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Properties

Label 16.0.11684193984...000.41
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}$
Root discriminant $100.98$
Ramified primes $2, 3, 5, 19$
Class number $163840$ (GRH)
Class group $[4, 4, 8, 16, 80]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![542004961, 0, 130137424, 0, 59620320, 0, 12966476, 0, 1842554, 0, 143756, 0, 5985, 0, 124, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 124*x^14 + 5985*x^12 + 143756*x^10 + 1842554*x^8 + 12966476*x^6 + 59620320*x^4 + 130137424*x^2 + 542004961)
 
gp: K = bnfinit(x^16 + 124*x^14 + 5985*x^12 + 143756*x^10 + 1842554*x^8 + 12966476*x^6 + 59620320*x^4 + 130137424*x^2 + 542004961, 1)
 

Normalized defining polynomial

\( x^{16} + 124 x^{14} + 5985 x^{12} + 143756 x^{10} + 1842554 x^{8} + 12966476 x^{6} + 59620320 x^{4} + 130137424 x^{2} + 542004961 \)

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:  \(116841939847873560576000000000000=2^{32}\cdot 3^{8}\cdot 5^{12}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $100.98$
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:  $2, 3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2280=2^{3}\cdot 3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{2280}(1,·)$, $\chi_{2280}(2051,·)$, $\chi_{2280}(1217,·)$, $\chi_{2280}(1673,·)$, $\chi_{2280}(77,·)$, $\chi_{2280}(911,·)$, $\chi_{2280}(1747,·)$, $\chi_{2280}(533,·)$, $\chi_{2280}(1369,·)$, $\chi_{2280}(2203,·)$, $\chi_{2280}(607,·)$, $\chi_{2280}(229,·)$, $\chi_{2280}(1063,·)$, $\chi_{2280}(2279,·)$, $\chi_{2280}(1139,·)$, $\chi_{2280}(1141,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{69361} a^{10} + \frac{90}{69361} a^{8} + \frac{2835}{69361} a^{6} - \frac{32911}{69361} a^{4} + \frac{25303}{69361} a^{2} + \frac{9982}{69361}$, $\frac{1}{69361} a^{11} + \frac{90}{69361} a^{9} + \frac{2835}{69361} a^{7} - \frac{32911}{69361} a^{5} + \frac{25303}{69361} a^{3} + \frac{9982}{69361} a$, $\frac{1}{22126159} a^{12} + \frac{108}{22126159} a^{10} - \frac{4365288}{22126159} a^{8} - \frac{3658014}{22126159} a^{6} - \frac{385748}{2011469} a^{4} + \frac{4419013}{22126159} a^{2} + \frac{6491527}{22126159}$, $\frac{1}{515119107679} a^{13} + \frac{2404411}{515119107679} a^{11} + \frac{1403432405}{3705892861} a^{9} - \frac{84125972499}{515119107679} a^{7} - \frac{3836699990}{46829009789} a^{5} - \frac{4696663408}{16616745409} a^{3} + \frac{162914270275}{515119107679} a$, $\frac{1}{26086146731972239} a^{14} - \frac{422101343}{26086146731972239} a^{12} - \frac{12045369782}{2371467884724749} a^{10} - \frac{296192805614724}{2371467884724749} a^{8} - \frac{5263003784854008}{26086146731972239} a^{6} + \frac{176856214989732}{841488604257169} a^{4} + \frac{7768972037597176}{26086146731972239} a^{2} - \frac{122998995798}{1120490817919}$, $\frac{1}{26086146731972239} a^{15} - \frac{8608}{26086146731972239} a^{13} + \frac{54551720381}{26086146731972239} a^{11} - \frac{3573041797383886}{26086146731972239} a^{9} + \frac{7916141346187865}{26086146731972239} a^{7} + \frac{12882785349638781}{26086146731972239} a^{5} + \frac{487186166211750}{2371467884724749} a^{3} - \frac{8300572295044549}{26086146731972239} a$

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

Class group and class number

$C_{4}\times C_{4}\times C_{8}\times C_{16}\times C_{80}$, which has order $163840$ (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:  \( -1 \) (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:  \( 16694.393243512957 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-285}) \), \(\Q(\sqrt{-570}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-57}) \), \(\Q(\sqrt{-114}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{-285})\), \(\Q(\sqrt{5}, \sqrt{-57})\), \(\Q(\sqrt{10}, \sqrt{-114})\), \(\Q(\sqrt{5}, \sqrt{-114})\), \(\Q(\sqrt{10}, \sqrt{-57})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-57})\), 4.4.72000.1, 4.0.2888000.1, 4.0.722000.3, \(\Q(\zeta_{15})^+\), 8.0.432373800960000.208, 8.0.10809345024000000.231, 8.0.42224004000000.63, 8.0.10809345024000000.162, 8.0.675584064000000.194, 8.8.5184000000.1, 8.0.133448704000000.60

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$