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Properties

Label 16.0.11700885803...5625.4
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 61^{8}$
Root discriminant $31.93$
Ramified primes $5, 61$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T157)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![226391, -2395, -66026, 143460, 8932, -2305, 46967, -17865, 17940, -3595, 3133, -250, 342, -5, 26, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 26*x^14 - 5*x^13 + 342*x^12 - 250*x^11 + 3133*x^10 - 3595*x^9 + 17940*x^8 - 17865*x^7 + 46967*x^6 - 2305*x^5 + 8932*x^4 + 143460*x^3 - 66026*x^2 - 2395*x + 226391)
 
gp: K = bnfinit(x^16 + 26*x^14 - 5*x^13 + 342*x^12 - 250*x^11 + 3133*x^10 - 3595*x^9 + 17940*x^8 - 17865*x^7 + 46967*x^6 - 2305*x^5 + 8932*x^4 + 143460*x^3 - 66026*x^2 - 2395*x + 226391, 1)
 

Normalized defining polynomial

\( x^{16} + 26 x^{14} - 5 x^{13} + 342 x^{12} - 250 x^{11} + 3133 x^{10} - 3595 x^{9} + 17940 x^{8} - 17865 x^{7} + 46967 x^{6} - 2305 x^{5} + 8932 x^{4} + 143460 x^{3} - 66026 x^{2} - 2395 x + 226391 \)

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:  \(1170088580305670166015625=5^{14}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.93$
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:  $5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{22} a^{12} + \frac{2}{11} a^{11} + \frac{1}{11} a^{10} + \frac{5}{11} a^{9} + \frac{7}{22} a^{8} + \frac{7}{22} a^{7} + \frac{7}{22} a^{6} + \frac{1}{22} a^{5} - \frac{4}{11} a^{4} + \frac{9}{22} a^{3} - \frac{3}{11} a^{2} - \frac{5}{22} a - \frac{1}{2}$, $\frac{1}{22} a^{13} - \frac{3}{22} a^{11} + \frac{1}{11} a^{10} + \frac{1}{22} a^{8} + \frac{1}{22} a^{7} + \frac{3}{11} a^{6} - \frac{1}{22} a^{5} - \frac{3}{22} a^{4} - \frac{9}{22} a^{3} + \frac{4}{11} a^{2} - \frac{1}{11} a$, $\frac{1}{768370702} a^{14} + \frac{8706564}{384185351} a^{13} - \frac{1232189}{69851882} a^{12} + \frac{182337841}{768370702} a^{11} - \frac{14579203}{384185351} a^{10} - \frac{76680726}{384185351} a^{9} + \frac{57242049}{768370702} a^{8} + \frac{104957441}{384185351} a^{7} - \frac{149248904}{384185351} a^{6} + \frac{139500154}{384185351} a^{5} + \frac{285887009}{768370702} a^{4} - \frac{148480217}{768370702} a^{3} + \frac{249095421}{768370702} a^{2} + \frac{346042855}{768370702} a - \frac{8444263}{34925941}$, $\frac{1}{3162911814253767330932122} a^{15} + \frac{1696942437486721}{3162911814253767330932122} a^{14} + \frac{63607208415355378470365}{3162911814253767330932122} a^{13} + \frac{16102546638009577091166}{1581455907126883665466061} a^{12} - \frac{8602438995976109441491}{287537437659433393721102} a^{11} + \frac{726980787571188827477551}{3162911814253767330932122} a^{10} + \frac{802732115866424970680891}{3162911814253767330932122} a^{9} - \frac{280385425707819557758703}{1581455907126883665466061} a^{8} - \frac{228019321179664059533544}{1581455907126883665466061} a^{7} - \frac{375801498536611688341297}{1581455907126883665466061} a^{6} + \frac{366558860983847737986463}{1581455907126883665466061} a^{5} + \frac{1467189841576788506746489}{3162911814253767330932122} a^{4} + \frac{355156187621505720201989}{1581455907126883665466061} a^{3} + \frac{667663605440361778271941}{3162911814253767330932122} a^{2} - \frac{643464112802818700816039}{1581455907126883665466061} a + \frac{4949526599744666726797}{26139767059948490338282}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{127758556963096127}{38572095295777650377221} a^{15} + \frac{8742703357063022661}{77144190591555300754442} a^{14} + \frac{1626245520566490017}{38572095295777650377221} a^{13} + \frac{77879039234505411678}{38572095295777650377221} a^{12} - \frac{1008373918376359732}{3506554117797968216111} a^{11} + \frac{1663536769976838009925}{77144190591555300754442} a^{10} - \frac{1708816843673902063993}{77144190591555300754442} a^{9} + \frac{13195510246337361419599}{77144190591555300754442} a^{8} - \frac{17244281908694853411363}{77144190591555300754442} a^{7} + \frac{24667222981520740429647}{38572095295777650377221} a^{6} - \frac{22816683097729358356841}{77144190591555300754442} a^{5} + \frac{9554832339830262613947}{38572095295777650377221} a^{4} + \frac{188128886625705554547187}{77144190591555300754442} a^{3} - \frac{170112589272049483971045}{77144190591555300754442} a^{2} + \frac{6615479372678235580055}{38572095295777650377221} a + \frac{1406877758611101409057}{318777647072542565101} \) (order $10$)
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:  \( 628180.26955 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T157):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.17732890625.2 x2, 8.0.58140625.1

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
61Data not computed