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Properties

Label 16.0.21767823360...0000.8
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 5^{12}$
Root discriminant $21.56$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4 \times D_4$ (as 16T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![916, 2296, 5130, -2072, -579, -2994, 1437, -178, 498, -282, -123, 234, -94, 2, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 15*x^14 + 2*x^13 - 94*x^12 + 234*x^11 - 123*x^10 - 282*x^9 + 498*x^8 - 178*x^7 + 1437*x^6 - 2994*x^5 - 579*x^4 - 2072*x^3 + 5130*x^2 + 2296*x + 916)
 
gp: K = bnfinit(x^16 - 6*x^15 + 15*x^14 + 2*x^13 - 94*x^12 + 234*x^11 - 123*x^10 - 282*x^9 + 498*x^8 - 178*x^7 + 1437*x^6 - 2994*x^5 - 579*x^4 - 2072*x^3 + 5130*x^2 + 2296*x + 916, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 15 x^{14} + 2 x^{13} - 94 x^{12} + 234 x^{11} - 123 x^{10} - 282 x^{9} + 498 x^{8} - 178 x^{7} + 1437 x^{6} - 2994 x^{5} - 579 x^{4} - 2072 x^{3} + 5130 x^{2} + 2296 x + 916 \)

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:  \(2176782336000000000000=2^{24}\cdot 3^{12}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.56$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{20} a^{12} - \frac{1}{5} a^{11} - \frac{1}{10} a^{10} + \frac{1}{10} a^{9} - \frac{1}{4} a^{8} + \frac{1}{5} a^{7} - \frac{1}{10} a^{6} - \frac{1}{5} a^{5} + \frac{1}{4} a^{4} - \frac{1}{10} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{20} a^{13} + \frac{1}{10} a^{11} + \frac{1}{5} a^{10} + \frac{3}{20} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{10} a^{6} - \frac{1}{20} a^{5} - \frac{1}{10} a^{4} - \frac{1}{2} a^{3} + \frac{3}{10} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{260} a^{14} + \frac{3}{130} a^{13} + \frac{1}{65} a^{12} - \frac{8}{65} a^{11} - \frac{17}{260} a^{10} - \frac{11}{65} a^{9} + \frac{29}{130} a^{8} + \frac{1}{26} a^{7} + \frac{3}{260} a^{6} - \frac{9}{65} a^{5} + \frac{7}{65} a^{4} - \frac{2}{65} a^{3} + \frac{1}{65} a^{2} + \frac{11}{65} a - \frac{1}{65}$, $\frac{1}{167793613863075231198700} a^{15} + \frac{2124326641135800939}{12907201066390402399900} a^{14} + \frac{3631511873314550508381}{167793613863075231198700} a^{13} - \frac{407006915453401336807}{33558722772615046239740} a^{12} + \frac{10072596857938555195651}{167793613863075231198700} a^{11} + \frac{38628365057772308737067}{167793613863075231198700} a^{10} - \frac{37985529966070510334137}{167793613863075231198700} a^{9} - \frac{2166106482561794739241}{12907201066390402399900} a^{8} + \frac{26102331374033386601709}{167793613863075231198700} a^{7} - \frac{38248070782238566914771}{167793613863075231198700} a^{6} + \frac{40198409106263304336619}{167793613863075231198700} a^{5} - \frac{38269039538640464301157}{167793613863075231198700} a^{4} - \frac{50248622148815241573}{1290720106639040239990} a^{3} - \frac{15407078504941798075773}{41948403465768807799675} a^{2} - \frac{1318629428965970084914}{41948403465768807799675} a + \frac{9801387698922406485047}{41948403465768807799675}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( \frac{129396703096}{258744287374975} a^{15} - \frac{1534529025061}{517488574749950} a^{14} + \frac{1916339570651}{258744287374975} a^{13} + \frac{20537607921}{20699542989998} a^{12} - \frac{11746569860249}{258744287374975} a^{11} + \frac{28878354871457}{258744287374975} a^{10} - \frac{12809328951357}{258744287374975} a^{9} - \frac{74659954422731}{517488574749950} a^{8} + \frac{57373119681729}{258744287374975} a^{7} + \frac{8172027276619}{258744287374975} a^{6} + \frac{149844442054969}{258744287374975} a^{5} - \frac{721733873486769}{517488574749950} a^{4} - \frac{3210090208327}{51748857474995} a^{3} - \frac{490194099669809}{517488574749950} a^{2} + \frac{596979056669124}{258744287374975} a + \frac{242317491367788}{258744287374975} \) (order $6$)
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:  \( 15797.6605593 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times D_4$ (as 16T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_4 \times D_4$
Character table for $C_4 \times D_4$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), 4.4.8000.1, 4.0.72000.2, 4.0.5400.2, \(\Q(\sqrt{-3}, \sqrt{5})\), 4.0.5400.1, 8.0.29160000.1, 8.0.5184000000.4, 8.0.729000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 siblings: data not computed

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
3Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$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}$