\cdot 7^{30}\cdot 13^{30}$
Root discriminant $74.32$
Ramified primes $3, 7, 13$
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, 15, -1563, -2330, 94085, 35416, -1913861, -12482, 18093496, -2758730, -89499925, 20933394, 250492399, -63559658, -430753452, 103172083, 482872954, -100868241, -366929496, 63027785, 193701628, -25930789, -72013209, 7104729, 18942377, -1291483, -3511489, 152549, 452485, -11171, -39419, 458, 2203, -8, -71, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 71*x^34 - 8*x^33 + 2203*x^32 + 458*x^31 - 39419*x^30 - 11171*x^29 + 452485*x^28 + 152549*x^27 - 3511489*x^26 - 1291483*x^25 + 18942377*x^24 + 7104729*x^23 - 72013209*x^22 - 25930789*x^21 + 193701628*x^20 + 63027785*x^19 - 366929496*x^18 - 100868241*x^17 + 482872954*x^16 + 103172083*x^15 - 430753452*x^14 - 63559658*x^13 + 250492399*x^12 + 20933394*x^11 - 89499925*x^10 - 2758730*x^9 + 18093496*x^8 - 12482*x^7 - 1913861*x^6 + 35416*x^5 + 94085*x^4 - 2330*x^3 - 1563*x^2 + 15*x + 1)
 
gp: K = bnfinit(x^36 - 71*x^34 - 8*x^33 + 2203*x^32 + 458*x^31 - 39419*x^30 - 11171*x^29 + 452485*x^28 + 152549*x^27 - 3511489*x^26 - 1291483*x^25 + 18942377*x^24 + 7104729*x^23 - 72013209*x^22 - 25930789*x^21 + 193701628*x^20 + 63027785*x^19 - 366929496*x^18 - 100868241*x^17 + 482872954*x^16 + 103172083*x^15 - 430753452*x^14 - 63559658*x^13 + 250492399*x^12 + 20933394*x^11 - 89499925*x^10 - 2758730*x^9 + 18093496*x^8 - 12482*x^7 - 1913861*x^6 + 35416*x^5 + 94085*x^4 - 2330*x^3 - 1563*x^2 + 15*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 71 x^{34} - 8 x^{33} + 2203 x^{32} + 458 x^{31} - 39419 x^{30} - 11171 x^{29} + 452485 x^{28} + 152549 x^{27} - 3511489 x^{26} - 1291483 x^{25} + 18942377 x^{24} + 7104729 x^{23} - 72013209 x^{22} - 25930789 x^{21} + 193701628 x^{20} + 63027785 x^{19} - 366929496 x^{18} - 100868241 x^{17} + 482872954 x^{16} + 103172083 x^{15} - 430753452 x^{14} - 63559658 x^{13} + 250492399 x^{12} + 20933394 x^{11} - 89499925 x^{10} - 2758730 x^{9} + 18093496 x^{8} - 12482 x^{7} - 1913861 x^{6} + 35416 x^{5} + 94085 x^{4} - 2330 x^{3} - 1563 x^{2} + 15 x + 1 \)

