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Properties

Label 36.0.23001609434...6752.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 3^{54}\cdot 13^{33}$
Root discriminant $109.10$
Ramified primes $2, 3, 13$
Class number Not computed
Class group Not computed
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![1601613, 0, 33633873, 0, 269070984, 0, 1160101683, 0, 3105527607, 0, 5568808401, 0, 7002607950, 0, 6356268126, 0, 4241564028, 0, 2103342228, 0, 778931478, 0, 215382726, 0, 44226585, 0, 6665490, 0, 722475, 0, 54444, 0, 2691, 0, 78, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 78*x^34 + 2691*x^32 + 54444*x^30 + 722475*x^28 + 6665490*x^26 + 44226585*x^24 + 215382726*x^22 + 778931478*x^20 + 2103342228*x^18 + 4241564028*x^16 + 6356268126*x^14 + 7002607950*x^12 + 5568808401*x^10 + 3105527607*x^8 + 1160101683*x^6 + 269070984*x^4 + 33633873*x^2 + 1601613)
 
gp: K = bnfinit(x^36 + 78*x^34 + 2691*x^32 + 54444*x^30 + 722475*x^28 + 6665490*x^26 + 44226585*x^24 + 215382726*x^22 + 778931478*x^20 + 2103342228*x^18 + 4241564028*x^16 + 6356268126*x^14 + 7002607950*x^12 + 5568808401*x^10 + 3105527607*x^8 + 1160101683*x^6 + 269070984*x^4 + 33633873*x^2 + 1601613, 1)
 

