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Properties

Label 16.0.11256645592...6249.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 53^{14}$
Root discriminant $116.34$
Ramified primes $13, 53$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![281867, -3587532, 12988290, -1401668, -1554137, -557115, -10053, 46024, 86739, 39721, 11517, 1918, 709, -13, 36, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 36*x^14 - 13*x^13 + 709*x^12 + 1918*x^11 + 11517*x^10 + 39721*x^9 + 86739*x^8 + 46024*x^7 - 10053*x^6 - 557115*x^5 - 1554137*x^4 - 1401668*x^3 + 12988290*x^2 - 3587532*x + 281867)
 
gp: K = bnfinit(x^16 - 3*x^15 + 36*x^14 - 13*x^13 + 709*x^12 + 1918*x^11 + 11517*x^10 + 39721*x^9 + 86739*x^8 + 46024*x^7 - 10053*x^6 - 557115*x^5 - 1554137*x^4 - 1401668*x^3 + 12988290*x^2 - 3587532*x + 281867, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 36 x^{14} - 13 x^{13} + 709 x^{12} + 1918 x^{11} + 11517 x^{10} + 39721 x^{9} + 86739 x^{8} + 46024 x^{7} - 10053 x^{6} - 557115 x^{5} - 1554137 x^{4} - 1401668 x^{3} + 12988290 x^{2} - 3587532 x + 281867 \)

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:  \(1125664559289128829386632937086249=13^{8}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $116.34$
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:  $13, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} - \frac{2}{13} a^{12} - \frac{6}{13} a^{11} + \frac{1}{13} a^{9} + \frac{2}{13} a^{8} + \frac{5}{13} a^{6} + \frac{6}{13} a^{5} + \frac{6}{13} a^{4} - \frac{2}{13} a^{3} + \frac{3}{13} a - \frac{1}{13}$, $\frac{1}{1420831510963471167413624917941975677299243245651924583} a^{15} - \frac{18886571700041678854288720763405439635659544434865974}{1420831510963471167413624917941975677299243245651924583} a^{14} + \frac{575211044969979135213262790429519884884978040379535928}{1420831510963471167413624917941975677299243245651924583} a^{13} - \frac{221453789137616837352631027018917112434870175984108803}{1420831510963471167413624917941975677299243245651924583} a^{12} + \frac{63388159522135161569965105111125950106583524858020610}{1420831510963471167413624917941975677299243245651924583} a^{11} + \frac{126111914044776207329948636369826957414153046856126991}{1420831510963471167413624917941975677299243245651924583} a^{10} + \frac{241305986742206692560767855179578747809068319363804342}{1420831510963471167413624917941975677299243245651924583} a^{9} + \frac{668562734198924293859184178865565357880305034798990389}{1420831510963471167413624917941975677299243245651924583} a^{8} - \frac{312593366259945550104308991284423552253274595460959272}{1420831510963471167413624917941975677299243245651924583} a^{7} + \frac{106889698588971069117236168002543943917352146744028966}{1420831510963471167413624917941975677299243245651924583} a^{6} + \frac{258574533999670001877377101658164587021510676557662497}{1420831510963471167413624917941975677299243245651924583} a^{5} - \frac{246326149129151040709387316566564305051368266073857143}{1420831510963471167413624917941975677299243245651924583} a^{4} - \frac{224252517171217097588265859883942755007954396706588532}{1420831510963471167413624917941975677299243245651924583} a^{3} + \frac{348318568535954912208062218650694778497872984343350068}{1420831510963471167413624917941975677299243245651924583} a^{2} - \frac{612707815815941261484094203155490585232746288222783985}{1420831510963471167413624917941975677299243245651924583} a + \frac{631182336070598317473587731847668054284486030475634000}{1420831510963471167413624917941975677299243245651924583}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 379837172.404 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{53}) \), 4.0.148877.1, 8.0.48695101400413.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$53$53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$