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Properties

Label 16.0.66607370372...8321.3
Degree $16$
Signature $[0, 8]$
Discriminant $13^{6}\cdot 53^{14}$
Root discriminant $84.42$
Ramified primes $13, 53$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![999947, -3500744, 4335539, -3154489, 2952893, -2147609, 1163887, -498936, 237730, -94769, 27213, -3517, -295, 199, -14, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 14*x^14 + 199*x^13 - 295*x^12 - 3517*x^11 + 27213*x^10 - 94769*x^9 + 237730*x^8 - 498936*x^7 + 1163887*x^6 - 2147609*x^5 + 2952893*x^4 - 3154489*x^3 + 4335539*x^2 - 3500744*x + 999947)
 
gp: K = bnfinit(x^16 - 7*x^15 - 14*x^14 + 199*x^13 - 295*x^12 - 3517*x^11 + 27213*x^10 - 94769*x^9 + 237730*x^8 - 498936*x^7 + 1163887*x^6 - 2147609*x^5 + 2952893*x^4 - 3154489*x^3 + 4335539*x^2 - 3500744*x + 999947, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} - 14 x^{14} + 199 x^{13} - 295 x^{12} - 3517 x^{11} + 27213 x^{10} - 94769 x^{9} + 237730 x^{8} - 498936 x^{7} + 1163887 x^{6} - 2147609 x^{5} + 2952893 x^{4} - 3154489 x^{3} + 4335539 x^{2} - 3500744 x + 999947 \)

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:  \(6660737037213780055542206728321=13^{6}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.42$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{960865914888900374202149599361544543689213613042497} a^{15} + \frac{375623772449781530201694754082234134900691262423639}{960865914888900374202149599361544543689213613042497} a^{14} + \frac{431297066498192138767567990814003501741301440789659}{960865914888900374202149599361544543689213613042497} a^{13} + \frac{457360564982499604409535554735025793553098332987463}{960865914888900374202149599361544543689213613042497} a^{12} - \frac{9379723586952462951542682081755686351666129638053}{960865914888900374202149599361544543689213613042497} a^{11} - \frac{432572755012004398312151412082067083307114248980584}{960865914888900374202149599361544543689213613042497} a^{10} - \frac{93945237509577373961710447850928798308044769527964}{960865914888900374202149599361544543689213613042497} a^{9} - \frac{339220226215806692219198821754234197302412307529461}{960865914888900374202149599361544543689213613042497} a^{8} + \frac{168890915490304769169552811091213559336684503554406}{960865914888900374202149599361544543689213613042497} a^{7} + \frac{93643247641993094589593904592775250927375499177211}{960865914888900374202149599361544543689213613042497} a^{6} - \frac{432061838352784074383426266728315638912771816618317}{960865914888900374202149599361544543689213613042497} a^{5} - \frac{3409137397162230767992032369045015768776434164099}{960865914888900374202149599361544543689213613042497} a^{4} - \frac{90299854362700793292102196004026194705361894615}{73912762683761567246319199950888041822247201003269} a^{3} + \frac{33738886003652509878948098277505715555931707571039}{73912762683761567246319199950888041822247201003269} a^{2} - \frac{17146083569792637134273240605138886559612040883666}{73912762683761567246319199950888041822247201003269} a - \frac{16842471310116164212596023747322817879153473667002}{73912762683761567246319199950888041822247201003269}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  \( -1 \) (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:  \( 92366828.0466 \) (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, 8.0.288136694677.1

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$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{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}$