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:  \(22878331820822683097801634238807198405761132789439230168181892642289=3^{18}\cdot 7^{30}\cdot 13^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.32$
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:  \(273=3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{273}(256,·)$, $\chi_{273}(1,·)$, $\chi_{273}(131,·)$, $\chi_{273}(4,·)$, $\chi_{273}(257,·)$, $\chi_{273}(269,·)$, $\chi_{273}(142,·)$, $\chi_{273}(16,·)$, $\chi_{273}(17,·)$, $\chi_{273}(146,·)$, $\chi_{273}(22,·)$, $\chi_{273}(152,·)$, $\chi_{273}(25,·)$, $\chi_{273}(38,·)$, $\chi_{273}(43,·)$, $\chi_{273}(172,·)$, $\chi_{273}(173,·)$, $\chi_{273}(185,·)$, $\chi_{273}(62,·)$, $\chi_{273}(64,·)$, $\chi_{273}(194,·)$, $\chi_{273}(68,·)$, $\chi_{273}(205,·)$, $\chi_{273}(79,·)$, $\chi_{273}(209,·)$, $\chi_{273}(211,·)$, $\chi_{273}(88,·)$, $\chi_{273}(272,·)$, $\chi_{273}(100,·)$, $\chi_{273}(101,·)$, $\chi_{273}(230,·)$, $\chi_{273}(235,·)$, $\chi_{273}(248,·)$, $\chi_{273}(121,·)$, $\chi_{273}(251,·)$, $\chi_{273}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{6} a^{30} + \frac{1}{6} a^{29} - \frac{1}{6} a^{28} - \frac{1}{2} a^{27} - \frac{1}{6} a^{26} + \frac{1}{6} a^{25} - \frac{1}{6} a^{24} - \frac{1}{2} a^{23} + \frac{1}{6} a^{22} - \frac{1}{6} a^{21} - \frac{1}{2} a^{20} - \frac{1}{6} a^{19} - \frac{1}{2} a^{18} + \frac{1}{6} a^{17} + \frac{1}{6} a^{16} - \frac{1}{2} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{31} - \frac{1}{3} a^{29} - \frac{1}{3} a^{28} + \frac{1}{3} a^{27} + \frac{1}{3} a^{26} - \frac{1}{3} a^{25} - \frac{1}{3} a^{24} - \frac{1}{3} a^{23} - \frac{1}{3} a^{22} - \frac{1}{3} a^{21} + \frac{1}{3} a^{20} - \frac{1}{3} a^{19} - \frac{1}{3} a^{18} + \frac{1}{3} a^{16} - \frac{1}{6} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{6} a^{32} + \frac{1}{3} a^{27} + \frac{1}{3} a^{26} + \frac{1}{3} a^{24} - \frac{1}{3} a^{23} - \frac{1}{3} a^{20} + \frac{1}{3} a^{19} - \frac{1}{3} a^{17} + \frac{1}{6} a^{16} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{2} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{6} a^{33} + \frac{1}{3} a^{28} + \frac{1}{3} a^{27} + \frac{1}{3} a^{25} - \frac{1}{3} a^{24} - \frac{1}{3} a^{21} + \frac{1}{3} a^{20} - \frac{1}{3} a^{18} + \frac{1}{6} a^{17} - \frac{1}{3} a^{16} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{2} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{34} + \frac{1}{3} a^{29} + \frac{1}{3} a^{28} + \frac{1}{3} a^{26} - \frac{1}{3} a^{25} - \frac{1}{3} a^{22} + \frac{1}{3} a^{21} - \frac{1}{3} a^{19} + \frac{1}{6} a^{18} - \frac{1}{3} a^{17} - \frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{2} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{195072093957497200073391048453809980846945412418970116106034088318505414723333329020652615399168738} a^{35} + \frac{4834454496823990902989628631779519402542589460507639569758766814224640356245601960643677390325201}{195072093957497200073391048453809980846945412418970116106034088318505414723333329020652615399168738} a^{34} + \frac{3663448767462806965255916299559226563911294730071094181383897073656169143125925306273066775791714}{97536046978748600036695524226904990423472706209485058053017044159252707361666664510326307699584369} a^{33} + \frac{3331020462160927545046557013626828835466743387842265601366069358858975756213697817819996537865713}{65024031319165733357797016151269993615648470806323372035344696106168471574444443006884205133056246} a^{32} - \frac{2399665699771900226811341166380835791401066618305050648723506860217483453254344069999259406632283}{65024031319165733357797016151269993615648470806323372035344696106168471574444443006884205133056246} a^{31} + \frac{9206546247332405276472557050048638725538730290732791064547250755355494942404682428630130297057541}{195072093957497200073391048453809980846945412418970116106034088318505414723333329020652615399168738} a^{30} + \frac{84196806255181057016609759829715379320719680413170127973955242937436093113225927603958330605569251}{195072093957497200073391048453809980846945412418970116106034088318505414723333329020652615399168738} a^{29} + \frac{17428157302912773094688590211976998277292174144460261068603333765027378719652432680058237481831893}{195072093957497200073391048453809980846945412418970116106034088318505414723333329020652615399168738} a^{28} - \frac{63620577339854180568607278916541142446779645854929964907538065613334869576404770910295612434029143}{195072093957497200073391048453809980846945412418970116106034088318505414723333329020652615399168738} a^{27} + \frac{20630971994871639382687135332752705053055818497172259210562670887610860548241515185402520562104101}{65024031319165733357797016151269993615648470806323372035344696106168471574444443006884205133056246} a^{26} - \frac{18999533212278019135126563962801067308094901958091154252796310018750758094699606081315013623962609}{65024031319165733357797016151269993615648470806323372035344696106168471574444443006884205133056246} a^{25} - \frac{42936905518951368829847262627731308906475311690002029400392793739022926522120728402688658962704527}{195072093957497200073391048453809980846945412418970116106034088318505414723333329020652615399168738} a^{24} + \frac{75993085547353782285445871967120345485378759482782269691287416085241113246253302602992888070748875}{195072093957497200073391048453809980846945412418970116106034088318505414723333329020652615399168738} a^{23} - \frac{23054814595161249527241531879009937693693793820890354261898204703268067195796916031736433889456301}{65024031319165733357797016151269993615648470806323372035344696106168471574444443006884205133056246} a^{22} - \frac{6995133918823