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Properties

Label 36.36.7578154066...1808.1
Degree $36$
Signature $[36, 0]$
Discriminant $2^{36}\cdot 7^{24}\cdot 13^{33}$
Root discriminant $76.83$
Ramified primes $2, 7, 13$
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![2197, 0, -92274, 0, 1337973, 0, -9967789, 0, 44665010, 0, -130409526, 0, 259562875, 0, -361633805, 0, 358202936, 0, -254526168, 0, 130412230, 0, -48291581, 0, 12905893, 0, -2472002, 0, 334113, 0, -30940, 0, 1859, 0, -65, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 65*x^34 + 1859*x^32 - 30940*x^30 + 334113*x^28 - 2472002*x^26 + 12905893*x^24 - 48291581*x^22 + 130412230*x^20 - 254526168*x^18 + 358202936*x^16 - 361633805*x^14 + 259562875*x^12 - 130409526*x^10 + 44665010*x^8 - 9967789*x^6 + 1337973*x^4 - 92274*x^2 + 2197)
 
gp: K = bnfinit(x^36 - 65*x^34 + 1859*x^32 - 30940*x^30 + 334113*x^28 - 2472002*x^26 + 12905893*x^24 - 48291581*x^22 + 130412230*x^20 - 254526168*x^18 + 358202936*x^16 - 361633805*x^14 + 259562875*x^12 - 130409526*x^10 + 44665010*x^8 - 9967789*x^6 + 1337973*x^4 - 92274*x^2 + 2197, 1)
 

Normalized defining polynomial

\( x^{36} - 65 x^{34} + 1859 x^{32} - 30940 x^{30} + 334113 x^{28} - 2472002 x^{26} + 12905893 x^{24} - 48291581 x^{22} + 130412230 x^{20} - 254526168 x^{18} + 358202936 x^{16} - 361633805 x^{14} + 259562875 x^{12} - 130409526 x^{10} + 44665010 x^{8} - 9967789 x^{6} + 1337973 x^{4} - 92274 x^{2} + 2197 \)

