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Properties

Label 36.36.1339527222...8125.1
Degree $36$
Signature $[36, 0]$
Discriminant $3^{48}\cdot 5^{27}\cdot 7^{30}$
Root discriminant $73.22$
Ramified primes $3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times C_{12}$ (as 36T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 69, 1272, 1113, -108390, -118797, 3622138, -4534752, -29887359, 63028536, 88315116, -281930013, -78776304, 608220999, -79242273, -763692862, 246176712, 616092369, -259925101, -337769112, 159349053, 129922520, -63680853, -35601552, 17329413, 6959373, -3261144, -958293, 423339, 90237, -37141, -5496, 2100, 194, -69, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 3*x^35 - 69*x^34 + 194*x^33 + 2100*x^32 - 5496*x^31 - 37141*x^30 + 90237*x^29 + 423339*x^28 - 958293*x^27 - 3261144*x^26 + 6959373*x^25 + 17329413*x^24 - 35601552*x^23 - 63680853*x^22 + 129922520*x^21 + 159349053*x^20 - 337769112*x^19 - 259925101*x^18 + 616092369*x^17 + 246176712*x^16 - 763692862*x^15 - 79242273*x^14 + 608220999*x^13 - 78776304*x^12 - 281930013*x^11 + 88315116*x^10 + 63028536*x^9 - 29887359*x^8 - 4534752*x^7 + 3622138*x^6 - 118797*x^5 - 108390*x^4 + 1113*x^3 + 1272*x^2 + 69*x + 1)
 
gp: K = bnfinit(x^36 - 3*x^35 - 69*x^34 + 194*x^33 + 2100*x^32 - 5496*x^31 - 37141*x^30 + 90237*x^29 + 423339*x^28 - 958293*x^27 - 3261144*x^26 + 6959373*x^25 + 17329413*x^24 - 35601552*x^23 - 63680853*x^22 + 129922520*x^21 + 159349053*x^20 - 337769112*x^19 - 259925101*x^18 + 616092369*x^17 + 246176712*x^16 - 763692862*x^15 - 79242273*x^14 + 608220999*x^13 - 78776304*x^12 - 281930013*x^11 + 88315116*x^10 + 63028536*x^9 - 29887359*x^8 - 4534752*x^7 + 3622138*x^6 - 118797*x^5 - 108390*x^4 + 1113*x^3 + 1272*x^2 + 69*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 3 x^{35} - 69 x^{34} + 194 x^{33} + 2100 x^{32} - 5496 x^{31} - 37141 x^{30} + 90237 x^{29} + 423339 x^{28} - 958293 x^{27} - 3261144 x^{26} + 6959373 x^{25} + 17329413 x^{24} - 35601552 x^{23} - 63680853 x^{22} + 129922520 x^{21} + 159349053 x^{20} - 337769112 x^{19} - 259925101 x^{18} + 616092369 x^{17} + 246176712 x^{16} - 763692862 x^{15} - 79242273 x^{14} + 608220999 x^{13} - 78776304 x^{12} - 281930013 x^{11} + 88315116 x^{10} + 63028536 x^{9} - 29887359 x^{8} - 4534752 x^{7} + 3622138 x^{6} - 118797 x^{5} - 108390 x^{4} + 1113 x^{3} + 1272 x^{2} + 69 x + 1 \)

