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Properties

Label 36.36.7873400697...0000.1
Degree $36$
Signature $[36, 0]$
Discriminant $2^{36}\cdot 5^{18}\cdot 19^{34}$
Root discriminant $72.15$
Ramified primes $2, 5, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![361, 0, -16245, 0, 248007, 0, -1929906, 0, 9002257, 0, -27413979, 0, 57368315, 0, -85215855, 0, 91675950, 0, -72233250, 0, 41855195, 0, -17801385, 0, 5515738, 0, -1228749, 0, 192812, 0, -20634, 0, 1425, 0, -57, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 57*x^34 + 1425*x^32 - 20634*x^30 + 192812*x^28 - 1228749*x^26 + 5515738*x^24 - 17801385*x^22 + 41855195*x^20 - 72233250*x^18 + 91675950*x^16 - 85215855*x^14 + 57368315*x^12 - 27413979*x^10 + 9002257*x^8 - 1929906*x^6 + 248007*x^4 - 16245*x^2 + 361)
 
gp: K = bnfinit(x^36 - 57*x^34 + 1425*x^32 - 20634*x^30 + 192812*x^28 - 1228749*x^26 + 5515738*x^24 - 17801385*x^22 + 41855195*x^20 - 72233250*x^18 + 91675950*x^16 - 85215855*x^14 + 57368315*x^12 - 27413979*x^10 + 9002257*x^8 - 1929906*x^6 + 248007*x^4 - 16245*x^2 + 361, 1)
 

Normalized defining polynomial

\( x^{36} - 57 x^{34} + 1425 x^{32} - 20634 x^{30} + 192812 x^{28} - 1228749 x^{26} + 5515738 x^{24} - 17801385 x^{22} + 41855195 x^{20} - 72233250 x^{18} + 91675950 x^{16} - 85215855 x^{14} + 57368315 x^{12} - 27413979 x^{10} + 9002257 x^{8} - 1929906 x^{6} + 248007 x^{4} - 16245 x^{2} + 361 \)

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:  \(7873400697252840110803083874402469866192691789824000000000000000000=2^{36}\cdot 5^{18}\cdot 19^{34}\)
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$$\chi_{380}(1,·)$, $\chi_{380}(259,·)$, $\chi_{380}(51,·)$, $\chi_{380}(9,·)$, $\chi_{380}(279,·)$, $\chi_{380}(149,·)$, $\chi_{380}(151,·)$, $\chi_{380}(31,·)$, $\chi_{380}(289,·)$, $\chi_{380}(91,·)$, $\chi_{380}(169,·)$, $\chi_{380}(299,·)$, $\chi_{380}(301,·)$, $\chi_{380}(49,·)$, $\chi_{380}(179,·)$, $\chi_{380}(309,·)$, $\chi_{380}(329,·)$, $\chi_{380}(59,·)$, $\chi_{380}(61,·)$, $\chi_{380}(319,·)$, $\chi_{380}(321,·)$, $\chi_{380}(71,·)$, $\chi_{380}(201,·)$, $\chi_{380}(331,·)$, $\chi_{380}(79,·)$, $\chi_{380}(81,·)$, $\chi_{380}(211,·)$, $\chi_{380}(219,·)$, $\chi_{380}(349,·)$, $\chi_{380}(101,·)$, $\chi_{380}(229,·)$, $\chi_{380}(231,·)$, $\chi_{380}(161,·)$, $\chi_{380}(371,·)$, $\chi_{380}(121,·)$, $\chi_{380}(379,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19} a^{18}$, $\frac{1}{19} a^{19}$, $\frac{1}{19} a^{20}$, $\frac{1}{19} a^{21}$, $\frac{1}{19} a^{22}$, $\frac{1}{19} a^{23}$, $\frac{1}{19} a^{24}$, $\frac{1}{19} a^{25}$, $\frac{1}{19} a^{26}$, $\frac{1}{19} a^{27}$, $\frac{1}{19} a^{28}$, $\frac{1}{19} a^{29}$, $\frac{1}{19} a^{30}$, $\frac{1}{19} a^{31}$, $\frac{1}{19} a^{32}$, $\frac{1}{19} a^{33}$, $\frac{1}{1133457480649920669921619} a^{34} - \frac{1129833074843767550910}{59655656876311614206401} a^{32} - \frac{584420845311264169230}{59655656876311614206401} a^{30} + \frac{516536903470855556280}{59655656876311614206401} a^{28} + \frac{426485966142478410406}{59655656876311614206401} a^{26} - \frac{1089644795199465390648}{59655656876311614206401} a^{24} + \frac{1212202568178481045603}{59655656876311614206401} a^{22} + \frac{611824504041416485143}{59655656876311614206401} a^{20} + \frac{1497818353771826636807}{59655656876311614206401} a^{18} + \frac{20704750119749466256}{59655656876311614206401} a^{16} + \frac{75036984028491332423}{3139771414542716537179} a^{14} + \frac{1065171716273097185112}{3139771414542716537179} a^{12} + \frac{1288958628036690657545}{3139771414542716537179} a^{10} - \frac{207916354302602213365}{3139771414542716537179} a^{8} - \frac{709837311936048233360}{3139771414542716537179} a^{6} - \frac{389131999125917862248}{3139771414542716537179} a^{4} + \frac{676390076979427056924}{3139771414542716537179} a^{2} - \frac{483349771766279855467}{3139771414542716537179}$, $\frac{1}{1133457480649920669921619} a^{35} - \frac{1129833074843767550910}{59655656876311614206401} a^{33} - \frac{584420845311264169230}{59655656876311614206401} a^{31} + \frac{516536903470855556280}{59655656876311614206401} a^{29} + \frac{426485966142478410406}{59655656876311614206401} a^{27} - \frac{1089644795199465390648}{59655656876311614206401} a^{25} + \frac{1212202568178481045603}{59655656876311614206401} a^{23} + \frac{611824504041416485143}{59655656876311614206401} a^{21} + \frac{1497818353771826636807}{59655656876311614206401} a^{19} + \frac{20704750119749466256}{59655656876311614206401} a^{17} + \frac{75036984028491332423}{3139771414542716537179} a^{15} + \frac{1065171716273097185112}{3139771414542716537179} a^{13} + \frac{1288958628036690657545}{3139771414542716537179} a^{11} - \frac{207916354302602213365}{3139771414542716537179} a^{9} - \frac{709837311936048233360}{3139771414542716537179} a^{7} - \frac{389131999125917862248}{3139771414542716537179} a^{5} + \frac{676390076979427056924}{3139771414542716537179} a^{3} - \frac{483349771766279855467}{3139771414542716537179} a$

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:  \( 12611979161433870000000 \) (assuming GRH)
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, \(\Q(\sqrt{5}, \sqrt{19})\), 6.6.158470336.1, 6.6.19808792000.1, 6.6.16290125.1, \(\Q(\zeta_{19})^+\), 12.12.392388240499264000000.1, \(\Q(\zeta_{76})^+\), 18.18.2805958071185818719200768000000000.1, 18.18.563362135874260093126953125.1

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ R $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$ $18^{2}$ $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{4}$

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