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:  \(75781540662375302882102919875947077697351951905720209039476085751808=2^{36}\cdot 7^{24}\cdot 13^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.83$
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:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(364=2^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{364}(1,·)$, $\chi_{364}(261,·)$, $\chi_{364}(135,·)$, $\chi_{364}(9,·)$, $\chi_{364}(11,·)$, $\chi_{364}(15,·)$, $\chi_{364}(275,·)$, $\chi_{364}(277,·)$, $\chi_{364}(151,·)$, $\chi_{364}(25,·)$, $\chi_{364}(29,·)$, $\chi_{364}(289,·)$, $\chi_{364}(291,·)$, $\chi_{364}(165,·)$, $\chi_{364}(309,·)$, $\chi_{364}(53,·)$, $\chi_{364}(67,·)$, $\chi_{364}(71,·)$, $\chi_{364}(331,·)$, $\chi_{364}(205,·)$, $\chi_{364}(337,·)$, $\chi_{364}(163,·)$, $\chi_{364}(267,·)$, $\chi_{364}(219,·)$, $\chi_{364}(225,·)$, $\chi_{364}(99,·)$, $\chi_{364}(81,·)$, $\chi_{364}(361,·)$, $\chi_{364}(323,·)$, $\chi_{364}(359,·)$, $\chi_{364}(239,·)$, $\chi_{364}(113,·)$, $\chi_{364}(123,·)$, $\chi_{364}(233,·)$, $\chi_{364}(121,·)$, $\chi_{364}(319,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12}$, $\frac{1}{13} a^{13}$, $\frac{1}{13} a^{14}$, $\frac{1}{13} a^{15}$, $\frac{1}{13} a^{16}$, $\frac{1}{13} a^{17}$, $\frac{1}{13} a^{18}$, $\frac{1}{13} a^{19}$, $\frac{1}{39} a^{20} + \frac{1}{39} a^{18} + \frac{1}{39} a^{16} - \frac{1}{39} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{39} a^{21} + \frac{1}{39} a^{19} + \frac{1}{39} a^{17} - \frac{1}{39} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{507} a^{22} - \frac{1}{39} a^{16} - \frac{1}{39} a^{14} - \frac{1}{39} a^{12} - \frac{4}{13} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{507} a^{23} - \frac{1}{39} a^{17} - \frac{1}{39} a^{15} - \frac{1}{39} a^{13} - \frac{4}{13} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{507} a^{24} - \frac{1}{39} a^{18} - \frac{1}{39} a^{16} - \frac{1}{39} a^{14} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{507} a^{25} - \frac{1}{39} a^{19} - \frac{1}{39} a^{17} - \frac{1}{39} a^{15} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{507} a^{26} + \frac{1}{3}$, $\frac{1}{507} a^{27} + \frac{1}{3} a$, $\frac{1}{507} a^{28} + \frac{1}{3} a^{2}$, $\frac{1}{507} a^{29} + \frac{1}{3} a^{3}$, $\frac{1}{507} a^{30} + \frac{1}{3} a^{4}$, $\frac{1}{507} a^{31} + \frac{1}{3} a^{5}$, $\frac{1}{6591} a^{32} + \frac{2}{507} a^{20} - \frac{1}{39} a^{18} - \frac{1}{39} a^{16} + \frac{1}{39} a^{12} - \frac{1}{3} a^{10} + \frac{14}{39} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{6591} a^{33} + \frac{2}{507} a^{21} - \frac{1}{39} a^{19} - \frac{1}{39} a^{17} + \frac{1}{39} a^{13} - \frac{1}{3} a^{11} + \frac{14}{39} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{57004153970267099349} a^{34} + \frac{835376656865344}{19001384656755699783} a^{32} - \frac{1975901377215895}{4384934920789776873} a^{30} - \frac{66987409448248}{112434228738199407} a^{28} - \frac{214087842904330}{1461644973596592291} a^{26} - \frac{521637265503872}{4384934920789776873} a^{24} + \frac{297244727940534}{487214991198864097} a^{22} - \frac{28461665252057321}{4384934920789776873} a^{20} + \frac{1333865769938608}{112434228738199407} a^{18} + \frac{1603013925018998}{112434228738199407} a^{16} - \frac{4172190872434261}{337302686214598221} a^{14} + \frac{2015926383968474}{337302686214598221} a^{12} + \frac{133704245389143557}{337302686214598221} a^{10} - \frac{103359272919070271}{337302686214598221} a^{8} - \frac{486020040748208}{25946360478046017} a^{6} + \frac{26792179590853}{143350057889757} a^{4} - \frac{5384586204151294}{25946360478046017} a^{2} + \frac{5932909379668054}{25946360478046017}$, $\frac{1}{57004153970267099349} a^{35} + \frac{835376656865344}{19001384656755699783} a^{33} - \frac{1975901377215895}{4384934920789776873} a^{31} - \frac{66987409448248}{112434228738199407} a^{29} - \frac{214087842904330}{1461644973596592291} a^{27} - \frac{521637265503872}{4384934920789776873} a^{25} + \frac{297244727940534}{487214991198864097} a^{23} - \frac{28461665252057321}{4384934920789776873} a^{21} + \frac{1333865769938608}{112434228738199407} a^{19} + \frac{1603013925018998}{112434228738199407} a^{17} - \frac{4172190872434261}{337302686214598221} a^{15} + \frac{2015926383968474}{337302686214598221} a^{13} + \frac{133704245389143557}{337302686214598221} a^{11} - \frac{103359272919070271}{337302686214598221} a^{9} - \frac{486020040748208}{25946360478046017} a^{7} + \frac{26792179590853}{143350057889757} a^{5} - \frac{5384586204151294}{25946360478046017} a^{3} + \frac{5932909379668054}{25946360478046017} a$

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:  \( 31901682379687710000000 \) (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

\(\Q(\sqrt{13}) \), 3.3.169.1, 3.3.8281.2, \(\Q(\zeta_{7})^+\), 3.3.8281.1, 4.4.35152.1, \(\Q(\zeta_{13})^+\), 6.6.891474493.2, 6.6.5274997.1, 6.6.891474493.1, 9.9.567869252041.1, \(\Q(\zeta_{52})^+\), 12.12.42317611137863236145152.2, 12.12.250400065904516190208.1, 12.12.42317611137863236145152.1, 18.18.708478645847689707516501157.1

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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
$2$2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
7Data not computed
$13$13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$