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Properties

Label 16.0.23672572938...0761.1
Degree $16$
Signature $[0, 8]$
Discriminant $11^{2}\cdot 89^{14}$
Root discriminant $68.53$
Ramified primes $11, 89$
Class number $113$ (GRH)
Class group $[113]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T258)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![99328, 60416, 291264, -207376, 268176, -308607, 254636, -164769, 106997, -38020, 18563, -4029, 1548, -203, 63, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 63*x^14 - 203*x^13 + 1548*x^12 - 4029*x^11 + 18563*x^10 - 38020*x^9 + 106997*x^8 - 164769*x^7 + 254636*x^6 - 308607*x^5 + 268176*x^4 - 207376*x^3 + 291264*x^2 + 60416*x + 99328)
 
gp: K = bnfinit(x^16 - 4*x^15 + 63*x^14 - 203*x^13 + 1548*x^12 - 4029*x^11 + 18563*x^10 - 38020*x^9 + 106997*x^8 - 164769*x^7 + 254636*x^6 - 308607*x^5 + 268176*x^4 - 207376*x^3 + 291264*x^2 + 60416*x + 99328, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 63 x^{14} - 203 x^{13} + 1548 x^{12} - 4029 x^{11} + 18563 x^{10} - 38020 x^{9} + 106997 x^{8} - 164769 x^{7} + 254636 x^{6} - 308607 x^{5} + 268176 x^{4} - 207376 x^{3} + 291264 x^{2} + 60416 x + 99328 \)

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:  \(236725729383493411014646970761=11^{2}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.53$
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:  $11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{16} a^{11} + \frac{1}{16} a^{8} - \frac{1}{4} a^{6} - \frac{1}{16} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{32} a^{12} + \frac{1}{32} a^{9} - \frac{1}{8} a^{7} + \frac{7}{32} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{9}{32} a^{3} - \frac{3}{8} a^{2}$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{12} + \frac{1}{64} a^{10} - \frac{1}{64} a^{9} + \frac{1}{16} a^{8} - \frac{5}{64} a^{7} + \frac{13}{64} a^{6} + \frac{1}{8} a^{5} - \frac{5}{64} a^{4} - \frac{19}{64} a^{3} - \frac{3}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{256} a^{14} + \frac{3}{256} a^{12} - \frac{3}{256} a^{11} + \frac{7}{256} a^{9} + \frac{27}{256} a^{8} - \frac{1}{32} a^{7} - \frac{63}{256} a^{6} - \frac{41}{256} a^{5} + \frac{5}{32} a^{4} - \frac{51}{256} a^{3} - \frac{1}{32} a^{2} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{11122819602176426105731084988416} a^{15} + \frac{211944395367186048380711937}{1390352450272053263216385623552} a^{14} + \frac{45594771731837790804109820767}{11122819602176426105731084988416} a^{13} - \frac{152012784016898612851831696663}{11122819602176426105731084988416} a^{12} + \frac{1767810283075109744158629695}{1390352450272053263216385623552} a^{11} + \frac{317963701556023513469000798883}{11122819602176426105731084988416} a^{10} - \frac{272575220256386802697273325721}{11122819602176426105731084988416} a^{9} - \frac{12021285858261973602528777339}{695176225136026631608192811776} a^{8} + \frac{443355224130820493413120178165}{11122819602176426105731084988416} a^{7} + \frac{1348472484339976082469715574171}{11122819602176426105731084988416} a^{6} + \frac{87973181262196551663197975855}{695176225136026631608192811776} a^{5} + \frac{2242454356873398858515586235393}{11122819602176426105731084988416} a^{4} - \frac{1015247295801037018897333604521}{2780704900544106526432771247104} a^{3} + \frac{23266222198492444848960742929}{173794056284006657902048202944} a^{2} + \frac{4467874974901633525408520403}{173794056284006657902048202944} a + \frac{193595527177582103175532837}{447922825474243963665072688}$

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

Class group and class number

$C_{113}$, which has order $113$ (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:  \( 52312146.3689 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T258):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.704969.1, 8.0.44231334895529.1, 8.2.5466794200571.1, 8.6.486544683850819.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.8.7.3$x^{8} - 7209$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.7.3$x^{8} - 7209$$8$$1$$7$$C_8$$[\ ]_{8}$