928877902577967903821160473738016895583125208050315237384659611963495221761249167343}{65024031319165733357797016151269993615648470806323372035344696106168471574444443006884205133056246} a^{21} + \frac{7467821186725959181747492632830108133610497878617024186385532806314151640711801493450251170490659}{65024031319165733357797016151269993615648470806323372035344696106168471574444443006884205133056246} a^{20} + \frac{48347776770791664169466725239200026970621891581829149117141997870357401139539753617162773695591921}{97536046978748600036695524226904990423472706209485058053017044159252707361666664510326307699584369} a^{19} + \frac{953982267299199779106968947736875565915795238181078553895584516685733722446666464844483932452092}{32512015659582866678898508075634996807824235403161686017672348053084235787222221503442102566528123} a^{18} + \frac{818017180649242880533506820141148904909100075263699881785846169990877312517837967242663136610381}{65024031319165733357797016151269993615648470806323372035344696106168471574444443006884205133056246} a^{17} - \frac{15818365619823837687558764769309934308538213886296618637218899431374498191285906577254972184888885}{32512015659582866678898508075634996807824235403161686017672348053084235787222221503442102566528123} a^{16} + \frac{39729872910293093951639340871110769076710815381928824371695170518410034276433643432604151500961305}{97536046978748600036695524226904990423472706209485058053017044159252707361666664510326307699584369} a^{15} + \frac{12188592709294795744762221153602143129847398153302381355233273160340606212640254548985655363796293}{97536046978748600036695524226904990423472706209485058053017044159252707361666664510326307699584369} a^{14} - \frac{11472508172525431128245047186037065532916558849496425276036195557525941072833619001107172529876415}{97536046978748600036695524226904990423472706209485058053017044159252707361666664510326307699584369} a^{13} - \frac{25897086974802929170378182929588005405698163881229894207351710556201669601489787763863316232578436}{97536046978748600036695524226904990423472706209485058053017044159252707361666664510326307699584369} a^{12} - \frac{2026827694537735138112944617830697170925113450550908382748503184148267293440684914065210652215397}{195072093957497200073391048453809980846945412418970116106034088318505414723333329020652615399168738} a^{11} - \frac{48260077272020414623921764247339542668091842055247012913368906482852458833814536187835747395627497}{195072093957497200073391048453809980846945412418970116106034088318505414723333329020652615399168738} a^{10} - \frac{6677623840620688626242341956469585364899145476880079605347798910441683036469057847114752448769312}{32512015659582866678898508075634996807824235403161686017672348053084235787222221503442102566528123} a^{9} - \frac{9295828621981384284306843086626617291155234534559991753003673653053546605211044823908530466547479}{195072093957497200073391048453809980846945412418970116106034088318505414723333329020652615399168738} a^{8} + \frac{4989861351214128726758639917899716524797696690443203210909767703717662266921391725418463884779735}{32512015659582866678898508075634996807824235403161686017672348053084235787222221503442102566528123} a^{7} - \frac{38159107591342956875174983451038474142788126911368007715281685664978497727649454975970183091628456}{97536046978748600036695524226904990423472706209485058053017044159252707361666664510326307699584369} a^{6} - \frac{5182603171617442572830123711084872362396241454817113729008553861383694929390905131684544878632872}{32512015659582866678898508075634996807824235403161686017672348053084235787222221503442102566528123} a^{5} - \frac{2256660869078466838463797691680851637352884253679524984593662080864694296269933162291683837799109}{97536046978748600036695524226904990423472706209485058053017044159252707361666664510326307699584369} a^{4} + \frac{85592812776586691979022580413919535011739013591413793371070894495963435005107109657241490228552801}{195072093957497200073391048453809980846945412418970116106034088318505414723333329020652615399168738} a^{3} + \frac{46506208912486874944340468334666481531238992893230594779224086431305382577137055182541584190032853}{195072093957497200073391048453809980846945412418970116106034088318505414723333329020652615399168738} a^{2} - \frac{23972274301914994370958777827445210171086188303747978139237967152700310061627279332475164138660127}{97536046978748600036695524226904990423472706209485058053017044159252707361666664510326307699584369} a - \frac{8308054799852073569845359269596327511124376133958557705921683294939236323756630434026838832057877}{32512015659582866678898508075634996807824235403161686017672348053084235787222221503442102566528123}$

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:  \( 20922797578851710000000 \) (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

\(\Q(\sqrt{273}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{21}) \), 3.3.169.1, 3.3.8281.1, \(\Q(\zeta_{7})^+\), 3.3.8281.2, \(\Q(\sqrt{13}, \sqrt{21})\), 6.6.3438544473.1, 6.6.168488679177.2, 6.6.996974433.1, 6.6.168488679177.1, \(\Q(\zeta_{13})^+\), 6.6.264503421.1, 6.6.891474493.1, 6.6.12960667629.1, 6.6.5274997.1, \(\Q(\zeta_{21})^+\), 6.6.891474493.2, 6.6.12960667629.2, 9.9.567869252041.1, 12.12.11823588092798847729.1, 12.12.28388435010810033397329.1, 12.12.993958020055671489.1, 12.12.28388435010810033397329.2, 18.18.4783129918873486243975221249718233.1, 18.18.708478645847689707516501157.1, 18.18.2177118761435360147462549499189.1

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 ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed
$13$13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$