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Properties

Label 16.16.3233912675...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{24}\cdot 5^{12}\cdot 29^{6}\cdot 41^{2}\cdot 281^{2}$
Root discriminant $107.61$
Ramified primes $2, 5, 29, 41, 281$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1086

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![84272481, 70705872, -364246632, -74291988, 234002138, 46017268, -57282828, -12019788, 6387559, 1315740, -361460, -63596, 11182, 1300, -180, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 180*x^14 + 1300*x^13 + 11182*x^12 - 63596*x^11 - 361460*x^10 + 1315740*x^9 + 6387559*x^8 - 12019788*x^7 - 57282828*x^6 + 46017268*x^5 + 234002138*x^4 - 74291988*x^3 - 364246632*x^2 + 70705872*x + 84272481)
 
gp: K = bnfinit(x^16 - 8*x^15 - 180*x^14 + 1300*x^13 + 11182*x^12 - 63596*x^11 - 361460*x^10 + 1315740*x^9 + 6387559*x^8 - 12019788*x^7 - 57282828*x^6 + 46017268*x^5 + 234002138*x^4 - 74291988*x^3 - 364246632*x^2 + 70705872*x + 84272481, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 180 x^{14} + 1300 x^{13} + 11182 x^{12} - 63596 x^{11} - 361460 x^{10} + 1315740 x^{9} + 6387559 x^{8} - 12019788 x^{7} - 57282828 x^{6} + 46017268 x^{5} + 234002138 x^{4} - 74291988 x^{3} - 364246632 x^{2} + 70705872 x + 84272481 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(323391267567114489856000000000000=2^{24}\cdot 5^{12}\cdot 29^{6}\cdot 41^{2}\cdot 281^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $107.61$
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:  $2, 5, 29, 41, 281$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{12} a^{11} + \frac{1}{12} a^{9} + \frac{5}{12} a^{7} - \frac{1}{3} a^{6} + \frac{1}{12} a^{5} - \frac{1}{6} a^{4} + \frac{1}{12} a^{3} - \frac{1}{2} a^{2} - \frac{5}{12} a - \frac{1}{2}$, $\frac{1}{12} a^{12} + \frac{1}{12} a^{10} - \frac{1}{12} a^{8} - \frac{1}{3} a^{7} + \frac{1}{12} a^{6} - \frac{1}{6} a^{5} - \frac{5}{12} a^{4} - \frac{1}{2} a^{3} - \frac{5}{12} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{12} a^{13} + \frac{1}{12} a^{9} - \frac{1}{12} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{4} a^{5} - \frac{1}{12} a^{4} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{4}$, $\frac{1}{10116} a^{14} - \frac{125}{10116} a^{13} - \frac{49}{3372} a^{12} - \frac{299}{10116} a^{11} - \frac{611}{10116} a^{10} - \frac{899}{10116} a^{9} + \frac{491}{5058} a^{8} + \frac{1357}{3372} a^{7} + \frac{4321}{10116} a^{6} + \frac{49}{1124} a^{5} + \frac{268}{843} a^{4} + \frac{523}{10116} a^{3} - \frac{287}{5058} a^{2} + \frac{15}{281} a + \frac{373}{1124}$, $\frac{1}{1864965313400840811695784282111887591733511055266692} a^{15} + \frac{15213495510904902234097003203632084055853428985}{1864965313400840811695784282111887591733511055266692} a^{14} + \frac{6288426521928297192649897020617220452058464882555}{621655104466946937231928094037295863911170351755564} a^{13} - \frac{76074552992625473938127953425061995777476339156165}{1864965313400840811695784282111887591733511055266692} a^{12} - \frac{36467634897754945927804596208114250693806665306325}{932482656700420405847892141055943795866755527633346} a^{11} - \frac{62908124809755166728369878309095660993510419733993}{932482656700420405847892141055943795866755527633346} a^{10} - \frac{54106620199039872386795640535017488120955397032134}{466241328350210202923946070527971897933377763816673} a^{9} - \frac{1259302587704038668743837712315972197132141264233}{17268197346304081589775780389924885108643620882099} a^{8} - \frac{51688718549988600356174403929707184797687264051189}{932482656700420405847892141055943795866755527633346} a^{7} + \frac{47274594022996921547133679545307688889399988878779}{155413776116736734307982023509323965977792587938891} a^{6} + \frac{10475851990507664283814244647395084595234567618213}{155413776116736734307982023509323965977792587938891} a^{5} - \frac{227439800226708511253038162733077713254629645417175}{466241328350210202923946070527971897933377763816673} a^{4} - \frac{349811741246376203565117606444463532931698752182325}{1864965313400840811695784282111887591733511055266692} a^{3} + \frac{32415929485121744804835581717745949595462766979525}{69072789385216326359103121559699540434574483528396} a^{2} - \frac{68302102568952241316436175160862445326608128860509}{207218368155648979077309364679098621303723450585188} a + \frac{10655652928691583155160901097618441566161980403821}{69072789385216326359103121559699540434574483528396}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 115209190933 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1086:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 97 conjugacy class representatives for t16n1086 are not computed
Character table for t16n1086 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 4.4.58000.1, 4.4.725.1, 8.8.3364000000.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
5Data not computed
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed
281Data not computed