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Properties

Label 16.0.50345559205...8125.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{6}\cdot 181^{5}$
Root discriminant $95.80$
Ramified primes $5, 101, 181$
Class number $175584$ (GRH)
Class group $[2, 2, 2, 2, 10974]$ (GRH)
Galois group 16T994

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1187058641, -794857148, 959344879, -428392404, 311549332, -95434883, 53717753, -11468099, 5362186, -802681, 314218, -32647, 10352, -721, 169, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 169*x^14 - 721*x^13 + 10352*x^12 - 32647*x^11 + 314218*x^10 - 802681*x^9 + 5362186*x^8 - 11468099*x^7 + 53717753*x^6 - 95434883*x^5 + 311549332*x^4 - 428392404*x^3 + 959344879*x^2 - 794857148*x + 1187058641)
 
gp: K = bnfinit(x^16 - 7*x^15 + 169*x^14 - 721*x^13 + 10352*x^12 - 32647*x^11 + 314218*x^10 - 802681*x^9 + 5362186*x^8 - 11468099*x^7 + 53717753*x^6 - 95434883*x^5 + 311549332*x^4 - 428392404*x^3 + 959344879*x^2 - 794857148*x + 1187058641, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 169 x^{14} - 721 x^{13} + 10352 x^{12} - 32647 x^{11} + 314218 x^{10} - 802681 x^{9} + 5362186 x^{8} - 11468099 x^{7} + 53717753 x^{6} - 95434883 x^{5} + 311549332 x^{4} - 428392404 x^{3} + 959344879 x^{2} - 794857148 x + 1187058641 \)

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:  \(50345559205004654867065673828125=5^{12}\cdot 101^{6}\cdot 181^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.80$
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:  $5, 101, 181$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{9} - \frac{1}{5} a^{6} - \frac{2}{5} a^{3} + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{10} - \frac{1}{5} a^{7} - \frac{2}{5} a^{4} + \frac{1}{5} a$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{11} - \frac{1}{5} a^{8} - \frac{2}{5} a^{5} + \frac{1}{5} a^{2}$, $\frac{1}{538820043693974274098984830632498195682045240320268399195} a^{15} + \frac{3733686060564122586940948531020099863833901460798657493}{107764008738794854819796966126499639136409048064053679839} a^{14} + \frac{8900348904666767013698551990929263931905560209547180358}{107764008738794854819796966126499639136409048064053679839} a^{13} + \frac{5106006542706654788886277132141892029328282666801045417}{107764008738794854819796966126499639136409048064053679839} a^{12} - \frac{19666729329717509605168851870809323252263861333728835062}{107764008738794854819796966126499639136409048064053679839} a^{11} - \frac{31594617661381115023644581110055210742847568338331723244}{107764008738794854819796966126499639136409048064053679839} a^{10} + \frac{26763177884188414247812912728745162387057159884087055001}{107764008738794854819796966126499639136409048064053679839} a^{9} + \frac{14044751410441201524958565635130584066456185370427217572}{107764008738794854819796966126499639136409048064053679839} a^{8} - \frac{32599559548118307676458345641894460570102364380302731212}{107764008738794854819796966126499639136409048064053679839} a^{7} - \frac{15927813857264484825148784367866951257224208016385263808}{107764008738794854819796966126499639136409048064053679839} a^{6} + \frac{25065093594773003225420907687254962289458542303913529519}{107764008738794854819796966126499639136409048064053679839} a^{5} - \frac{21693179982875796384267841414899874487599963704359053480}{107764008738794854819796966126499639136409048064053679839} a^{4} - \frac{23483968505074497626122266318461099145875611842635941923}{107764008738794854819796966126499639136409048064053679839} a^{3} + \frac{12130904811045582849081867832005000620327304643373127611}{107764008738794854819796966126499639136409048064053679839} a^{2} + \frac{21088101576983393240456621965967363690989002660153618176}{107764008738794854819796966126499639136409048064053679839} a - \frac{195125038696807852567886749382930548655589534375986106822}{538820043693974274098984830632498195682045240320268399195}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{10974}$, which has order $175584$ (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:  \( 6824.94222592 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T994:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 29 conjugacy class representatives for t16n994
Character table for t16n994 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.8.1153988125.1

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
Arithmetically equvalently 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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
101Data not computed
181Data not computed