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Properties

Label 16.0.68668573773...0625.8
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 109^{12}$
Root discriminant $112.80$
Ramified primes $5, 109$
Class number $200$ (GRH)
Class group $[2, 10, 10]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27982932961, 0, 27886579105, 0, 8211857189, 0, 976886210, 0, 57640466, 0, 1840490, 0, 32109, 0, 285, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 285*x^14 + 32109*x^12 + 1840490*x^10 + 57640466*x^8 + 976886210*x^6 + 8211857189*x^4 + 27886579105*x^2 + 27982932961)
 
gp: K = bnfinit(x^16 + 285*x^14 + 32109*x^12 + 1840490*x^10 + 57640466*x^8 + 976886210*x^6 + 8211857189*x^4 + 27886579105*x^2 + 27982932961, 1)
 

Normalized defining polynomial

\( x^{16} + 285 x^{14} + 32109 x^{12} + 1840490 x^{10} + 57640466 x^{8} + 976886210 x^{6} + 8211857189 x^{4} + 27886579105 x^{2} + 27982932961 \)

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:  \(686685737739964474054511962890625=5^{12}\cdot 109^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.80$
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:  $5, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{2} a^{2} + \frac{3}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{3}{8}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{6} - \frac{1}{8} a^{3} + \frac{1}{8}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{7} + \frac{1}{16} a^{6} - \frac{1}{4} a^{5} + \frac{3}{16} a^{4} + \frac{1}{16} a^{3} + \frac{1}{4} a^{2} - \frac{3}{16} a - \frac{1}{16}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{4} a^{4} - \frac{3}{16} a^{3} + \frac{1}{16} a^{2} + \frac{1}{4} a + \frac{3}{16}$, $\frac{1}{838976} a^{12} + \frac{6139}{209744} a^{10} - \frac{10943}{209744} a^{8} + \frac{21701}{419488} a^{6} - \frac{28515}{209744} a^{4} + \frac{50523}{209744} a^{2} - \frac{1}{2} a + \frac{23509}{838976}$, $\frac{1}{343141184} a^{13} - \frac{1802903}{85785296} a^{11} - \frac{4572875}{85785296} a^{9} + \frac{1280165}{171570592} a^{7} - \frac{1}{8} a^{6} - \frac{9021289}{85785296} a^{5} - \frac{12796297}{85785296} a^{3} + \frac{71755957}{343141184} a - \frac{3}{8}$, $\frac{1}{836096529648277468022912} a^{14} - \frac{1}{686282368} a^{13} + \frac{491566755437312495}{836096529648277468022912} a^{12} + \frac{1802903}{171570592} a^{11} + \frac{49973206747898642851}{6532004137877167718929} a^{10} + \frac{4572875}{171570592} a^{9} + \frac{23655777218050089365087}{418048264824138734011456} a^{8} + \frac{20166159}{343141184} a^{7} + \frac{37044340396849967058825}{418048264824138734011456} a^{6} - \frac{33871359}{171570592} a^{5} + \frac{4143852002523852327383}{52256033103017341751432} a^{4} - \frac{30096351}{171570592} a^{3} - \frac{308646927574167765992751}{836096529648277468022912} a^{2} + \frac{56921987}{686282368} a - \frac{453932724365489041}{4998155974965940352}$, $\frac{1}{341963480626145484421371008} a^{15} - \frac{3358591076719743}{170981740313072742210685504} a^{13} - \frac{1}{1677952} a^{12} - \frac{1621396848670474464134519}{85490870156536371105342752} a^{11} - \frac{6139}{419488} a^{10} - \frac{9423780772169977569787939}{170981740313072742210685504} a^{9} - \frac{15275}{419488} a^{8} - \frac{543782902971043938385625}{10686358769567046388167844} a^{7} + \frac{30735}{838976} a^{6} - \frac{19303948273625920745982093}{85490870156536371105342752} a^{5} + \frac{28515}{419488} a^{4} + \frac{79542384522526774864353957}{341963480626145484421371008} a^{3} + \frac{185439}{419488} a^{2} + \frac{26915493339864100239}{1022122896880534801984} a - \frac{547869}{1677952}$

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

Class group and class number

$C_{2}\times C_{10}\times C_{10}$, which has order $200$ (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:  \( 30581868.8278 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{545}) \), \(\Q(\sqrt{109}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{109})\), 4.0.59405.1 x2, 4.0.2725.1 x2, 4.4.161878625.1, 4.4.161878625.2, 8.0.88223850625.1, 8.8.26204689231890625.1, 8.0.26204689231890625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.8.6.1$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}$
$109$109.4.3.1$x^{4} - 109$$4$$1$$3$$C_4$$[\ ]_{4}$
109.4.3.1$x^{4} - 109$$4$$1$$3$$C_4$$[\ ]_{4}$
109.4.3.1$x^{4} - 109$$4$$1$$3$$C_4$$[\ ]_{4}$
109.4.3.1$x^{4} - 109$$4$$1$$3$$C_4$$[\ ]_{4}$