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Properties

Label 36.0.23610692285...6704.2
Degree $36$
Signature $[0, 18]$
Discriminant $2^{54}\cdot 3^{54}\cdot 7^{30}$
Root discriminant $74.38$
Ramified primes $2, 3, 7$
Class number Not computed
Class group Not computed
Galois group $C_6^2$ (as 36T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![262144, 0, 56623104, 0, 804519936, 0, 4012965888, 0, 10498277376, 0, 16929153024, 0, 18332004352, 0, 14038161408, 0, 7858649088, 0, 3285680128, 0, 1039332096, 0, 250210560, 0, 45809920, 0, 6327072, 0, 647760, 0, 47624, 0, 2376, 0, 72, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 72*x^34 + 2376*x^32 + 47624*x^30 + 647760*x^28 + 6327072*x^26 + 45809920*x^24 + 250210560*x^22 + 1039332096*x^20 + 3285680128*x^18 + 7858649088*x^16 + 14038161408*x^14 + 18332004352*x^12 + 16929153024*x^10 + 10498277376*x^8 + 4012965888*x^6 + 804519936*x^4 + 56623104*x^2 + 262144)
 
gp: K = bnfinit(x^36 + 72*x^34 + 2376*x^32 + 47624*x^30 + 647760*x^28 + 6327072*x^26 + 45809920*x^24 + 250210560*x^22 + 1039332096*x^20 + 3285680128*x^18 + 7858649088*x^16 + 14038161408*x^14 + 18332004352*x^12 + 16929153024*x^10 + 10498277376*x^8 + 4012965888*x^6 + 804519936*x^4 + 56623104*x^2 + 262144, 1)
 

Normalized defining polynomial

\( x^{36} + 72 x^{34} + 2376 x^{32} + 47624 x^{30} + 647760 x^{28} + 6327072 x^{26} + 45809920 x^{24} + 250210560 x^{22} + 1039332096 x^{20} + 3285680128 x^{18} + 7858649088 x^{16} + 14038161408 x^{14} + 18332004352 x^{12} + 16929153024 x^{10} + 10498277376 x^{8} + 4012965888 x^{6} + 804519936 x^{4} + 56623104 x^{2} + 262144 \)

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:  \(23610692285332399309092778573219694177406932269512586359440570056704=2^{54}\cdot 3^{54}\cdot 7^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.38$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(504=2^{3}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{504}(1,·)$, $\chi_{504}(131,·)$, $\chi_{504}(257,·)$, $\chi_{504}(395,·)$, $\chi_{504}(17,·)$, $\chi_{504}(403,·)$, $\chi_{504}(25,·)$, $\chi_{504}(289,·)$, $\chi_{504}(163,·)$, $\chi_{504}(41,·)$, $\chi_{504}(299,·)$, $\chi_{504}(185,·)$, $\chi_{504}(59,·)$, $\chi_{504}(193,·)$, $\chi_{504}(83,·)$, $\chi_{504}(67,·)$, $\chi_{504}(211,·)$, $\chi_{504}(457,·)$, $\chi_{504}(331,·)$, $\chi_{504}(467,·)$, $\chi_{504}(337,·)$, $\chi_{504}(419,·)$, $\chi_{504}(43,·)$, $\chi_{504}(377,·)$, $\chi_{504}(89,·)$, $\chi_{504}(353,·)$, $\chi_{504}(227,·)$, $\chi_{504}(425,·)$, $\chi_{504}(209,·)$, $\chi_{504}(379,·)$, $\chi_{504}(361,·)$, $\chi_{504}(235,·)$, $\chi_{504}(499,·)$, $\chi_{504}(169,·)$, $\chi_{504}(121,·)$, $\chi_{504}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{4096} a^{24}$, $\frac{1}{4096} a^{25}$, $\frac{1}{8192} a^{26}$, $\frac{1}{8192} a^{27}$, $\frac{1}{16384} a^{28}$, $\frac{1}{16384} a^{29}$, $\frac{1}{32768} a^{30}$, $\frac{1}{32768} a^{31}$, $\frac{1}{65536} a^{32}$, $\frac{1}{65536} a^{33}$, $\frac{1}{131072} a^{34}$, $\frac{1}{131072} a^{35}$

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

$C_6^2$ (as 36T4):

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 $C_6^2$
Character table for $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\sqrt{-2}, \sqrt{21})\), 6.0.3359232.1, 6.0.1229312.1, 6.0.8065516032.1, 6.0.8065516032.2, 6.0.3456649728.1, 6.6.6751269.1, 6.0.232339968.1, \(\Q(\zeta_{21})^+\), 6.0.169375836672.2, 6.6.330812181.2, 6.0.169375836672.1, 6.6.330812181.1, 9.9.62523502209.1, 12.0.11948427342082473984.6, 12.0.53981860730241024.4, 12.0.28688174048340020035584.1, 12.0.28688174048340020035584.2, 18.0.524682375772545974113841184768.4, 18.0.4859083482029548266268283212136448.1, \(\Q(\zeta_{63})^+\)

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{12}$

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
$2$2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
3Data not computed
7Data not computed