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Properties

Label 16.0.62802105505...6369.2
Degree $16$
Signature $[0, 8]$
Discriminant $17^{12}\cdot 47^{6}$
Root discriminant $35.47$
Ramified primes $17, 47$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.D_4$ (as 16T330)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13, -124, 546, -1320, 1441, -260, -163, 547, -394, -430, 494, 12, -157, 46, 11, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 11*x^14 + 46*x^13 - 157*x^12 + 12*x^11 + 494*x^10 - 430*x^9 - 394*x^8 + 547*x^7 - 163*x^6 - 260*x^5 + 1441*x^4 - 1320*x^3 + 546*x^2 - 124*x + 13)
 
gp: K = bnfinit(x^16 - 7*x^15 + 11*x^14 + 46*x^13 - 157*x^12 + 12*x^11 + 494*x^10 - 430*x^9 - 394*x^8 + 547*x^7 - 163*x^6 - 260*x^5 + 1441*x^4 - 1320*x^3 + 546*x^2 - 124*x + 13, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 11 x^{14} + 46 x^{13} - 157 x^{12} + 12 x^{11} + 494 x^{10} - 430 x^{9} - 394 x^{8} + 547 x^{7} - 163 x^{6} - 260 x^{5} + 1441 x^{4} - 1320 x^{3} + 546 x^{2} - 124 x + 13 \)

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:  \(6280210550563314266206369=17^{12}\cdot 47^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{8} a^{11} + \frac{1}{8} a^{9} + \frac{3}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{3}{8} a^{5} - \frac{3}{8} a^{4} + \frac{3}{8} a^{3} - \frac{3}{8} a^{2} + \frac{3}{8} a + \frac{1}{8}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} + \frac{1}{16} a^{10} + \frac{1}{8} a^{9} + \frac{3}{8} a^{8} - \frac{3}{16} a^{7} - \frac{3}{16} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{8} a + \frac{7}{16}$, $\frac{1}{32} a^{13} + \frac{3}{32} a^{10} - \frac{1}{4} a^{9} - \frac{13}{32} a^{8} + \frac{5}{16} a^{7} - \frac{1}{32} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{3}{8} a^{2} + \frac{5}{32} a + \frac{7}{32}$, $\frac{1}{6545344} a^{14} + \frac{84723}{6545344} a^{13} + \frac{23395}{1636336} a^{12} - \frac{95665}{6545344} a^{11} + \frac{452045}{6545344} a^{10} - \frac{765861}{6545344} a^{9} + \frac{1289283}{6545344} a^{8} - \frac{1874271}{6545344} a^{7} + \frac{1230025}{6545344} a^{6} + \frac{87327}{204542} a^{5} - \frac{301983}{818168} a^{4} + \frac{755323}{1636336} a^{3} + \frac{244081}{6545344} a^{2} - \frac{996997}{3272672} a - \frac{158747}{503488}$, $\frac{1}{7553326976} a^{15} + \frac{7}{1888331744} a^{14} + \frac{32539175}{7553326976} a^{13} - \frac{37077909}{7553326976} a^{12} - \frac{91309895}{1888331744} a^{11} - \frac{22482583}{472082936} a^{10} - \frac{848494565}{3776663488} a^{9} + \frac{404971243}{1888331744} a^{8} + \frac{1313988145}{3776663488} a^{7} + \frac{1526587345}{7553326976} a^{6} + \frac{372278277}{944165872} a^{5} + \frac{53336917}{1888331744} a^{4} + \frac{1448662845}{7553326976} a^{3} + \frac{522662255}{7553326976} a^{2} - \frac{219176621}{581025152} a - \frac{171504051}{581025152}$

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:  \( -1 \) (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:  \( 398194.082095 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.D_4$ (as 16T330):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4.D_4$
Character table for $C_2^4.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.2.13583.1, 8.2.2506034826287.2, 8.0.3136464113.1, 8.2.147413813311.2

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

Sibling fields

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$47$47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$