↑

Properties

Label 16.8.54566491401...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{8}\cdot 13^{6}\cdot 29^{7}$
Root discriminant $72.20$
Ramified primes $2, 5, 13, 29$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T1558

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![703921, 805466, -877846, -1275944, 436787, 856700, -174982, -294992, 69843, 47012, -14066, -3082, 1143, 88, -36, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 36*x^14 + 88*x^13 + 1143*x^12 - 3082*x^11 - 14066*x^10 + 47012*x^9 + 69843*x^8 - 294992*x^7 - 174982*x^6 + 856700*x^5 + 436787*x^4 - 1275944*x^3 - 877846*x^2 + 805466*x + 703921)
 
gp: K = bnfinit(x^16 - 4*x^15 - 36*x^14 + 88*x^13 + 1143*x^12 - 3082*x^11 - 14066*x^10 + 47012*x^9 + 69843*x^8 - 294992*x^7 - 174982*x^6 + 856700*x^5 + 436787*x^4 - 1275944*x^3 - 877846*x^2 + 805466*x + 703921, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 36 x^{14} + 88 x^{13} + 1143 x^{12} - 3082 x^{11} - 14066 x^{10} + 47012 x^{9} + 69843 x^{8} - 294992 x^{7} - 174982 x^{6} + 856700 x^{5} + 436787 x^{4} - 1275944 x^{3} - 877846 x^{2} + 805466 x + 703921 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(545664914012032080281600000000=2^{24}\cdot 5^{8}\cdot 13^{6}\cdot 29^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.20$
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:  $2, 5, 13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{781} a^{14} + \frac{259}{781} a^{13} - \frac{195}{781} a^{12} + \frac{267}{781} a^{11} - \frac{375}{781} a^{10} + \frac{233}{781} a^{9} + \frac{30}{71} a^{8} + \frac{12}{71} a^{7} - \frac{106}{781} a^{6} - \frac{9}{781} a^{5} - \frac{333}{781} a^{4} - \frac{327}{781} a^{3} + \frac{334}{781} a^{2} + \frac{35}{71} a - \frac{42}{781}$, $\frac{1}{199648135347243241272087504906962833109} a^{15} + \frac{68950640394668612497521941232716703}{199648135347243241272087504906962833109} a^{14} - \frac{84595747631545478268901922469772265456}{199648135347243241272087504906962833109} a^{13} + \frac{20851588870451478236600695446011558344}{199648135347243241272087504906962833109} a^{12} + \frac{66402410095298468095051550704392051478}{199648135347243241272087504906962833109} a^{11} - \frac{43946101565457096842357863735302179166}{199648135347243241272087504906962833109} a^{10} + \frac{6423766855366747348046103219957989111}{199648135347243241272087504906962833109} a^{9} - \frac{8462784942679694699740861140372522599}{18149830486113021933826136809723893919} a^{8} + \frac{22981770444732542223567652122546508883}{199648135347243241272087504906962833109} a^{7} + \frac{54460223711052825626103034581156359894}{199648135347243241272087504906962833109} a^{6} - \frac{80004261162588862210323958566901591212}{199648135347243241272087504906962833109} a^{5} + \frac{75672886867365392240527177534410547364}{199648135347243241272087504906962833109} a^{4} - \frac{186201294470234846269451561247078772}{2811945568271031567212500069112152579} a^{3} + \frac{24021167698924225392328577254567324419}{199648135347243241272087504906962833109} a^{2} + \frac{49125917709847516674501471557211074624}{199648135347243241272087504906962833109} a - \frac{21504533256023037769861544309326379433}{199648135347243241272087504906962833109}$

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

Class group and class number

$C_{4}$, 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:  $11$
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:  \( 291693993.029 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1558:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 94 conjugacy class representatives for t16n1558 are not computed
Character table for t16n1558 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.4.659478560000.2

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 $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ R $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
13.8.6.4$x^{8} - 13 x^{4} + 338$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
$29$29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
29.8.7.3$x^{8} + 58$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$