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Properties

Label 16.8.70699102466...7153.1
Degree $16$
Signature $[8, 4]$
Discriminant $17^{15}\cdot 89^{12}$
Root discriminant $412.66$
Ramified primes $17, 89$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5465405110997, -8682678868227, -3911934284598, -71370632807, 255680761873, 10794765535, -13587856083, -1037559794, 480433550, 69162204, -5293289, -1381795, -17594, 7377, 188, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 188*x^14 + 7377*x^13 - 17594*x^12 - 1381795*x^11 - 5293289*x^10 + 69162204*x^9 + 480433550*x^8 - 1037559794*x^7 - 13587856083*x^6 + 10794765535*x^5 + 255680761873*x^4 - 71370632807*x^3 - 3911934284598*x^2 - 8682678868227*x - 5465405110997)
 
gp: K = bnfinit(x^16 - x^15 + 188*x^14 + 7377*x^13 - 17594*x^12 - 1381795*x^11 - 5293289*x^10 + 69162204*x^9 + 480433550*x^8 - 1037559794*x^7 - 13587856083*x^6 + 10794765535*x^5 + 255680761873*x^4 - 71370632807*x^3 - 3911934284598*x^2 - 8682678868227*x - 5465405110997, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 188 x^{14} + 7377 x^{13} - 17594 x^{12} - 1381795 x^{11} - 5293289 x^{10} + 69162204 x^{9} + 480433550 x^{8} - 1037559794 x^{7} - 13587856083 x^{6} + 10794765535 x^{5} + 255680761873 x^{4} - 71370632807 x^{3} - 3911934284598 x^{2} - 8682678868227 x - 5465405110997 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(706991024666918541719408792174287819307153=17^{15}\cdot 89^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $412.66$
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:  $17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} + \frac{2}{13} a^{13} - \frac{1}{13} a^{12} + \frac{3}{13} a^{11} + \frac{4}{13} a^{10} - \frac{1}{13} a^{8} + \frac{4}{13} a^{7} - \frac{2}{13} a^{6} - \frac{4}{13} a^{5} + \frac{5}{13} a^{4} - \frac{3}{13} a^{3} - \frac{5}{13} a^{2} + \frac{5}{13} a$, $\frac{1}{1001680701549845148377796996894445811726976617538661887178265725094238890300487900415403891} a^{15} + \frac{1981348421955726390205492202266598880670615679929441743495356752520839974684793589278681}{77052361657680396029061307453418908594382816733743222090635825007249145407729838493492607} a^{14} + \frac{95319547156957378953028410663129499659722951386521008038389901438038617278587170733775467}{1001680701549845148377796996894445811726976617538661887178265725094238890300487900415403891} a^{13} + \frac{185066096212105525168947385219686066805192491904140710668291113262131084886629763415187422}{1001680701549845148377796996894445811726976617538661887178265725094238890300487900415403891} a^{12} + \frac{406836471914817562770545060340465173658815018322045115484707498872775891578121484658937776}{1001680701549845148377796996894445811726976617538661887178265725094238890300487900415403891} a^{11} + \frac{134238590032544160332042062384757794449829415023596687515315289696489063963455424007776236}{1001680701549845148377796996894445811726976617538661887178265725094238890300487900415403891} a^{10} + \frac{59170111756085851479277563754770943773051723928839415267602308457403501503556317310964411}{1001680701549845148377796996894445811726976617538661887178265725094238890300487900415403891} a^{9} + \frac{424390765413270694009803941476342095496140707377378054299157636124003469173682551461796057}{1001680701549845148377796996894445811726976617538661887178265725094238890300487900415403891} a^{8} - \frac{454981052986605183277239466674391651917508510634107672909596886453439479830836928078517107}{1001680701549845148377796996894445811726976617538661887178265725094238890300487900415403891} a^{7} - \frac{17744525481961574747215682369558039950779125685939148599070070954034018502397576470678111}{77052361657680396029061307453418908594382816733743222090635825007249145407729838493492607} a^{6} - \frac{31830811372394975410613443389186895097961162651003415479498021646702396549687714692592094}{77052361657680396029061307453418908594382816733743222090635825007249145407729838493492607} a^{5} - \frac{25686231490911872934774044990955324873035799534319554798373540869448827875022764987999811}{77052361657680396029061307453418908594382816733743222090635825007249145407729838493492607} a^{4} - \frac{331888153591359535260828078935544600502355532851490475433977476573973711830011559746400963}{1001680701549845148377796996894445811726976617538661887178265725094238890300487900415403891} a^{3} - \frac{126713798772093643890302079210273880263377318375117521370468540449548219601443647000716642}{1001680701549845148377796996894445811726976617538661887178265725094238890300487900415403891} a^{2} - \frac{205520625052019860163224395320013240632795478664381141146832097487331837494544312965248877}{1001680701549845148377796996894445811726976617538661887178265725094238890300487900415403891} a - \frac{27983909027511563534388701149358627523776295551850122670246439452407808583945158110472830}{77052361657680396029061307453418908594382816733743222090635825007249145407729838493492607}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $11$
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:  \( 160504965712000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.8.25745567912986193.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
$89$89.4.3.3$x^{4} + 267$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.3$x^{4} + 267$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.4$x^{4} + 2403$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.4$x^{4} + 2403$$4$$1$$3$$C_4$$[\ ]_{4}$