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Properties

Label 16.0.40011163903...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{8}\cdot 181^{2}$
Root discriminant $34.49$
Ramified primes $5, 29, 181$
Class number $68$ (GRH)
Class group $[2, 34]$ (GRH)
Galois group $C_4^2:C_2^2.C_2$ (as 16T406)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10141, -20529, 31185, -28760, 25069, -16384, 11867, -7225, 4956, -2534, 1262, -398, 134, -23, 11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 11*x^14 - 23*x^13 + 134*x^12 - 398*x^11 + 1262*x^10 - 2534*x^9 + 4956*x^8 - 7225*x^7 + 11867*x^6 - 16384*x^5 + 25069*x^4 - 28760*x^3 + 31185*x^2 - 20529*x + 10141)
 
gp: K = bnfinit(x^16 - 2*x^15 + 11*x^14 - 23*x^13 + 134*x^12 - 398*x^11 + 1262*x^10 - 2534*x^9 + 4956*x^8 - 7225*x^7 + 11867*x^6 - 16384*x^5 + 25069*x^4 - 28760*x^3 + 31185*x^2 - 20529*x + 10141, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 11 x^{14} - 23 x^{13} + 134 x^{12} - 398 x^{11} + 1262 x^{10} - 2534 x^{9} + 4956 x^{8} - 7225 x^{7} + 11867 x^{6} - 16384 x^{5} + 25069 x^{4} - 28760 x^{3} + 31185 x^{2} - 20529 x + 10141 \)

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:  \(4001116390384599853515625=5^{12}\cdot 29^{8}\cdot 181^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.49$
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:  $5, 29, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{91} a^{14} - \frac{27}{91} a^{13} + \frac{40}{91} a^{12} + \frac{3}{7} a^{11} - \frac{18}{91} a^{10} - \frac{2}{7} a^{9} - \frac{19}{91} a^{8} - \frac{5}{91} a^{7} - \frac{2}{7} a^{6} + \frac{22}{91} a^{5} - \frac{6}{91} a^{4} + \frac{3}{7} a^{3} + \frac{33}{91} a^{2} + \frac{3}{91} a - \frac{36}{91}$, $\frac{1}{20456411003417472230864171} a^{15} - \frac{70534990186330671148778}{20456411003417472230864171} a^{14} + \frac{6529887877801075275819481}{20456411003417472230864171} a^{13} + \frac{6083916825840935333041602}{20456411003417472230864171} a^{12} - \frac{110214988141833304526999}{889409174061629227428877} a^{11} - \frac{7281852144778396745584826}{20456411003417472230864171} a^{10} - \frac{8532531136332539515452513}{20456411003417472230864171} a^{9} - \frac{5978463774493892147938194}{20456411003417472230864171} a^{8} + \frac{8452964229754403565437131}{20456411003417472230864171} a^{7} + \frac{1465522767250396517951817}{20456411003417472230864171} a^{6} + \frac{6217899143265387447501442}{20456411003417472230864171} a^{5} - \frac{5956482224179713225833720}{20456411003417472230864171} a^{4} + \frac{10225461221223092938164117}{20456411003417472230864171} a^{3} + \frac{209691840528891797158}{224795725312279914624881} a^{2} - \frac{373535687269685421131993}{2922344429059638890123453} a + \frac{6640911649164397217555665}{20456411003417472230864171}$

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

Class group and class number

$C_{2}\times C_{34}$, which has order $68$ (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:  \( 3793.72993285 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2:C_2^2.C_2$ (as 16T406):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_4^2:C_2^2.C_2$
Character table for $C_4^2:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$181$181.4.0.1$x^{4} - x + 54$$1$$4$$0$$C_4$$[\ ]^{4}$
181.4.0.1$x^{4} - x + 54$$1$$4$$0$$C_4$$[\ ]^{4}$
181.4.2.2$x^{4} - 181 x^{2} + 589698$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
181.4.0.1$x^{4} - x + 54$$1$$4$$0$$C_4$$[\ ]^{4}$