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Properties

Label 16.4.10704914143...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{32}\cdot 5^{10}\cdot 761^{5}$
Root discriminant $86.97$
Ramified primes $2, 5, 761$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1360

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![69429119, -34630408, 53236782, -27521876, 19279562, -7603764, 2305578, -208908, -205841, 114852, -44354, 8548, -1408, -36, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 36*x^13 - 1408*x^12 + 8548*x^11 - 44354*x^10 + 114852*x^9 - 205841*x^8 - 208908*x^7 + 2305578*x^6 - 7603764*x^5 + 19279562*x^4 - 27521876*x^3 + 53236782*x^2 - 34630408*x + 69429119)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 36*x^13 - 1408*x^12 + 8548*x^11 - 44354*x^10 + 114852*x^9 - 205841*x^8 - 208908*x^7 + 2305578*x^6 - 7603764*x^5 + 19279562*x^4 - 27521876*x^3 + 53236782*x^2 - 34630408*x + 69429119, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 36 x^{13} - 1408 x^{12} + 8548 x^{11} - 44354 x^{10} + 114852 x^{9} - 205841 x^{8} - 208908 x^{7} + 2305578 x^{6} - 7603764 x^{5} + 19279562 x^{4} - 27521876 x^{3} + 53236782 x^{2} - 34630408 x + 69429119 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10704914143082750935040000000000=2^{32}\cdot 5^{10}\cdot 761^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.97$
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:  $2, 5, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{60} a^{12} + \frac{3}{20} a^{10} + \frac{1}{15} a^{9} + \frac{1}{10} a^{8} - \frac{1}{15} a^{7} - \frac{7}{20} a^{6} - \frac{1}{3} a^{5} + \frac{7}{30} a^{4} - \frac{1}{20} a^{2} + \frac{1}{15} a - \frac{1}{60}$, $\frac{1}{180} a^{13} + \frac{1}{180} a^{12} - \frac{11}{180} a^{11} + \frac{11}{60} a^{10} - \frac{2}{9} a^{9} + \frac{1}{90} a^{8} - \frac{17}{36} a^{7} - \frac{9}{20} a^{6} - \frac{14}{45} a^{5} + \frac{37}{90} a^{4} + \frac{17}{180} a^{3} - \frac{19}{180} a^{2} - \frac{7}{180} a + \frac{19}{180}$, $\frac{1}{180} a^{14} + \frac{7}{90} a^{11} - \frac{5}{36} a^{10} - \frac{1}{6} a^{9} - \frac{1}{12} a^{8} + \frac{23}{90} a^{7} + \frac{73}{180} a^{6} - \frac{5}{18} a^{5} - \frac{23}{60} a^{4} - \frac{1}{30} a^{3} + \frac{1}{5} a^{2} + \frac{7}{90} a + \frac{89}{180}$, $\frac{1}{190041037664984167694273318723376202126292381328540} a^{15} - \frac{76394867370243868891538525876686657754028295423}{38008207532996833538854663744675240425258476265708} a^{14} + \frac{50426215032902929253134983997279406799042609538}{47510259416246041923568329680844050531573095332135} a^{13} - \frac{59706515989933784475179508991533990661194952293}{21115670851664907521585924302597355791810264592060} a^{12} - \frac{3734429061495177380855617766770712302954597282957}{190041037664984167694273318723376202126292381328540} a^{11} - \frac{2285930135966201281183077276747353488857347332504}{47510259416246041923568329680844050531573095332135} a^{10} - \frac{39055564681084002069536424760459012371409292828707}{190041037664984167694273318723376202126292381328540} a^{9} + \frac{5390647980215315111686165153315774943312877909869}{63347012554994722564757772907792067375430793776180} a^{8} - \frac{15474705467930995295131064924532664368030486061435}{38008207532996833538854663744675240425258476265708} a^{7} + \frac{21859932961910185028923757255893420998616304379779}{47510259416246041923568329680844050531573095332135} a^{6} + \frac{92642457524065688766493543520660307433956257810999}{190041037664984167694273318723376202126292381328540} a^{5} + \frac{4980188204828871388846219139835175147541671551829}{38008207532996833538854663744675240425258476265708} a^{4} - \frac{15337556794950817553181620608274840413601143652}{49233429446887090076236611068232176716656057339} a^{3} + \frac{13522475601029671869447339240146882846684772983957}{38008207532996833538854663744675240425258476265708} a^{2} - \frac{139698198794137828074925791995721865727806818353}{328222862979247267174910740454881178111040382260} a - \frac{2735385560850365049585497865358979141295847994563}{10557835425832453760792962151298677895905132296030}$

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:  $9$
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:  \( 476480610.902 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1360:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 65 conjugacy class representatives for t16n1360 are not computed
Character table for t16n1360 is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.1948160000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
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}$
761Data not computed