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Properties

Label 36.36.7949113468...0625.1
Degree $36$
Signature $[36, 0]$
Discriminant $5^{18}\cdot 37^{34}$
Root discriminant $67.70$
Ramified primes $5, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![1, -9, -216, 75, 7575, 1398, -104688, -24948, 755073, 180587, -3261522, -759461, 9135431, 2049918, -17429913, -3730447, 23361262, 4709558, -22401208, -4188542, 15506227, 2637723, -7758928, -1171922, 2791702, 363060, -713805, -76859, 127184, 10771, -15301, -947, 1177, 47, -52, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 52*x^34 + 47*x^33 + 1177*x^32 - 947*x^31 - 15301*x^30 + 10771*x^29 + 127184*x^28 - 76859*x^27 - 713805*x^26 + 363060*x^25 + 2791702*x^24 - 1171922*x^23 - 7758928*x^22 + 2637723*x^21 + 15506227*x^20 - 4188542*x^19 - 22401208*x^18 + 4709558*x^17 + 23361262*x^16 - 3730447*x^15 - 17429913*x^14 + 2049918*x^13 + 9135431*x^12 - 759461*x^11 - 3261522*x^10 + 180587*x^9 + 755073*x^8 - 24948*x^7 - 104688*x^6 + 1398*x^5 + 7575*x^4 + 75*x^3 - 216*x^2 - 9*x + 1)
 
gp: K = bnfinit(x^36 - x^35 - 52*x^34 + 47*x^33 + 1177*x^32 - 947*x^31 - 15301*x^30 + 10771*x^29 + 127184*x^28 - 76859*x^27 - 713805*x^26 + 363060*x^25 + 2791702*x^24 - 1171922*x^23 - 7758928*x^22 + 2637723*x^21 + 15506227*x^20 - 4188542*x^19 - 22401208*x^18 + 4709558*x^17 + 23361262*x^16 - 3730447*x^15 - 17429913*x^14 + 2049918*x^13 + 9135431*x^12 - 759461*x^11 - 3261522*x^10 + 180587*x^9 + 755073*x^8 - 24948*x^7 - 104688*x^6 + 1398*x^5 + 7575*x^4 + 75*x^3 - 216*x^2 - 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 52 x^{34} + 47 x^{33} + 1177 x^{32} - 947 x^{31} - 15301 x^{30} + 10771 x^{29} + 127184 x^{28} - 76859 x^{27} - 713805 x^{26} + 363060 x^{25} + 2791702 x^{24} - 1171922 x^{23} - 7758928 x^{22} + 2637723 x^{21} + 15506227 x^{20} - 4188542 x^{19} - 22401208 x^{18} + 4709558 x^{17} + 23361262 x^{16} - 3730447 x^{15} - 17429913 x^{14} + 2049918 x^{13} + 9135431 x^{12} - 759461 x^{11} - 3261522 x^{10} + 180587 x^{9} + 755073 x^{8} - 24948 x^{7} - 104688 x^{6} + 1398 x^{5} + 7575 x^{4} + 75 x^{3} - 216 x^{2} - 9 x + 1 \)

