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Properties

Label 36.0.78734006972...0000.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 5^{18}\cdot 19^{34}$
Root discriminant $72.15$
Ramified primes $2, 5, 19$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{18}$ (as 36T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3814697265625, 0, -762939453125, 0, 152587890625, 0, -30517578125, 0, 6103515625, 0, -1220703125, 0, 244140625, 0, -48828125, 0, 9765625, 0, -1953125, 0, 390625, 0, -78125, 0, 15625, 0, -3125, 0, 625, 0, -125, 0, 25, 0, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 5*x^34 + 25*x^32 - 125*x^30 + 625*x^28 - 3125*x^26 + 15625*x^24 - 78125*x^22 + 390625*x^20 - 1953125*x^18 + 9765625*x^16 - 48828125*x^14 + 244140625*x^12 - 1220703125*x^10 + 6103515625*x^8 - 30517578125*x^6 + 152587890625*x^4 - 762939453125*x^2 + 3814697265625)
 
gp: K = bnfinit(x^36 - 5*x^34 + 25*x^32 - 125*x^30 + 625*x^28 - 3125*x^26 + 15625*x^24 - 78125*x^22 + 390625*x^20 - 1953125*x^18 + 9765625*x^16 - 48828125*x^14 + 244140625*x^12 - 1220703125*x^10 + 6103515625*x^8 - 30517578125*x^6 + 152587890625*x^4 - 762939453125*x^2 + 3814697265625, 1)
 

Normalized defining polynomial

\( x^{36} - 5 x^{34} + 25 x^{32} - 125 x^{30} + 625 x^{28} - 3125 x^{26} + 15625 x^{24} - 78125 x^{22} + 390625 x^{20} - 1953125 x^{18} + 9765625 x^{16} - 48828125 x^{14} + 244140625 x^{12} - 1220703125 x^{10} + 6103515625 x^{8} - 30517578125 x^{6} + 152587890625 x^{4} - 762939453125 x^{2} + 3814697265625 \)

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:  \(7873400697252840110803083874402469866192691789824000000000000000000=2^{36}\cdot 5^{18}\cdot 19^{34}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.15$
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:  $2, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(380=2^{2}\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{380}(1,·)$, $\chi_{380}(259,·)$, $\chi_{380}(261,·)$, $\chi_{380}(139,·)$, $\chi_{380}(141,·)$, $\chi_{380}(21,·)$, $\chi_{380}(279,·)$, $\chi_{380}(281,·)$, $\chi_{380}(159,·)$, $\chi_{380}(161,·)$, $\chi_{380}(39,·)$, $\chi_{380}(41,·)$, $\chi_{380}(299,·)$, $\chi_{380}(301,·)$, $\chi_{380}(179,·)$, $\chi_{380}(181,·)$, $\chi_{380}(59,·)$, $\chi_{380}(61,·)$, $\chi_{380}(319,·)$, $\chi_{380}(321,·)$, $\chi_{380}(199,·)$, $\chi_{380}(201,·)$, $\chi_{380}(79,·)$, $\chi_{380}(81,·)$, $\chi_{380}(339,·)$, $\chi_{380}(341,·)$, $\chi_{380}(219,·)$, $\chi_{380}(221,·)$, $\chi_{380}(99,·)$, $\chi_{380}(101,·)$, $\chi_{380}(359,·)$, $\chi_{380}(239,·)$, $\chi_{380}(241,·)$, $\chi_{380}(119,·)$, $\chi_{380}(121,·)$, $\chi_{380}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{25} a^{4}$, $\frac{1}{25} a^{5}$, $\frac{1}{125} a^{6}$, $\frac{1}{125} a^{7}$, $\frac{1}{625} a^{8}$, $\frac{1}{625} a^{9}$, $\frac{1}{3125} a^{10}$, $\frac{1}{3125} a^{11}$, $\frac{1}{15625} a^{12}$, $\frac{1}{15625} a^{13}$, $\frac{1}{78125} a^{14}$, $\frac{1}{78125} a^{15}$, $\frac{1}{390625} a^{16}$, $\frac{1}{390625} a^{17}$, $\frac{1}{1953125} a^{18}$, $\frac{1}{1953125} a^{19}$, $\frac{1}{9765625} a^{20}$, $\frac{1}{9765625} a^{21}$, $\frac{1}{48828125} a^{22}$, $\frac{1}{48828125} a^{23}$, $\frac{1}{244140625} a^{24}$, $\frac{1}{244140625} a^{25}$, $\frac{1}{1220703125} a^{26}$, $\frac{1}{1220703125} a^{27}$, $\frac{1}{6103515625} a^{28}$, $\frac{1}{6103515625} a^{29}$, $\frac{1}{30517578125} a^{30}$, $\frac{1}{30517578125} a^{31}$, $\frac{1}{152587890625} a^{32}$, $\frac{1}{152587890625} a^{33}$, $\frac{1}{762939453125} a^{34}$, $\frac{1}{762939453125} a^{35}$

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:  \( \frac{1}{762939453125} a^{34} \) (order $38$)
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

$C_2\times C_{18}$ (as 36T2):

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 $C_2\times C_{18}$
Character table for $C_2\times C_{18}$ is not computed

Intermediate fields

\(\Q(\sqrt{-19}) \), \(\Q(\sqrt{95}) \), \(\Q(\sqrt{-5}) \), 3.3.361.1, \(\Q(\sqrt{-5}, \sqrt{-19})\), 6.0.2476099.1, 6.6.19808792000.1, 6.0.1042568000.1, \(\Q(\zeta_{19})^+\), 12.0.392388240499264000000.3, \(\Q(\zeta_{19})\), 18.18.2805958071185818719200768000000000.1, 18.0.147682003746622037852672000000000.1

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 $18^{2}$ R ${\href{/LocalNumberField/7.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{4}$ $18^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$ $18^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{4}$ $18^{2}$ $18^{2}$

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
2Data not computed
5Data not computed
19Data not computed