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Properties

Label 16.8.79361092911...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 5^{10}\cdot 29^{6}\cdot 751^{4}$
Root discriminant $202.41$
Ramified primes $2, 5, 29, 751$
Class number $14$ (GRH)
Class group $[14]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T456)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14401865752759, -26945103359324, -13351517441880, -570322900656, -438868218257, -8528696844, 7975385568, -205481584, 194964715, 7185744, -1384598, 34408, -27589, -324, 52, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 52*x^14 - 324*x^13 - 27589*x^12 + 34408*x^11 - 1384598*x^10 + 7185744*x^9 + 194964715*x^8 - 205481584*x^7 + 7975385568*x^6 - 8528696844*x^5 - 438868218257*x^4 - 570322900656*x^3 - 13351517441880*x^2 - 26945103359324*x + 14401865752759)
 
gp: K = bnfinit(x^16 + 52*x^14 - 324*x^13 - 27589*x^12 + 34408*x^11 - 1384598*x^10 + 7185744*x^9 + 194964715*x^8 - 205481584*x^7 + 7975385568*x^6 - 8528696844*x^5 - 438868218257*x^4 - 570322900656*x^3 - 13351517441880*x^2 - 26945103359324*x + 14401865752759, 1)
 

Normalized defining polynomial

\( x^{16} + 52 x^{14} - 324 x^{13} - 27589 x^{12} + 34408 x^{11} - 1384598 x^{10} + 7185744 x^{9} + 194964715 x^{8} - 205481584 x^{7} + 7975385568 x^{6} - 8528696844 x^{5} - 438868218257 x^{4} - 570322900656 x^{3} - 13351517441880 x^{2} - 26945103359324 x + 14401865752759 \)

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:  \(7936109291110060736213155840000000000=2^{32}\cdot 5^{10}\cdot 29^{6}\cdot 751^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $202.41$
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:  $2, 5, 29, 751$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{495} a^{14} + \frac{29}{495} a^{13} - \frac{116}{495} a^{12} + \frac{24}{55} a^{11} + \frac{184}{495} a^{10} - \frac{38}{165} a^{8} - \frac{4}{165} a^{7} - \frac{107}{495} a^{6} - \frac{59}{495} a^{5} - \frac{31}{99} a^{4} - \frac{203}{495} a^{3} - \frac{34}{495} a^{2} - \frac{3}{11} a - \frac{224}{495}$, $\frac{1}{13518786669053693927740442866372768926042538037260185825890174174348403545936407603519928340438645} a^{15} - \frac{7415809690084533619377686063370626727984698521547409696488730723189834337496718743209730985503}{13518786669053693927740442866372768926042538037260185825890174174348403545936407603519928340438645} a^{14} - \frac{440994767412444679273569768141880387947267297191509945183553775780782023874155423140800746272634}{13518786669053693927740442866372768926042538037260185825890174174348403545936407603519928340438645} a^{13} - \frac{1376308956887569095558717090620119130732671314709348816528835700057308581964960837601961830066682}{13518786669053693927740442866372768926042538037260185825890174174348403545936407603519928340438645} a^{12} - \frac{197717342865960352253592072668328709386227815590437103627795399148489484563529831505626935618094}{1931255238436241989677206123767538418006076862465740832270024882049771935133772514788561191491235} a^{11} - \frac{6723393362754846526745011785214134448578531349536219011983166582025666153891172691790230377831978}{13518786669053693927740442866372768926042538037260185825890174174348403545936407603519928340438645} a^{10} - \frac{282853530172794843209302478345418384020954477410872219453560237325714966130368668324797684366469}{643751746145413996559068707922512806002025620821913610756674960683257311711257504929520397163745} a^{9} - \frac{74998654924751291112380350338156833565014417113226317394417751286831777903562892015511284235823}{4506262223017897975913480955457589642014179345753395275296724724782801181978802534506642780146215} a^{8} - \frac{5043692880067634936858661173109114023699407715430612950951448368875784049254314366834256289787253}{13518786669053693927740442866372768926042538037260185825890174174348403545936407603519928340438645} a^{7} + \frac{287450049265662179816596790682233224961798623516292406480266797565794296352222115967394138841539}{901252444603579595182696191091517928402835869150679055059344944956560236395760506901328556029243} a^{6} + \frac{1331000175003781529200240651320859094582004913788700785815791208233153789277390751056825534692121}{4506262223017897975913480955457589642014179345753395275296724724782801181978802534506642780146215} a^{5} + \frac{3368674695145116033070169280024620905056920013026166024563913559814363973259160222014327965526}{45517800232504019958722029853106966080951306522761568437340653785684860424028308429360028082285} a^{4} - \frac{5428054871237226739660229905944170268977410613270474524765930837105048454710087543060990898654168}{13518786669053693927740442866372768926042538037260185825890174174348403545936407603519928340438645} a^{3} - \frac{5942077286076576293539525822212550151672340828192688679076068723265723668926443930334209593981}{1931255238436241989677206123767538418006076862465740832270024882049771935133772514788561191491235} a^{2} + \frac{610501821349313868130334020240157894962802899995997678409393729306169241822901233480714650075341}{1228980606277608538885494806033888084185685276114562347808197652213491231448764327592720758221695} a + \frac{2748174625901303465499562837351571268255067476186897860950812421580204953702519159631756685148643}{13518786669053693927740442866372768926042538037260185825890174174348403545936407603519928340438645}$

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

Class group and class number

$C_{14}$, which has order $14$ (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:  \( 262571450736 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T456):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.4.46400.1, 4.4.725.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.2152960000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
751Data not computed