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:  \(794911346891760916131484470981274061649585725862903171539306640625=5^{18}\cdot 37^{34}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.70$
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:  $5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(185=5\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{185}(1,·)$, $\chi_{185}(4,·)$, $\chi_{185}(136,·)$, $\chi_{185}(9,·)$, $\chi_{185}(11,·)$, $\chi_{185}(141,·)$, $\chi_{185}(16,·)$, $\chi_{185}(21,·)$, $\chi_{185}(151,·)$, $\chi_{185}(26,·)$, $\chi_{185}(159,·)$, $\chi_{185}(34,·)$, $\chi_{185}(36,·)$, $\chi_{185}(41,·)$, $\chi_{185}(44,·)$, $\chi_{185}(46,·)$, $\chi_{185}(176,·)$, $\chi_{185}(49,·)$, $\chi_{185}(181,·)$, $\chi_{185}(184,·)$, $\chi_{185}(64,·)$, $\chi_{185}(139,·)$, $\chi_{185}(71,·)$, $\chi_{185}(81,·)$, $\chi_{185}(84,·)$, $\chi_{185}(86,·)$, $\chi_{185}(164,·)$, $\chi_{185}(144,·)$, $\chi_{185}(99,·)$, $\chi_{185}(101,·)$, $\chi_{185}(104,·)$, $\chi_{185}(174,·)$, $\chi_{185}(114,·)$, $\chi_{185}(169,·)$, $\chi_{185}(121,·)$, $\chi_{185}(149,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $\frac{1}{13886401539812541861565334433263835657093338236742302705687} a^{35} + \frac{2181185730114800308726401878425883303431579530438291140712}{13886401539812541861565334433263835657093338236742302705687} a^{34} + \frac{5721561089897440669601075013004627531573956032676073072535}{13886401539812541861565334433263835657093338236742302705687} a^{33} - \frac{1460003755409034002815779097277203566386836758802250077369}{13886401539812541861565334433263835657093338236742302705687} a^{32} - \frac{6882201112333404080301163772835271109781105011484943540989}{13886401539812541861565334433263835657093338236742302705687} a^{31} - \frac{6044301391038932181566019486161366419882051408548492558741}{13886401539812541861565334433263835657093338236742302705687} a^{30} - \frac{3723940432539540625187889249786691838780072143152911467613}{13886401539812541861565334433263835657093338236742302705687} a^{29} - \frac{5019133374706325358925592483959109056623806177623127826066}{13886401539812541861565334433263835657093338236742302705687} a^{28} + \frac{112519915031983801953824322636047788554769133359736123625}{13886401539812541861565334433263835657093338236742302705687} a^{27} + \frac{1672609239091802217087315025194196918048799522320457629693}{13886401539812541861565334433263835657093338236742302705687} a^{26} + \frac{1815657754267725313090613630532221859905853333379116188026}{13886401539812541861565334433263835657093338236742302705687} a^{25} + \frac{3679907901962411956798144343318070636838156949814177736540}{13886401539812541861565334433263835657093338236742302705687} a^{24} + \frac{6090529190879362203299364906980226138329170949053477484600}{13886401539812541861565334433263835657093338236742302705687} a^{23} + \frac{4715799611624400111278671922257475018627874266441419684126}{13886401539812541861565334433263835657093338236742302705687} a^{22} - \frac{5178261741991002489015072581518940072492085469094802056557}{13886401539812541861565334433263835657093338236742302705687} a^{21} + \frac{3833926942586797159488611261236026777570318650662488161334}{13886401539812541861565334433263835657093338236742302705687} a^{20} + \frac{646138642162059801737386564752873925462804503755508926100}{13886401539812541861565334433263835657093338236742302705687} a^{19} + \frac{2070845987800775078107360982473832068354248530268713391925}{13886401539812541861565334433263835657093338236742302705687} a^{18} + \frac{3615485625304069683940453948772513052719735036065464882194}{13886401539812541861565334433263835657093338236742302705687} a^{17} - \frac{5508327870738665650408770268991714313114830803768529334161}{13886401539812541861565334433263835657093338236742302705687} a^{16} + \frac{1111472866662268860831992264217775329395074252748348503949}{13886401539812541861565334433263835657093338236742302705687} a^{15} + \frac{3314797814198062622675671553490072089936985504839565678916}{13886401539812541861565334433263835657093338236742302705687} a^{14} + \frac{2082635525973232151732829783791140045214451407676608828761}{13886401539812541861565334433263835657093338236742302705687} a^{13} - \frac{2302533799436978464729647646899638025807681778319439301480}{13886401539812541861565334433263835657093338236742302705687} a^{12} - \frac{6880784228610786681856909745019551163233663878912347509648}{13886401539812541861565334433263835657093338236742302705687} a^{11} - \frac{3901806144208877153267902279140376627369410212907083577009}{13886401539812541861565334433263835657093338236742302705687} a^{10} + \frac{5153130928658622841286472221764428713514085089200757596322}{13886401539812541861565334433263835657093338236742302705687} a^{9} + \frac{1140421422245377869984362485074746795055128244067333200257}{13886401539812541861565334433263835657093338236742302705687} a^{8} - \frac{890659475003117147319782588390301441619671124298532507630}{13886401539812541861565334433263835657093338236742302705687} a^{7} - \frac{3652732980667335133597140706548999436687465825396925185715}{13886401539812541861565334433263835657093338236742302705687} a^{6} - \frac{3712596824170530520045489452374865330458577541947722573950}{13886401539812541861565334433263835657093338236742302705687} a^{5} - \frac{6428373482528634200866528105667272185647723773019467060762}{13886401539812541861565334433263835657093338236742302705687} a^{4} - \frac{3079791461676216205649856347906038951127779654903736406296}{13886401539812541861565334433263835657093338236742302705687} a^{3} - \frac{3503224656212104276955089159836766715790692641363052031180}{13886401539812541861565334433263835657093338236742302705687} a^{2} + \frac{3629667515305959887341602094356388791136253206263685039851}{13886401539812541861565334433263835657093338236742302705687} a + \frac{1435317828544409033669468733781742543523909318541468770404}{13886401539812541861565334433263835657093338236742302705687}$

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:  \( 2894219484422715000000 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{185}) \), \(\Q(\sqrt{37}) \), 3.3.1369.1, \(\Q(\sqrt{5}, \sqrt{37})\), 6.6.234270125.1, 6.6.8667994625.1, 6.6.69343957.1, 9.9.3512479453921.1, 12.12.75134130819028890625.1, 18.18.24096702957455403051316876953125.1, 18.18.891578009425849912898724447265625.1, \(\Q(\zeta_{37})^+\)

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 $18^{2}$ $18^{2}$ R $18^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$ $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{18}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{18}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$

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
5Data not computed
37Data not computed