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:  \(13395272227285527394091512575412214059512641429074108600616455078125=3^{48}\cdot 5^{27}\cdot 7^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.22$
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$$\chi_{315}(256,·)$, $\chi_{315}(1,·)$, $\chi_{315}(4,·)$, $\chi_{315}(262,·)$, $\chi_{315}(13,·)$, $\chi_{315}(16,·)$, $\chi_{315}(274,·)$, $\chi_{315}(151,·)$, $\chi_{315}(283,·)$, $\chi_{315}(157,·)$, $\chi_{315}(289,·)$, $\chi_{315}(292,·)$, $\chi_{315}(169,·)$, $\chi_{315}(46,·)$, $\chi_{315}(178,·)$, $\chi_{315}(307,·)$, $\chi_{315}(52,·)$, $\chi_{315}(184,·)$, $\chi_{315}(313,·)$, $\chi_{315}(187,·)$, $\chi_{315}(64,·)$, $\chi_{315}(73,·)$, $\chi_{315}(202,·)$, $\chi_{315}(79,·)$, $\chi_{315}(208,·)$, $\chi_{315}(82,·)$, $\chi_{315}(211,·)$, $\chi_{315}(214,·)$, $\chi_{315}(223,·)$, $\chi_{315}(97,·)$, $\chi_{315}(226,·)$, $\chi_{315}(103,·)$, $\chi_{315}(106,·)$, $\chi_{315}(109,·)$, $\chi_{315}(118,·)$, $\chi_{315}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{71} a^{30} + \frac{23}{71} a^{29} + \frac{31}{71} a^{28} + \frac{7}{71} a^{27} - \frac{2}{71} a^{26} - \frac{15}{71} a^{25} - \frac{32}{71} a^{24} + \frac{1}{71} a^{23} - \frac{2}{71} a^{22} - \frac{9}{71} a^{21} + \frac{20}{71} a^{20} - \frac{32}{71} a^{19} - \frac{29}{71} a^{18} + \frac{3}{71} a^{17} - \frac{1}{71} a^{16} + \frac{22}{71} a^{15} + \frac{20}{71} a^{14} - \frac{17}{71} a^{13} + \frac{13}{71} a^{12} - \frac{31}{71} a^{11} + \frac{17}{71} a^{10} + \frac{3}{71} a^{9} + \frac{17}{71} a^{8} - \frac{14}{71} a^{7} + \frac{9}{71} a^{6} - \frac{11}{71} a^{5} - \frac{33}{71} a^{4} - \frac{34}{71} a^{3} - \frac{29}{71} a^{2} - \frac{17}{71} a - \frac{21}{71}$, $\frac{1}{71} a^{31} - \frac{1}{71} a^{29} + \frac{4}{71} a^{28} - \frac{21}{71} a^{27} + \frac{31}{71} a^{26} + \frac{29}{71} a^{25} + \frac{27}{71} a^{24} - \frac{25}{71} a^{23} - \frac{34}{71} a^{22} + \frac{14}{71} a^{21} + \frac{5}{71} a^{20} - \frac{3}{71} a^{19} + \frac{31}{71} a^{18} + \frac{1}{71} a^{17} - \frac{26}{71} a^{16} + \frac{11}{71} a^{15} + \frac{20}{71} a^{14} - \frac{22}{71} a^{13} + \frac{25}{71} a^{12} + \frac{20}{71} a^{11} - \frac{33}{71} a^{10} + \frac{19}{71} a^{9} + \frac{21}{71} a^{8} - \frac{24}{71} a^{7} - \frac{5}{71} a^{6} + \frac{7}{71} a^{5} + \frac{15}{71} a^{4} - \frac{28}{71} a^{3} + \frac{11}{71} a^{2} + \frac{15}{71} a - \frac{14}{71}$, $\frac{1}{71} a^{32} + \frac{27}{71} a^{29} + \frac{10}{71} a^{28} - \frac{33}{71} a^{27} + \frac{27}{71} a^{26} + \frac{12}{71} a^{25} + \frac{14}{71} a^{24} - \frac{33}{71} a^{23} + \frac{12}{71} a^{22} - \frac{4}{71} a^{21} + \frac{17}{71} a^{20} - \frac{1}{71} a^{19} - \frac{28}{71} a^{18} - \frac{23}{71} a^{17} + \frac{10}{71} a^{16} - \frac{29}{71} a^{15} - \frac{2}{71} a^{14} + \frac{8}{71} a^{13} + \frac{33}{71} a^{12} + \frac{7}{71} a^{11} - \frac{35}{71} a^{10} + \frac{24}{71} a^{9} - \frac{7}{71} a^{8} - \frac{19}{71} a^{7} + \frac{16}{71} a^{6} + \frac{4}{71} a^{5} + \frac{10}{71} a^{4} - \frac{23}{71} a^{3} - \frac{14}{71} a^{2} - \frac{31}{71} a - \frac{21}{71}$, $\frac{1}{71} a^{33} + \frac{28}{71} a^{29} - \frac{18}{71} a^{28} - \frac{20}{71} a^{27} - \frac{5}{71} a^{26} - \frac{7}{71} a^{25} - \frac{21}{71} a^{24} - \frac{15}{71} a^{23} - \frac{21}{71} a^{22} - \frac{24}{71} a^{21} + \frac{27}{71} a^{20} - \frac{16}{71} a^{19} - \frac{21}{71} a^{18} - \frac{2}{71} a^{16} - \frac{28}{71} a^{15} - \frac{35}{71} a^{14} - \frac{5}{71} a^{13} + \frac{11}{71} a^{12} + \frac{21}{71} a^{11} - \frac{9}{71} a^{10} - \frac{17}{71} a^{9} + \frac{19}{71} a^{8} - \frac{32}{71} a^{7} - \frac{26}{71} a^{6} + \frac{23}{71} a^{5} + \frac{16}{71} a^{4} - \frac{19}{71} a^{3} - \frac{29}{71} a^{2} + \frac{12}{71} a - \frac{1}{71}$, $\frac{1}{71} a^{34} - \frac{23}{71} a^{29} + \frac{35}{71} a^{28} + \frac{12}{71} a^{27} - \frac{22}{71} a^{26} - \frac{27}{71} a^{25} + \frac{29}{71} a^{24} + \frac{22}{71} a^{23} + \frac{32}{71} a^{22} - \frac{5}{71} a^{21} - \frac{8}{71} a^{20} + \frac{23}{71} a^{19} + \frac{31}{71} a^{18} - \frac{15}{71} a^{17} - \frac{12}{71} a^{15} + \frac{3}{71} a^{14} - \frac{10}{71} a^{13} + \frac{12}{71} a^{12} + \frac{7}{71} a^{11} + \frac{4}{71} a^{10} + \frac{6}{71} a^{9} - \frac{11}{71} a^{8} + \frac{11}{71} a^{7} - \frac{16}{71} a^{6} - \frac{31}{71} a^{5} - \frac{18}{71} a^{4} - \frac{28}{71} a^{2} - \frac{22}{71} a + \frac{20}{71}$, $\frac{1}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{35} + \frac{75776606117635529788007831520524297878661266615829257811243857958789751708830144900300542715}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{34} + \frac{69507435226125531280587744445181457516799483182196047456422022547540723979117967558556156771}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{33} + \frac{109262810288747558006504136411945084416942407053066224183641275262153902425213143609166557084}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{32} - \frac{128208133512226981887783171407642193465506469236906216593575265326391815358109083157887392268}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{31} + \frac{35375887280236786212219253009764757359310594491056963032588543739968889362349491243298091696}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{30} - \frac{7789860381177172379317116676601618345935130062662380872021871585102922513542943904396221057722}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{29} - \frac{3175706255919369903263619682285297200317357132728023261153788007