↑

Properties

Label 16.0.17003500263...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 101^{6}$
Root discriminant $32.69$
Ramified primes $3, 5, 101$
Class number $24$ (GRH)
Class group $[2, 12]$ (GRH)
Galois group $D_4:C_4$ (as 16T26)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 8, 67, 556, 3478, 7574, 8099, 3118, 4268, 426, 1061, -18, 93, 7, 3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 3*x^14 + 7*x^13 + 93*x^12 - 18*x^11 + 1061*x^10 + 426*x^9 + 4268*x^8 + 3118*x^7 + 8099*x^6 + 7574*x^5 + 3478*x^4 + 556*x^3 + 67*x^2 + 8*x + 1)
 
gp: K = bnfinit(x^16 - x^15 + 3*x^14 + 7*x^13 + 93*x^12 - 18*x^11 + 1061*x^10 + 426*x^9 + 4268*x^8 + 3118*x^7 + 8099*x^6 + 7574*x^5 + 3478*x^4 + 556*x^3 + 67*x^2 + 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 3 x^{14} + 7 x^{13} + 93 x^{12} - 18 x^{11} + 1061 x^{10} + 426 x^{9} + 4268 x^{8} + 3118 x^{7} + 8099 x^{6} + 7574 x^{5} + 3478 x^{4} + 556 x^{3} + 67 x^{2} + 8 x + 1 \)

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:  \(1700350026389931884765625=3^{8}\cdot 5^{12}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.69$
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:  $3, 5, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{33} a^{14} - \frac{1}{33} a^{13} + \frac{5}{33} a^{12} - \frac{2}{11} a^{11} + \frac{4}{33} a^{10} - \frac{8}{33} a^{9} + \frac{2}{33} a^{8} + \frac{1}{11} a^{7} + \frac{4}{33} a^{6} + \frac{1}{3} a^{4} - \frac{16}{33} a^{3} + \frac{13}{33} a^{2} - \frac{5}{11} a + \frac{16}{33}$, $\frac{1}{2278613091009978688019043} a^{15} + \frac{19433256861483341269295}{2278613091009978688019043} a^{14} + \frac{11252362321422423125640}{759537697003326229339681} a^{13} - \frac{7140169103179009787897}{2278613091009978688019043} a^{12} + \frac{19107835898901170837474}{2278613091009978688019043} a^{11} - \frac{17290257056203910431383}{69048881545756929939971} a^{10} - \frac{762002708233022292678656}{2278613091009978688019043} a^{9} - \frac{47362477011606769554304}{2278613091009978688019043} a^{8} - \frac{976699569613053716795003}{2278613091009978688019043} a^{7} + \frac{251231432965132766520461}{759537697003326229339681} a^{6} + \frac{26863080375503083613618}{69048881545756929939971} a^{5} + \frac{182941356911094984165970}{2278613091009978688019043} a^{4} + \frac{142529394806286447127171}{2278613091009978688019043} a^{3} + \frac{23334484900149377168482}{207146644637270789819913} a^{2} + \frac{135886939251430682658738}{759537697003326229339681} a - \frac{949036433782108764470126}{2278613091009978688019043}$

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

Class group and class number

$C_{2}\times C_{12}$, which has order $24$ (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:  \( -\frac{18000043627973370116545}{2278613091009978688019043} a^{15} + \frac{2827675218500170088339}{207146644637270789819913} a^{14} - \frac{74329490472383727266659}{2278613091009978688019043} a^{13} - \frac{26306263502125640064905}{759537697003326229339681} a^{12} - \frac{534856202508869444073573}{759537697003326229339681} a^{11} + \frac{498199398824692432111807}{759537697003326229339681} a^{10} - \frac{20004148786762565255639998}{2278613091009978688019043} a^{9} + \frac{6418630706892487690350478}{2278613091009978688019043} a^{8} - \frac{78919731788988231664799138}{2278613091009978688019043} a^{7} - \frac{2637490477824868946004302}{2278613091009978688019043} a^{6} - \frac{12339057071167873168539802}{207146644637270789819913} a^{5} - \frac{16546667198786974897483344}{759537697003326229339681} a^{4} - \frac{7051557556047561353170048}{759537697003326229339681} a^{3} - \frac{12921848791000252023507617}{2278613091009978688019043} a^{2} - \frac{5555092455934224333997565}{759537697003326229339681} a - \frac{79135943391822973503779}{759537697003326229339681} \) (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:  \( 50808.1466806 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.2525.1, 4.0.12625.1, 8.8.52158988125.1, 8.0.1303974703125.1, 8.0.159390625.1

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 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{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}$
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$