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Properties

Label 16.0.68157975826...473.10
Degree $16$
Signature $[0, 8]$
Discriminant $17^{15}\cdot 47^{8}$
Root discriminant $97.63$
Ramified primes $17, 47$
Class number $640$ (GRH)
Class group $[4, 160]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6662675, -13786253, 17704192, -14208448, 8773684, -4224297, 1972426, -553249, 182717, -42501, 11578, -2653, 562, -120, 18, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 18*x^14 - 120*x^13 + 562*x^12 - 2653*x^11 + 11578*x^10 - 42501*x^9 + 182717*x^8 - 553249*x^7 + 1972426*x^6 - 4224297*x^5 + 8773684*x^4 - 14208448*x^3 + 17704192*x^2 - 13786253*x + 6662675)
 
gp: K = bnfinit(x^16 - x^15 + 18*x^14 - 120*x^13 + 562*x^12 - 2653*x^11 + 11578*x^10 - 42501*x^9 + 182717*x^8 - 553249*x^7 + 1972426*x^6 - 4224297*x^5 + 8773684*x^4 - 14208448*x^3 + 17704192*x^2 - 13786253*x + 6662675, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 18 x^{14} - 120 x^{13} + 562 x^{12} - 2653 x^{11} + 11578 x^{10} - 42501 x^{9} + 182717 x^{8} - 553249 x^{7} + 1972426 x^{6} - 4224297 x^{5} + 8773684 x^{4} - 14208448 x^{3} + 17704192 x^{2} - 13786253 x + 6662675 \)

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:  \(68157975826732896644627006991473=17^{15}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.63$
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:  $17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{8} a^{10} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{3}{8} a^{5} + \frac{3}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{3}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{32} a^{12} + \frac{1}{32} a^{11} + \frac{1}{32} a^{10} - \frac{3}{32} a^{9} + \frac{1}{16} a^{8} - \frac{7}{32} a^{7} + \frac{7}{32} a^{6} - \frac{15}{32} a^{5} + \frac{3}{16} a^{4} - \frac{3}{32} a^{3} + \frac{13}{32} a^{2} + \frac{1}{32} a - \frac{1}{32}$, $\frac{1}{32} a^{13} - \frac{3}{32} a^{9} - \frac{1}{32} a^{8} + \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{7}{32} a^{5} + \frac{3}{32} a^{4} + \frac{3}{8} a^{3} - \frac{1}{8} a^{2} + \frac{7}{16} a + \frac{13}{32}$, $\frac{1}{256} a^{14} - \frac{1}{128} a^{13} - \frac{1}{256} a^{12} - \frac{13}{256} a^{11} - \frac{3}{64} a^{10} - \frac{1}{8} a^{9} - \frac{5}{128} a^{8} + \frac{5}{256} a^{7} + \frac{5}{128} a^{6} + \frac{3}{64} a^{5} + \frac{19}{64} a^{4} - \frac{17}{256} a^{3} + \frac{9}{256} a^{2} - \frac{17}{64} a - \frac{89}{256}$, $\frac{1}{136311358867398436000414204446212721746603776} a^{15} + \frac{101149628802377344611807969164081741295145}{136311358867398436000414204446212721746603776} a^{14} - \frac{847972746486053633967674387703584218513167}{136311358867398436000414204446212721746603776} a^{13} + \frac{12599922255044818609853669212241259433965}{8519459929212402250025887777888295109162736} a^{12} + \frac{653523311064037713516212772772772308722823}{12391941715218039636401291313292065613327616} a^{11} + \frac{1260280248354370238123455516872812231374409}{34077839716849609000103551111553180436650944} a^{10} + \frac{375719150644742025844139460781895402072281}{6195970857609019818200645656646032806663808} a^{9} - \frac{7958785697344706457063572312322651043472545}{136311358867398436000414204446212721746603776} a^{8} - \frac{29063172683155643665026641391203576036593607}{136311358867398436000414204446212721746603776} a^{7} + \frac{1344150083761148929416755904525033406555539}{6195970857609019818200645656646032806663808} a^{6} + \frac{1412214510476793147536110221911960209376481}{4259729964606201125012943888944147554581368} a^{5} - \frac{50071365205105266201132231821458517930234309}{136311358867398436000414204446212721746603776} a^{4} - \frac{8016237324599321418707353361921768404299941}{68155679433699218000207102223106360873301888} a^{3} + \frac{61261658217392259007004374185361629111810567}{136311358867398436000414204446212721746603776} a^{2} - \frac{50669311845127804360064649366639333341688717}{136311358867398436000414204446212721746603776} a - \frac{2099116894595411732028875253808214362508749}{4397140608625756000013361433748797475696896}$

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

Class group and class number

$C_{4}\times C_{160}$, which has order $640$ (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:  \( 28502147.7233 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-799}) \), 4.0.10852817.1, 8.0.2002321826203313.4

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
17Data not computed
$47$47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$