Normalized defining polynomial

\( x^{36} + 78 x^{34} + 2691 x^{32} + 54444 x^{30} + 722475 x^{28} + 6665490 x^{26} + 44226585 x^{24} + 215382726 x^{22} + 778931478 x^{20} + 2103342228 x^{18} + 4241564028 x^{16} + 6356268126 x^{14} + 7002607950 x^{12} + 5568808401 x^{10} + 3105527607 x^{8} + 1160101683 x^{6} + 269070984 x^{4} + 33633873 x^{2} + 1601613 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(23001609434531037317435795350729577139835655058997834746673027296371146752=2^{36}\cdot 3^{54}\cdot 13^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $109.10$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(468=2^{2}\cdot 3^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{468}(1,·)$, $\chi_{468}(133,·)$, $\chi_{468}(11,·)$, $\chi_{468}(275,·)$, $\chi_{468}(277,·)$, $\chi_{468}(25,·)$, $\chi_{468}(431,·)$, $\chi_{468}(157,·)$, $\chi_{468}(445,·)$, $\chi_{468}(289,·)$, $\chi_{468}(167,·)$, $\chi_{468}(47,·)$, $\chi_{468}(433,·)$, $\chi_{468}(181,·)$, $\chi_{468}(313,·)$, $\chi_{468}(59,·)$, $\chi_{468}(61,·)$, $\chi_{468}(395,·)$, $\chi_{468}(49,·)$, $\chi_{468}(71,·)$, $\chi_{468}(203,·)$, $\chi_{468}(205,·)$, $\chi_{468}(337,·)$, $\chi_{468}(83,·)$, $\chi_{468}(215,·)$, $\chi_{468}(217,·)$, $\chi_{468}(227,·)$, $\chi_{468}(359,·)$, $\chi_{468}(361,·)$, $\chi_{468}(323,·)$, $\chi_{468}(239,·)$, $\chi_{468}(371,·)$, $\chi_{468}(373,·)$, $\chi_{468}(119,·)$, $\chi_{468}(121,·)$, $\chi_{468}(383,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{117} a^{12}$, $\frac{1}{117} a^{13}$, $\frac{1}{117} a^{14}$, $\frac{1}{117} a^{15}$, $\frac{1}{117} a^{16}$, $\frac{1}{117} a^{17}$, $\frac{1}{351} a^{18}$, $\frac{1}{351} a^{19}$, $\frac{1}{351} a^{20}$, $\frac{1}{351} a^{21}$, $\frac{1}{351} a^{22}$, $\frac{1}{351} a^{23}$, $\frac{1}{13689} a^{24}$, $\frac{1}{13689} a^{25}$, $\frac{1}{13689} a^{26}$, $\frac{1}{13689} a^{27}$, $\frac{1}{13689} a^{28}$, $\frac{1}{13689} a^{29}$, $\frac{1}{2176551} a^{30} + \frac{2}{241839} a^{28} - \frac{7}{241839} a^{26} + \frac{7}{241839} a^{24} - \frac{20}{18603} a^{22} + \frac{8}{18603} a^{20} - \frac{22}{18603} a^{18} - \frac{22}{6201} a^{16} - \frac{8}{2067} a^{14} - \frac{1}{2067} a^{12} + \frac{2}{53} a^{10} - \frac{5}{53} a^{8} - \frac{2}{159} a^{6} - \frac{25}{53} a^{4} + \frac{22}{53} a^{2} + \frac{15}{53}$, $\frac{1}{2176551} a^{31} + \frac{2}{241839} a^{29} - \frac{7}{241839} a^{27} + \frac{7}{241839} a^{25} - \frac{20}{18603} a^{23} + \frac{8}{18603} a^{21} - \frac{22}{18603} a^{19} - \frac{22}{6201} a^{17} - \frac{8}{2067} a^{15} - \frac{1}{2067} a^{13} + \frac{2}{53} a^{11} - \frac{5}{53} a^{9} - \frac{2}{159} a^{7} - \frac{25}{53} a^{5} + \frac{22}{53} a^{3} + \frac{15}{53} a$, $\frac{1}{2176551} a^{32} - \frac{23}{725517} a^{28} - \frac{25}{725517} a^{26} + \frac{8}{725517} a^{24} - \frac{1}{6201} a^{22} - \frac{7}{18603} a^{20} + \frac{4}{6201} a^{18} + \frac{1}{6201} a^{16} + \frac{5}{6201} a^{14} + \frac{23}{6201} a^{12} - \frac{17}{159} a^{10} + \frac{1}{53} a^{8} + \frac{14}{159} a^{6} - \frac{5}{53} a^{4} - \frac{10}{53} a^{2} - \frac{5}{53}$, $\frac{1}{2176551} a^{33} - \frac{23}{725517} a^{29} - \frac{25}{725517} a^{27} + \frac{8}{725517} a^{25} - \frac{1}{6201} a^{23} - \frac{7}{18603} a^{21} + \frac{4}{6201} a^{19} + \frac{1}{6201} a^{17} + \frac{5}{6201} a^{15} + \frac{23}{6201} a^{13} - \frac{17}{159} a^{11} + \frac{1}{53} a^{9} + \frac{14}{159} a^{7} - \frac{5}{53} a^{5} - \frac{10}{53} a^{3} - \frac{5}{53} a$, $\frac{1}{15351258188919159} a^{34} - \frac{13377029}{568565118108117} a^{32} + \frac{554442149}{15351258188919159} a^{30} - \frac{81126882622}{5117086062973053} a^{28} + \frac{16168186726}{5117086062973053} a^{26} - \frac{167079525316}{5117086062973053} a^{24} + \frac{52197210626}{43735778316009} a^{22} + \frac{47403238}{280957890681} a^{20} - \frac{14605635619}{14578592772003} a^{18} - \frac{171832933361}{43735778316009} a^{16} - \frac{47392751696}{43735778316009} a^{14} + \frac{132543827678}{43735778316009} a^{12} - \frac{24926738848}{1121430213231} a^{10} + \frac{30938181614}{1121430213231} a^{8} - \frac{34797986597}{373810071077} a^{6} - \frac{180627227693}{373810071077} a^{4} + \frac{40002298397}{373810071077} a^{2} + \frac{119885653926}{373810071077}$, $\frac{1}{15351258188919159} a^{35} - \frac{13377029}{568565118108117} a^{33} + \frac{554442149}{15351258188919159} a^{31} - \frac{81126882622}{5117086062973053} a^{29} + \frac{16168186726}{5117086062973053} a^{27} - \frac{167079525316}{5117086062973053} a^{25} + \frac{52197210626}{43735778316009} a^{23} + \frac{47403238}{280957890681} a^{21} - \frac{14605635619}{14578592772003} a^{19} - \frac{171832933361}{43735778316009} a^{17} - \frac{47392751696}{43735778316009} a^{15} + \frac{132543827678}{43735778316009} a^{13} - \frac{24926738848}{1121430213231} a^{11} + \frac{30938181614}{1121430213231} a^{9} - \frac{34797986597}{373810071077} a^{7} - \frac{180627227693}{373810071077} a^{5} + \frac{40002298397}{373810071077} a^{3} + \frac{119885653926}{373810071077} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $17$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{9})^+\), 3.3.13689.2, 3.3.13689.1, 3.3.169.1, 4.0.316368.2, 6.6.14414517.1, 6.6.2436053373.2, 6.6.2436053373.1, \(\Q(\zeta_{13})^+\), 9.9.2565164201769.1, 12.0.16828007762689447514112.1, 12.0.2843933311894516629884928.1, 12.0.2843933311894516629884928.2, 12.0.5351362262028177408.1, 18.18.14456408038335708501176406117.1

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 R ${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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}$
3Data not computed
13Data not computed