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Properties

Label 16.0.48736115900...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 41^{8}$
Root discriminant $26.18$
Ramified primes $5, 41$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -128, 832, -544, 976, 112, 40, 538, -141, 269, 10, 14, 61, -17, 13, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 13*x^14 - 17*x^13 + 61*x^12 + 14*x^11 + 10*x^10 + 269*x^9 - 141*x^8 + 538*x^7 + 40*x^6 + 112*x^5 + 976*x^4 - 544*x^3 + 832*x^2 - 128*x + 256)
 
gp: K = bnfinit(x^16 - x^15 + 13*x^14 - 17*x^13 + 61*x^12 + 14*x^11 + 10*x^10 + 269*x^9 - 141*x^8 + 538*x^7 + 40*x^6 + 112*x^5 + 976*x^4 - 544*x^3 + 832*x^2 - 128*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 13 x^{14} - 17 x^{13} + 61 x^{12} + 14 x^{11} + 10 x^{10} + 269 x^{9} - 141 x^{8} + 538 x^{7} + 40 x^{6} + 112 x^{5} + 976 x^{4} - 544 x^{3} + 832 x^{2} - 128 x + 256 \)

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:  \(48736115900396728515625=5^{14}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.18$
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:  $5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} + \frac{3}{8} a^{8} + \frac{1}{8} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{3}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} + \frac{1}{16} a^{10} + \frac{3}{16} a^{9} - \frac{7}{16} a^{8} + \frac{1}{8} a^{7} + \frac{3}{8} a^{6} - \frac{3}{16} a^{5} - \frac{5}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{224} a^{13} + \frac{1}{224} a^{12} - \frac{5}{224} a^{11} + \frac{1}{224} a^{10} - \frac{13}{224} a^{9} - \frac{3}{14} a^{8} - \frac{1}{112} a^{7} - \frac{71}{224} a^{6} - \frac{83}{224} a^{5} - \frac{3}{7} a^{4} - \frac{27}{56} a^{3} + \frac{9}{28} a^{2} + \frac{1}{14} a + \frac{1}{7}$, $\frac{1}{44534336} a^{14} - \frac{92241}{44534336} a^{13} - \frac{730535}{44534336} a^{12} - \frac{2278365}{44534336} a^{11} + \frac{515961}{44534336} a^{10} - \frac{5051999}{22267168} a^{9} + \frac{6174939}{22267168} a^{8} - \frac{21761035}{44534336} a^{7} + \frac{5713019}{44534336} a^{6} + \frac{6134859}{22267168} a^{5} - \frac{1850881}{11133584} a^{4} + \frac{314213}{5566792} a^{3} - \frac{367593}{2783396} a^{2} + \frac{205980}{695849} a - \frac{198812}{695849}$, $\frac{1}{623480704} a^{15} - \frac{5}{623480704} a^{14} + \frac{170161}{623480704} a^{13} + \frac{10462723}{623480704} a^{12} + \frac{19123593}{623480704} a^{11} - \frac{1039911}{311740352} a^{10} + \frac{74862925}{311740352} a^{9} + \frac{182744477}{623480704} a^{8} - \frac{21555265}{623480704} a^{7} + \frac{115607279}{311740352} a^{6} - \frac{30799495}{77935088} a^{5} + \frac{22056555}{77935088} a^{4} + \frac{16848341}{38967544} a^{3} + \frac{2095378}{4870943} a^{2} - \frac{444314}{4870943} a - \frac{2392939}{4870943}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( 7717.57384586 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.210125.1, 4.4.5125.1, 4.0.1025.1, 8.4.131328125.1, 8.4.220762578125.1, 8.0.44152515625.2

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

Sibling fields

Degree 8 siblings: data not computed
Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$41$41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$