328656452008937238491610200366}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{28} + \frac{7260686838892839183469706599830953656658513799405148206384590088452408330106835788828816442752}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{27} + \frac{1930694025150104136489652943818933968341221735570723382061328468479807206080716769322164915614}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{26} + \frac{8333967374669127828467264020546000504796632489523277117500527553598191574383615980795819762834}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{25} + \frac{5220885261485565096923018884921273200693322685642576242724021415714718158173181347609983530860}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{24} + \frac{253906231416364135008145822889427177818264415656278136697538213427613205074945847096606592127}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{23} - \frac{6411625404913825385339119993094555789312834120849064369493705270231933523933303451203342496868}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{22} + \frac{8930742543029535435587186321699178438374384372915726572685204447259460579533514148972016912607}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{21} + \frac{5615019113093563908173690794571171367579913908637106641883335174607582459341171166199740534549}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{20} + \frac{7109409991167278126022079327040871412605281599549259820124002401298534814641440369991043135866}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{19} - \frac{7737759739232456998349568067683134929574684336827042523607777502003092575829778742973176572670}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{18} + \frac{1601256272412231684353705970445547944875139981328099503667998829624925013370531359554571324824}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{17} + \frac{5154404435041166593519329828462549054707889875721138759559577893478645877141461674898193544154}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{16} + \frac{1309421759385068513095261627535855363400834402207820624295509393968496500888001043291361215422}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{15} + \frac{5276225250393292883630389267268778700876266448338359023074301776863214187434034406087972462854}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{14} - \frac{6157359692203157028119222011184741248543419994144834942594648586035418198897860431191575414409}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{13} - \frac{7271211207394576315274150236759583837295117956666467465458312458113939300780232815387656692434}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{12} + \frac{8897503016521167178903081417871528186684770071008456301801907241165841271780609317710106848136}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{11} - \frac{4267355246861668290855906230105995888139596346832516882127884229738314138448026265699115806090}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{10} + \frac{244182733284682496584544939405332787305568654427608578025206314287243950764679131868197042179}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{9} + \frac{2510995797589750226131999166705180730615735671149988277155953561737899578707953283441262822957}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{8} + \frac{4535088744137318939728228705081678305171880076121588186838779571012545062496829692023598357711}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{7} - \frac{2652502810304111459956935638289619758653313705418901599038187584074502009539456921967562799419}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{6} + \frac{2089083681911036972628757317800597548691947758538502672342056929636588983047933864258375609770}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{5} + \frac{6309345415340937211212187110925117664289004250287889927433021849125896121472603549111915887710}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{4} - \frac{3920237716536579849746720600117828358919083452140597782676780275457507176505750581992665860133}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{3} - \frac{3289577172904634963832717809051368649981103859347268904765670431952591524998358189531237190429}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a^{2} - \frac{3560827921178537443917495164900818783711589967128980040523632876135555198289017254170918770036}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629} a - \frac{5563331479952183823228764476508581170067522387086446861524086153179228583902808424629507426994}{18354267156144129536093465374130701007825995491530700119103986070973727579814673100657612388629}$

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:  \( 16003335917605122000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_{12}$ (as 36T3):

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 $C_3\times C_{12}$
Character table for $C_3\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.2, 3.3.3969.1, 4.4.6125.1, 6.6.820125.1, 6.6.300125.1, 6.6.1969120125.2, 6.6.1969120125.1, 9.9.62523502209.1, 12.12.9891413435408203125.1, \(\Q(\zeta_{35})^+\), 12.12.23749283658415095703125.2, 12.12.23749283658415095703125.1, 18.18.7635133454060210702501953125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ R R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/13.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{12}$

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$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
7Data not computed