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Properties

Label 16.0.12551515611...1089.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{10}\cdot 53^{8}$
Root discriminant $42.77$
Ramified primes $17, 53$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_4\wr C_2$ (as 16T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2209, -235, 4714, -3277, 6562, -8942, 8534, -4461, 6745, -4583, 1511, -646, 287, -58, 13, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 13*x^14 - 58*x^13 + 287*x^12 - 646*x^11 + 1511*x^10 - 4583*x^9 + 6745*x^8 - 4461*x^7 + 8534*x^6 - 8942*x^5 + 6562*x^4 - 3277*x^3 + 4714*x^2 - 235*x + 2209)
 
gp: K = bnfinit(x^16 - 6*x^15 + 13*x^14 - 58*x^13 + 287*x^12 - 646*x^11 + 1511*x^10 - 4583*x^9 + 6745*x^8 - 4461*x^7 + 8534*x^6 - 8942*x^5 + 6562*x^4 - 3277*x^3 + 4714*x^2 - 235*x + 2209, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 13 x^{14} - 58 x^{13} + 287 x^{12} - 646 x^{11} + 1511 x^{10} - 4583 x^{9} + 6745 x^{8} - 4461 x^{7} + 8534 x^{6} - 8942 x^{5} + 6562 x^{4} - 3277 x^{3} + 4714 x^{2} - 235 x + 2209 \)

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:  \(125515156113146867692601089=17^{10}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.77$
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:  $17, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{1316} a^{14} + \frac{121}{658} a^{13} + \frac{26}{329} a^{12} - \frac{29}{329} a^{11} + \frac{27}{188} a^{10} - \frac{76}{329} a^{9} - \frac{257}{658} a^{8} + \frac{15}{1316} a^{7} + \frac{407}{1316} a^{6} - \frac{149}{658} a^{5} - \frac{275}{1316} a^{4} - \frac{313}{658} a^{3} - \frac{213}{1316} a^{2} - \frac{359}{1316} a + \frac{1}{28}$, $\frac{1}{209415058143207176758558640104} a^{15} + \frac{60190200017159303178855207}{209415058143207176758558640104} a^{14} - \frac{651609724368744820230370559}{3739554609700128156402832859} a^{13} + \frac{24669818636003411870789501625}{104707529071603588379279320052} a^{12} + \frac{47250927364975788596274127869}{209415058143207176758558640104} a^{11} - \frac{71520998105276003533468163381}{209415058143207176758558640104} a^{10} - \frac{26472712026685281835914115555}{104707529071603588379279320052} a^{9} - \frac{38881624215422295599045985845}{209415058143207176758558640104} a^{8} - \frac{21435202805008054648107160003}{52353764535801794189639660026} a^{7} + \frac{43587196749908683052180883147}{209415058143207176758558640104} a^{6} + \frac{4891608286658715862468886305}{209415058143207176758558640104} a^{5} + \frac{6936427454505707597461697117}{29916436877601025251222662872} a^{4} - \frac{31773159687710378409507214783}{209415058143207176758558640104} a^{3} + \frac{6950451242130720365476155321}{26176882267900897094819830013} a^{2} + \frac{27908355741329420663862095817}{104707529071603588379279320052} a - \frac{1039404234659455611165270203}{4455639534961854824650183832}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 452391.453967 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T42):

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 $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{901}) \), 4.4.15317.1 x2, 4.4.47753.1 x2, \(\Q(\sqrt{17}, \sqrt{53})\), 8.0.38765933153.1, 8.0.11203354681217.1, 8.8.659020863601.1

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 8 siblings: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$53$53.8.4.1$x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
53.8.4.1$x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$