↑

Properties

Label 16.0.93386520115...5801.1
Degree $16$
Signature $[0, 8]$
Discriminant $11^{12}\cdot 29^{14}$
Root discriminant $114.99$
Ramified primes $11, 29$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![373271957, 60659805, 32831156, 37380006, 37129668, 15382026, 7278998, 3857440, 1102409, -85395, -93881, -2421, 3888, 121, -88, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 88*x^14 + 121*x^13 + 3888*x^12 - 2421*x^11 - 93881*x^10 - 85395*x^9 + 1102409*x^8 + 3857440*x^7 + 7278998*x^6 + 15382026*x^5 + 37129668*x^4 + 37380006*x^3 + 32831156*x^2 + 60659805*x + 373271957)
 
gp: K = bnfinit(x^16 - x^15 - 88*x^14 + 121*x^13 + 3888*x^12 - 2421*x^11 - 93881*x^10 - 85395*x^9 + 1102409*x^8 + 3857440*x^7 + 7278998*x^6 + 15382026*x^5 + 37129668*x^4 + 37380006*x^3 + 32831156*x^2 + 60659805*x + 373271957, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 88 x^{14} + 121 x^{13} + 3888 x^{12} - 2421 x^{11} - 93881 x^{10} - 85395 x^{9} + 1102409 x^{8} + 3857440 x^{7} + 7278998 x^{6} + 15382026 x^{5} + 37129668 x^{4} + 37380006 x^{3} + 32831156 x^{2} + 60659805 x + 373271957 \)

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:  \(933865201156678930433597550025801=11^{12}\cdot 29^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $114.99$
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:  $11, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2587} a^{13} - \frac{1089}{2587} a^{12} - \frac{51}{2587} a^{11} + \frac{82}{2587} a^{10} - \frac{8}{199} a^{9} + \frac{473}{2587} a^{8} + \frac{294}{2587} a^{7} + \frac{484}{2587} a^{6} - \frac{1110}{2587} a^{5} - \frac{471}{2587} a^{4} + \frac{340}{2587} a^{3} + \frac{93}{199} a^{2} + \frac{1188}{2587} a - \frac{124}{2587}$, $\frac{1}{1984229} a^{14} - \frac{111}{1984229} a^{13} - \frac{946091}{1984229} a^{12} - \frac{494760}{1984229} a^{11} - \frac{287262}{1984229} a^{10} + \frac{178157}{1984229} a^{9} + \frac{682783}{1984229} a^{8} - \frac{278537}{1984229} a^{7} - \frac{878172}{1984229} a^{6} - \frac{899787}{1984229} a^{5} - \frac{729346}{1984229} a^{4} + \frac{134530}{1984229} a^{3} + \frac{78941}{1984229} a^{2} + \frac{910801}{1984229} a - \frac{444647}{1984229}$, $\frac{1}{308573199331176394856854994355077628317181544166834604977} a^{15} - \frac{72168521147634956473971254631633510908432885507414}{308573199331176394856854994355077628317181544166834604977} a^{14} - \frac{28524540057487849743817288527042821891728647147438945}{308573199331176394856854994355077628317181544166834604977} a^{13} + \frac{55172268935146124193327553205414741352856281480642689778}{308573199331176394856854994355077628317181544166834604977} a^{12} + \frac{65679486853052781818183191327514989731110020552087164862}{308573199331176394856854994355077628317181544166834604977} a^{11} - \frac{49380189955090760279302490246826557413720192828345891580}{308573199331176394856854994355077628317181544166834604977} a^{10} - \frac{11230755069036769817631514809492791403161315064771418661}{308573199331176394856854994355077628317181544166834604977} a^{9} - \frac{414133329611686976104571347676483901671120797253849005}{5822135836437290468997264044435426949380783852204426509} a^{8} + \frac{28335911778537334529777955632176304709845038620065779098}{308573199331176394856854994355077628317181544166834604977} a^{7} - \frac{17685793494890046145883979732436748536227555597337911745}{308573199331176394856854994355077628317181544166834604977} a^{6} + \frac{106604883553025794980916839986909768397034242319995141653}{308573199331176394856854994355077628317181544166834604977} a^{5} + \frac{72742499559148830406910160065531028193688278186360839989}{308573199331176394856854994355077628317181544166834604977} a^{4} + \frac{142941746482099548565472558911892141185817413253340997865}{308573199331176394856854994355077628317181544166834604977} a^{3} + \frac{22207291677052763496789523933778921360674159472659642707}{308573199331176394856854994355077628317181544166834604977} a^{2} + \frac{72507937900197060314791581225963307621334591310943834664}{308573199331176394856854994355077628317181544166834604977} a + \frac{3121660428153754387959137068402984841509922647748227979}{308573199331176394856854994355077628317181544166834604977}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 1253829574.02 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-319}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-11}, \sqrt{29})\), 4.4.2951069.1, 4.0.24389.1, 8.0.8708808242761.1, 8.4.30559208123848349.1 x2

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
11Data not computed
29Data not computed