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Properties

Label 16.0.11256645592...6249.5
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![3286543, 313326, 7697352, -2571426, 4631732, 13506, 1042184, -11026, 161280, -12012, 10037, -1287, 1262, -205, 79, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 79*x^14 - 205*x^13 + 1262*x^12 - 1287*x^11 + 10037*x^10 - 12012*x^9 + 161280*x^8 - 11026*x^7 + 1042184*x^6 + 13506*x^5 + 4631732*x^4 - 2571426*x^3 + 7697352*x^2 + 313326*x + 3286543)
 
gp: K = bnfinit(x^16 - 7*x^15 + 79*x^14 - 205*x^13 + 1262*x^12 - 1287*x^11 + 10037*x^10 - 12012*x^9 + 161280*x^8 - 11026*x^7 + 1042184*x^6 + 13506*x^5 + 4631732*x^4 - 2571426*x^3 + 7697352*x^2 + 313326*x + 3286543, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 79 x^{14} - 205 x^{13} + 1262 x^{12} - 1287 x^{11} + 10037 x^{10} - 12012 x^{9} + 161280 x^{8} - 11026 x^{7} + 1042184 x^{6} + 13506 x^{5} + 4631732 x^{4} - 2571426 x^{3} + 7697352 x^{2} + 313326 x + 3286543 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} + \frac{6}{13} a^{13} + \frac{1}{13} a^{12} + \frac{3}{13} a^{11} + \frac{1}{13} a^{10} + \frac{1}{13} a^{8} + \frac{2}{13} a^{6} - \frac{2}{13} a^{5} - \frac{1}{13} a^{3} + \frac{1}{13} a^{2}$, $\frac{1}{5803311450904546983580418219464961154269785561981374109} a^{15} + \frac{100569657162833771606391603170602690600490348657424741}{5803311450904546983580418219464961154269785561981374109} a^{14} - \frac{1872663650386474063842713809426302256345677348771675914}{5803311450904546983580418219464961154269785561981374109} a^{13} + \frac{433875126373980529221140687181006409219301953522295299}{5803311450904546983580418219464961154269785561981374109} a^{12} - \frac{1867404727652246436608046733092399209401772582696391486}{5803311450904546983580418219464961154269785561981374109} a^{11} + \frac{2373723940082366438855344811342396147782925106594939845}{5803311450904546983580418219464961154269785561981374109} a^{10} + \frac{600330008632932660085230770926328326653467924540464992}{5803311450904546983580418219464961154269785561981374109} a^{9} + \frac{812583789252289230298110508463882541981479113073139681}{5803311450904546983580418219464961154269785561981374109} a^{8} - \frac{2804813473752680358325400168947588050375466427933555933}{5803311450904546983580418219464961154269785561981374109} a^{7} - \frac{1953722773415652480249381722657902773125379235187074864}{5803311450904546983580418219464961154269785561981374109} a^{6} - \frac{1772212332690924605100617246848001283887787337633395004}{5803311450904546983580418219464961154269785561981374109} a^{5} + \frac{2264977824799271567085042252649840076175475272784665992}{5803311450904546983580418219464961154269785561981374109} a^{4} + \frac{2227447243210791269600457640811524783446361123796085384}{5803311450904546983580418219464961154269785561981374109} a^{3} + \frac{2658104678887008648137378563333586366851543210397960985}{5803311450904546983580418219464961154269785561981374109} a^{2} + \frac{73956835383712565567383353546768082843717053181627577}{446408573146503614121570632266535473405368120152413393} a - \frac{85526820795145868567997658099717570603291729504610991}{446408573146503614121570632266535473405368120152413393}$

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:  \( 392759812.742 \) (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 ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ 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}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$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}$