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Properties

Label 36.36.1297393824...3597.1
Degree $36$
Signature $[36, 0]$
Discriminant $7^{30}\cdot 13^{33}$
Root discriminant $53.13$
Ramified primes $7, 13$
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![1, 24, -216, -1922, 6018, 32327, -73571, -230058, 456052, 908896, -1655004, -2278575, 3879994, 3907332, -6263633, -4795428, 7259749, 4331906, -6208136, -2928983, 3983738, 1494700, -1934944, -576199, 712180, 166634, -197288, -35554, 40454, 5426, -5952, -560, 594, 35, -36, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 36*x^34 + 35*x^33 + 594*x^32 - 560*x^31 - 5952*x^30 + 5426*x^29 + 40454*x^28 - 35554*x^27 - 197288*x^26 + 166634*x^25 + 712180*x^24 - 576199*x^23 - 1934944*x^22 + 1494700*x^21 + 3983738*x^20 - 2928983*x^19 - 6208136*x^18 + 4331906*x^17 + 7259749*x^16 - 4795428*x^15 - 6263633*x^14 + 3907332*x^13 + 3879994*x^12 - 2278575*x^11 - 1655004*x^10 + 908896*x^9 + 456052*x^8 - 230058*x^7 - 73571*x^6 + 32327*x^5 + 6018*x^4 - 1922*x^3 - 216*x^2 + 24*x + 1)
 
gp: K = bnfinit(x^36 - x^35 - 36*x^34 + 35*x^33 + 594*x^32 - 560*x^31 - 5952*x^30 + 5426*x^29 + 40454*x^28 - 35554*x^27 - 197288*x^26 + 166634*x^25 + 712180*x^24 - 576199*x^23 - 1934944*x^22 + 1494700*x^21 + 3983738*x^20 - 2928983*x^19 - 6208136*x^18 + 4331906*x^17 + 7259749*x^16 - 4795428*x^15 - 6263633*x^14 + 3907332*x^13 + 3879994*x^12 - 2278575*x^11 - 1655004*x^10 + 908896*x^9 + 456052*x^8 - 230058*x^7 - 73571*x^6 + 32327*x^5 + 6018*x^4 - 1922*x^3 - 216*x^2 + 24*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 36 x^{34} + 35 x^{33} + 594 x^{32} - 560 x^{31} - 5952 x^{30} + 5426 x^{29} + 40454 x^{28} - 35554 x^{27} - 197288 x^{26} + 166634 x^{25} + 712180 x^{24} - 576199 x^{23} - 1934944 x^{22} + 1494700 x^{21} + 3983738 x^{20} - 2928983 x^{19} - 6208136 x^{18} + 4331906 x^{17} + 7259749 x^{16} - 4795428 x^{15} - 6263633 x^{14} + 3907332 x^{13} + 3879994 x^{12} - 2278575 x^{11} - 1655004 x^{10} + 908896 x^{9} + 456052 x^{8} - 230058 x^{7} - 73571 x^{6} + 32327 x^{5} + 6018 x^{4} - 1922 x^{3} - 216 x^{2} + 24 x + 1 \)

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:  \(129739382499069208406967320777552926214641189868504834496493597=7^{30}\cdot 13^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.13$
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:  $7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(91=7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{91}(1,·)$, $\chi_{91}(4,·)$, $\chi_{91}(5,·)$, $\chi_{91}(6,·)$, $\chi_{91}(9,·)$, $\chi_{91}(16,·)$, $\chi_{91}(19,·)$, $\chi_{91}(20,·)$, $\chi_{91}(22,·)$, $\chi_{91}(23,·)$, $\chi_{91}(24,·)$, $\chi_{91}(25,·)$, $\chi_{91}(29,·)$, $\chi_{91}(30,·)$, $\chi_{91}(31,·)$, $\chi_{91}(33,·)$, $\chi_{91}(34,·)$, $\chi_{91}(36,·)$, $\chi_{91}(41,·)$, $\chi_{91}(43,·)$, $\chi_{91}(45,·)$, $\chi_{91}(47,·)$, $\chi_{91}(51,·)$, $\chi_{91}(53,·)$, $\chi_{91}(54,·)$, $\chi_{91}(59,·)$, $\chi_{91}(64,·)$, $\chi_{91}(73,·)$, $\chi_{91}(74,·)$, $\chi_{91}(76,·)$, $\chi_{91}(79,·)$, $\chi_{91}(80,·)$, $\chi_{91}(81,·)$, $\chi_{91}(83,·)$, $\chi_{91}(88,·)$, $\chi_{91}(89,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$

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:  \( 48664971049465540000 \) (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, 3.3.8281.1, 3.3.8281.2, \(\Q(\zeta_{7})^+\), 4.4.107653.1, \(\Q(\zeta_{13})^+\), 6.6.891474493.1, 6.6.891474493.2, 6.6.5274997.1, 9.9.567869252041.1, 12.12.210845878198059013.1, 12.12.506240953553539690213.2, 12.12.506240953553539690213.1, 12.12.2995508600908518877.1, 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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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
7Data not computed
$13$13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$