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Properties

Label 16.0.17958563260...0625.4
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 5^{12}\cdot 7^{12}$
Root discriminant $32.80$
Ramified primes $3, 5, 7$
Class number $32$ (GRH)
Class group $[4, 8]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, -16384, -8192, 1024, -1792, -3904, 1760, 340, -59, 85, 110, -61, -7, 1, -2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 2*x^14 + x^13 - 7*x^12 - 61*x^11 + 110*x^10 + 85*x^9 - 59*x^8 + 340*x^7 + 1760*x^6 - 3904*x^5 - 1792*x^4 + 1024*x^3 - 8192*x^2 - 16384*x + 65536)
 
gp: K = bnfinit(x^16 - x^15 - 2*x^14 + x^13 - 7*x^12 - 61*x^11 + 110*x^10 + 85*x^9 - 59*x^8 + 340*x^7 + 1760*x^6 - 3904*x^5 - 1792*x^4 + 1024*x^3 - 8192*x^2 - 16384*x + 65536, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 2 x^{14} + x^{13} - 7 x^{12} - 61 x^{11} + 110 x^{10} + 85 x^{9} - 59 x^{8} + 340 x^{7} + 1760 x^{6} - 3904 x^{5} - 1792 x^{4} + 1024 x^{3} - 8192 x^{2} - 16384 x + 65536 \)

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:  \(1795856326022129150390625=3^{12}\cdot 5^{12}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.80$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} + \frac{1}{16} a^{7} - \frac{7}{16} a^{6} + \frac{3}{16} a^{5} - \frac{1}{8} a^{4} + \frac{5}{16} a^{3} + \frac{5}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{1}{32} a^{9} + \frac{1}{64} a^{8} - \frac{7}{64} a^{7} + \frac{3}{64} a^{6} - \frac{9}{32} a^{5} + \frac{21}{64} a^{4} + \frac{5}{64} a^{3} + \frac{5}{16} a^{2} - \frac{1}{2} a$, $\frac{1}{1280} a^{12} + \frac{7}{1280} a^{11} - \frac{1}{128} a^{10} - \frac{3}{256} a^{9} + \frac{1}{1280} a^{8} + \frac{79}{256} a^{7} - \frac{317}{640} a^{6} - \frac{63}{256} a^{5} + \frac{621}{1280} a^{4} + \frac{19}{64} a^{3} - \frac{1}{2} a^{2} + \frac{7}{20} a + \frac{1}{5}$, $\frac{1}{209920} a^{13} + \frac{67}{209920} a^{12} - \frac{23}{20992} a^{11} + \frac{5}{41984} a^{10} + \frac{381}{209920} a^{9} + \frac{14811}{41984} a^{8} - \frac{16307}{104960} a^{7} + \frac{7561}{41984} a^{6} - \frac{32359}{209920} a^{5} - \frac{2147}{5248} a^{4} - \frac{83}{2624} a^{3} - \frac{1553}{3280} a^{2} - \frac{71}{205} a + \frac{18}{41}$, $\frac{1}{839680} a^{14} - \frac{1}{839680} a^{13} - \frac{97}{419840} a^{12} - \frac{4671}{839680} a^{11} + \frac{5241}{839680} a^{10} + \frac{84227}{839680} a^{9} - \frac{39081}{419840} a^{8} - \frac{181483}{839680} a^{7} + \frac{415813}{839680} a^{6} - \frac{16667}{209920} a^{5} - \frac{5759}{26240} a^{4} - \frac{853}{13120} a^{3} + \frac{17}{80} a^{2} + \frac{81}{410} a - \frac{13}{205}$, $\frac{1}{16793600} a^{15} - \frac{9}{16793600} a^{14} - \frac{1}{1679360} a^{13} + \frac{801}{16793600} a^{12} - \frac{2119}{671744} a^{11} - \frac{2061}{409600} a^{10} + \frac{661219}{8396800} a^{9} + \frac{1294261}{16793600} a^{8} - \frac{7191747}{16793600} a^{7} - \frac{1768201}{4198400} a^{6} - \frac{218513}{1049600} a^{5} - \frac{7389}{26240} a^{4} - \frac{1547}{65600} a^{3} - \frac{147}{820} a^{2} - \frac{837}{4100} a + \frac{278}{1025}$

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

Class group and class number

$C_{4}\times C_{8}$, which has order $32$ (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:  \( \frac{1281}{3358720} a^{15} + \frac{831}{3358720} a^{14} - \frac{1281}{1679360} a^{13} + \frac{1281}{3358720} a^{12} - \frac{8967}{3358720} a^{11} - \frac{78141}{3358720} a^{10} - \frac{15337}{1679360} a^{9} + \frac{21777}{671744} a^{8} - \frac{75579}{3358720} a^{7} + \frac{21777}{167936} a^{6} + \frac{14091}{20992} a^{5} + \frac{6573}{26240} a^{4} - \frac{8967}{13120} a^{3} + \frac{1281}{3280} a^{2} - \frac{1281}{410} a - \frac{1281}{205} \) (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:  \( 103783.231276 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}, \sqrt{21})\), 4.0.46305.1 x2, 4.0.231525.1 x2, \(\Q(\zeta_{5})\), 4.0.55125.1, 8.0.53603825625.5, 8.0.3038765625.3, 8.8.1340095640625.1

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 R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
7Data not computed