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Properties

Label 16.0.35687701699...5625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{15}\cdot 61^{9}$
Root discriminant $45.66$
Ramified primes $5, 61$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![246011, -1186107, 2634030, -3615625, 3485545, -2539361, 1451972, -653290, 225995, -59860, 13942, -3709, 905, -105, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 105*x^13 + 905*x^12 - 3709*x^11 + 13942*x^10 - 59860*x^9 + 225995*x^8 - 653290*x^7 + 1451972*x^6 - 2539361*x^5 + 3485545*x^4 - 3615625*x^3 + 2634030*x^2 - 1186107*x + 246011)
 
gp: K = bnfinit(x^16 - 3*x^15 - 105*x^13 + 905*x^12 - 3709*x^11 + 13942*x^10 - 59860*x^9 + 225995*x^8 - 653290*x^7 + 1451972*x^6 - 2539361*x^5 + 3485545*x^4 - 3615625*x^3 + 2634030*x^2 - 1186107*x + 246011, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 105 x^{13} + 905 x^{12} - 3709 x^{11} + 13942 x^{10} - 59860 x^{9} + 225995 x^{8} - 653290 x^{7} + 1451972 x^{6} - 2539361 x^{5} + 3485545 x^{4} - 3615625 x^{3} + 2634030 x^{2} - 1186107 x + 246011 \)

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:  \(356877016993229400634765625=5^{15}\cdot 61^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.66$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{54048434947749486163134855839438} a^{15} + \frac{558422846146269536973293068859}{27024217473874743081567427919719} a^{14} + \frac{5070224442009969883960011180602}{27024217473874743081567427919719} a^{13} - \frac{4607676561374275814596863842605}{54048434947749486163134855839438} a^{12} + \frac{26029800520783060363627470047869}{54048434947749486163134855839438} a^{11} + \frac{1378274675776845695521580596985}{27024217473874743081567427919719} a^{10} + \frac{4349306740815768701412579890613}{54048434947749486163134855839438} a^{9} + \frac{9534383174187476794433425724855}{27024217473874743081567427919719} a^{8} + \frac{1478817544424279741197806829943}{54048434947749486163134855839438} a^{7} + \frac{10458549315972465744193500106980}{27024217473874743081567427919719} a^{6} - \frac{21290374569695548227409607832547}{54048434947749486163134855839438} a^{5} + \frac{16397638826816893152751011393135}{54048434947749486163134855839438} a^{4} + \frac{3562483439873006994772632379138}{27024217473874743081567427919719} a^{3} - \frac{9306971110429412571933691446729}{27024217473874743081567427919719} a^{2} + \frac{24415024216633835391642262870043}{54048434947749486163134855839438} a + \frac{11739607823160373155734220413912}{27024217473874743081567427919719}$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( \frac{122485541347720989418419806805}{27024217473874743081567427919719} a^{15} - \frac{285450128632873337373826377134}{27024217473874743081567427919719} a^{14} - \frac{280032530844391181112456922676}{27024217473874743081567427919719} a^{13} - \frac{13057582614754001084186232989458}{27024217473874743081567427919719} a^{12} + \frac{102284944623171376594814934011354}{27024217473874743081567427919719} a^{11} - \frac{375915516840968214386625889059788}{27024217473874743081567427919719} a^{10} + \frac{1405831049303540777682875810120678}{27024217473874743081567427919719} a^{9} - \frac{6239597768865351508851665092922426}{27024217473874743081567427919719} a^{8} + \frac{22850027960812255337263182870659262}{27024217473874743081567427919719} a^{7} - \frac{61778065809929853775006566299773026}{27024217473874743081567427919719} a^{6} + \frac{127038381706351063866324785529189509}{27024217473874743081567427919719} a^{5} - \frac{204056073428238999457880194688819485}{27024217473874743081567427919719} a^{4} + \frac{251286356862232001072324492813804516}{27024217473874743081567427919719} a^{3} - \frac{221165270874462485775574536270377342}{27024217473874743081567427919719} a^{2} + \frac{121808351511246140513849670933471858}{27024217473874743081567427919719} a - \frac{31088876645772954012288249121686236}{27024217473874743081567427919719} \) (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:  \( 2032499.09239 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

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

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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