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Properties

Label 16.0.37319027463...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 13^{12}$
Root discriminant $39.65$
Ramified primes $3, 5, 13$
Class number $52$ (GRH)
Class group $[2, 26]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -243, 1026, -3996, 15804, 15093, 13659, 10428, 8500, 3152, 1563, 559, 170, -32, 12, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 12*x^14 - 32*x^13 + 170*x^12 + 559*x^11 + 1563*x^10 + 3152*x^9 + 8500*x^8 + 10428*x^7 + 13659*x^6 + 15093*x^5 + 15804*x^4 - 3996*x^3 + 1026*x^2 - 243*x + 81)
 
gp: K = bnfinit(x^16 - x^15 + 12*x^14 - 32*x^13 + 170*x^12 + 559*x^11 + 1563*x^10 + 3152*x^9 + 8500*x^8 + 10428*x^7 + 13659*x^6 + 15093*x^5 + 15804*x^4 - 3996*x^3 + 1026*x^2 - 243*x + 81, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 12 x^{14} - 32 x^{13} + 170 x^{12} + 559 x^{11} + 1563 x^{10} + 3152 x^{9} + 8500 x^{8} + 10428 x^{7} + 13659 x^{6} + 15093 x^{5} + 15804 x^{4} - 3996 x^{3} + 1026 x^{2} - 243 x + 81 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(37319027463036582275390625=3^{8}\cdot 5^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.65$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(195=3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{195}(64,·)$, $\chi_{195}(1,·)$, $\chi_{195}(8,·)$, $\chi_{195}(142,·)$, $\chi_{195}(79,·)$, $\chi_{195}(83,·)$, $\chi_{195}(86,·)$, $\chi_{195}(157,·)$, $\chi_{195}(161,·)$, $\chi_{195}(164,·)$, $\chi_{195}(103,·)$, $\chi_{195}(44,·)$, $\chi_{195}(47,·)$, $\chi_{195}(181,·)$, $\chi_{195}(118,·)$, $\chi_{195}(122,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{12} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{12} a^{11} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{36} a^{12} - \frac{1}{36} a^{11} - \frac{1}{18} a^{9} + \frac{2}{9} a^{8} - \frac{1}{18} a^{7} + \frac{2}{9} a^{5} - \frac{7}{18} a^{4} - \frac{5}{12} a^{2} + \frac{1}{4} a$, $\frac{1}{1846471644} a^{13} - \frac{231259}{307745274} a^{12} + \frac{28258561}{923235822} a^{11} - \frac{71780225}{1846471644} a^{10} - \frac{18106606}{153872637} a^{9} - \frac{9602995}{102581758} a^{8} + \frac{38574868}{461617911} a^{7} + \frac{85561628}{461617911} a^{6} - \frac{21728767}{307745274} a^{5} - \frac{166998143}{461617911} a^{4} + \frac{113082215}{615490548} a^{3} + \frac{78603871}{307745274} a^{2} - \frac{12042145}{102581758} a + \frac{73603191}{205163516}$, $\frac{1}{5539414932} a^{14} - \frac{1}{5539414932} a^{13} - \frac{15018325}{1846471644} a^{12} - \frac{46131691}{2769707466} a^{11} + \frac{29138287}{2769707466} a^{10} + \frac{348939821}{2769707466} a^{9} - \frac{2736527}{31835718} a^{8} - \frac{196436971}{1384853733} a^{7} + \frac{117165197}{2769707466} a^{6} + \frac{116681969}{923235822} a^{5} + \frac{233434309}{615490548} a^{4} + \frac{34328483}{615490548} a^{3} - \frac{137772325}{615490548} a^{2} + \frac{13376573}{51290879} a + \frac{11310385}{51290879}$, $\frac{1}{11078829864} a^{15} + \frac{1}{5539414932} a^{13} + \frac{49903325}{5539414932} a^{12} + \frac{1742081}{1846471644} a^{11} + \frac{16098859}{1230981096} a^{10} + \frac{455560033}{2769707466} a^{9} + \frac{290177236}{1384853733} a^{8} + \frac{111201377}{923235822} a^{7} + \frac{324995893}{2769707466} a^{6} + \frac{915472129}{3692943288} a^{5} + \frac{76734461}{923235822} a^{4} + \frac{277051507}{615490548} a^{3} + \frac{80156379}{205163516} a^{2} - \frac{93512895}{205163516} a + \frac{101279331}{410327032}$

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

Class group and class number

$C_{2}\times C_{26}$, which has order $52$ (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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{7901839}{11078829864} a^{15} - \frac{3591745}{2769707466} a^{14} + \frac{16522027}{1846471644} a^{13} - \frac{164096815}{5539414932} a^{12} + \frac{762168289}{5539414932} a^{11} + \frac{3387015535}{11078829864} a^{10} + \frac{697516879}{923235822} a^{9} + \frac{1699613734}{1384853733} a^{8} + \frac{10881704647}{2769707466} a^{7} + \frac{860582102}{461617911} a^{6} + \frac{2466092117}{1230981096} a^{5} + \frac{117809236}{153872637} a^{4} - \frac{39509195}{205163516} a^{3} - \frac{9547825435}{615490548} a^{2} - \frac{2155047}{205163516} a + \frac{2155047}{410327032} \) (order $10$)
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:  \( 114579.196923 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), 4.0.2471625.2, \(\Q(\sqrt{5}, \sqrt{13})\), 4.0.2471625.1, 4.0.21125.1, \(\Q(\zeta_{5})\), 4.4.494325.1, 4.4.19773.1, 8.0.6108930140625.4, 8.0.446265625.1, 8.8.244357205625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
13Data not computed