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Properties

Label 16.0.34336147906...0625.7
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 11^{8}$
Root discriminant $19.21$
Ramified primes $3, 5, 11$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2 \times (C_2^2:C_4)$ (as 16T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, 0, 3000, -2625, 5175, -2800, 1805, -130, -379, 207, -134, -19, 24, -17, 9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 9*x^14 - 17*x^13 + 24*x^12 - 19*x^11 - 134*x^10 + 207*x^9 - 379*x^8 - 130*x^7 + 1805*x^6 - 2800*x^5 + 5175*x^4 - 2625*x^3 + 3000*x^2 + 625)
 
gp: K = bnfinit(x^16 - x^15 + 9*x^14 - 17*x^13 + 24*x^12 - 19*x^11 - 134*x^10 + 207*x^9 - 379*x^8 - 130*x^7 + 1805*x^6 - 2800*x^5 + 5175*x^4 - 2625*x^3 + 3000*x^2 + 625, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 9 x^{14} - 17 x^{13} + 24 x^{12} - 19 x^{11} - 134 x^{10} + 207 x^{9} - 379 x^{8} - 130 x^{7} + 1805 x^{6} - 2800 x^{5} + 5175 x^{4} - 2625 x^{3} + 3000 x^{2} + 625 \)

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:  \(343361479062744140625=3^{8}\cdot 5^{12}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.21$
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:  $3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{10} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{10} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{11} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{2} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{50} a^{12} - \frac{1}{50} a^{11} - \frac{1}{50} a^{10} + \frac{9}{25} a^{9} - \frac{8}{25} a^{8} + \frac{1}{50} a^{7} + \frac{1}{50} a^{6} - \frac{3}{50} a^{5} - \frac{7}{25} a^{4} - \frac{1}{10} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{50} a^{13} - \frac{1}{25} a^{11} + \frac{1}{25} a^{10} - \frac{4}{25} a^{9} - \frac{1}{2} a^{8} - \frac{9}{25} a^{7} - \frac{6}{25} a^{6} + \frac{9}{25} a^{5} - \frac{2}{25} a^{4} + \frac{3}{10} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{4750} a^{14} - \frac{8}{2375} a^{13} + \frac{7}{2375} a^{12} + \frac{104}{2375} a^{11} - \frac{211}{4750} a^{10} + \frac{2241}{4750} a^{9} + \frac{318}{2375} a^{8} + \frac{391}{2375} a^{7} - \frac{747}{2375} a^{6} - \frac{93}{950} a^{5} + \frac{439}{950} a^{4} + \frac{46}{95} a^{3} - \frac{16}{95} a^{2} - \frac{6}{19} a + \frac{17}{38}$, $\frac{1}{23492252179772092250} a^{15} + \frac{486966672787817}{11746126089886046125} a^{14} - \frac{55595153494997871}{23492252179772092250} a^{13} + \frac{52267954417405124}{11746126089886046125} a^{12} + \frac{1026214723876716119}{23492252179772092250} a^{11} - \frac{50786957781052901}{1236434325251162750} a^{10} + \frac{245105897702332472}{618217162625581375} a^{9} + \frac{1420357183987798517}{23492252179772092250} a^{8} - \frac{5586602614937809612}{11746126089886046125} a^{7} + \frac{1600240181910993699}{4698450435954418450} a^{6} - \frac{1254589311069796791}{4698450435954418450} a^{5} + \frac{443838869210047908}{2349225217977209225} a^{4} + \frac{28676612339975777}{187938017438176738} a^{3} + \frac{25383144111194076}{469845043595441845} a^{2} - \frac{63724685892344701}{187938017438176738} a + \frac{9285476478628844}{93969008719088369}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( -\frac{6623689572442746}{11746126089886046125} a^{15} + \frac{26264136515579601}{23492252179772092250} a^{14} - \frac{65045845659391496}{11746126089886046125} a^{13} + \frac{34581756210212438}{2349225217977209225} a^{12} - \frac{511644626198434761}{23492252179772092250} a^{11} + \frac{536056802857900619}{23492252179772092250} a^{10} + \frac{1560502944814546977}{23492252179772092250} a^{9} - \frac{483014656357952156}{2349225217977209225} a^{8} + \frac{3770977466577582553}{11746126089886046125} a^{7} - \frac{3301068180088322331}{23492252179772092250} a^{6} - \frac{11501301745756123}{9891474602009302} a^{5} + \frac{2603374912580958199}{939690087190883690} a^{4} - \frac{2074602117628186387}{469845043595441845} a^{3} + \frac{403042899987829255}{93969008719088369} a^{2} - \frac{330309634242928527}{187938017438176738} a + \frac{100815322642753339}{187938017438176738} \) (order $10$)
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:  \( 5965.81155927 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:C_4$ (as 16T21):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2 \times (C_2^2:C_4)$
Character table for $C_2 \times (C_2^2:C_4)$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{165}) \), \(\Q(\sqrt{33}) \), 4.0.136125.2, 4.2.12375.1, \(\Q(\zeta_{5})\), 4.2.1375.1, 4.2.275.1, \(\Q(\sqrt{5}, \sqrt{33})\), 4.2.2475.1, 8.4.741200625.1, 8.0.18530015625.8, 8.0.1890625.1, 8.0.18530015625.3, 8.4.18530015625.2, 8.0.153140625.2, 8.0.18530015625.7

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 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$