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Properties

Label 16.0.96313927189...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{8}\cdot 149^{4}$
Root discriminant $42.07$
Ramified primes $5, 29, 149$
Class number $192$ (GRH)
Class group $[2, 96]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T373)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7025, -26460, 52696, -76722, 93776, -85970, 58931, -32331, 16213, -7340, 3011, -735, 344, -66, 31, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 31*x^14 - 66*x^13 + 344*x^12 - 735*x^11 + 3011*x^10 - 7340*x^9 + 16213*x^8 - 32331*x^7 + 58931*x^6 - 85970*x^5 + 93776*x^4 - 76722*x^3 + 52696*x^2 - 26460*x + 7025)
 
gp: K = bnfinit(x^16 - 3*x^15 + 31*x^14 - 66*x^13 + 344*x^12 - 735*x^11 + 3011*x^10 - 7340*x^9 + 16213*x^8 - 32331*x^7 + 58931*x^6 - 85970*x^5 + 93776*x^4 - 76722*x^3 + 52696*x^2 - 26460*x + 7025, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 31 x^{14} - 66 x^{13} + 344 x^{12} - 735 x^{11} + 3011 x^{10} - 7340 x^{9} + 16213 x^{8} - 32331 x^{7} + 58931 x^{6} - 85970 x^{5} + 93776 x^{4} - 76722 x^{3} + 52696 x^{2} - 26460 x + 7025 \)

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:  \(96313927189328563031640625=5^{8}\cdot 29^{8}\cdot 149^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.07$
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:  $5, 29, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{8} - \frac{1}{3} a^{7} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{30} a^{11} - \frac{1}{10} a^{9} - \frac{4}{15} a^{8} + \frac{1}{5} a^{7} - \frac{11}{30} a^{6} + \frac{11}{30} a^{5} - \frac{4}{15} a^{4} - \frac{13}{30} a^{3} - \frac{1}{6} a^{2} + \frac{11}{30} a$, $\frac{1}{60} a^{12} - \frac{1}{60} a^{11} + \frac{1}{30} a^{10} - \frac{1}{12} a^{9} + \frac{29}{60} a^{8} + \frac{1}{20} a^{7} + \frac{11}{30} a^{6} - \frac{7}{30} a^{5} + \frac{1}{3} a^{4} + \frac{7}{15} a^{3} - \frac{19}{60} a^{2} - \frac{1}{10} a + \frac{5}{12}$, $\frac{1}{2100} a^{13} - \frac{1}{150} a^{12} + \frac{1}{100} a^{11} + \frac{19}{2100} a^{10} - \frac{22}{175} a^{9} - \frac{88}{525} a^{8} + \frac{659}{2100} a^{7} - \frac{23}{50} a^{6} - \frac{17}{350} a^{5} - \frac{73}{210} a^{4} + \frac{1019}{2100} a^{3} - \frac{199}{2100} a^{2} + \frac{499}{2100} a - \frac{19}{420}$, $\frac{1}{31222800} a^{14} + \frac{647}{15611400} a^{13} + \frac{4777}{743400} a^{12} - \frac{127429}{15611400} a^{11} + \frac{38356}{650475} a^{10} - \frac{2577929}{31222800} a^{9} - \frac{2101837}{5203800} a^{8} - \frac{14008639}{31222800} a^{7} - \frac{34729}{130095} a^{6} - \frac{7728083}{15611400} a^{5} + \frac{2819389}{31222800} a^{4} - \frac{495941}{4460400} a^{3} + \frac{7397291}{15611400} a^{2} - \frac{8498923}{31222800} a - \frac{136537}{6244560}$, $\frac{1}{1572578351685773485200} a^{15} + \frac{1972472175913}{314515670337154697040} a^{14} + \frac{12697331341148353}{98286146980360842825} a^{13} + \frac{1355628943178046017}{196572293960721685650} a^{12} + \frac{2262610080422195401}{157257835168577348520} a^{11} - \frac{57118312981538979941}{1572578351685773485200} a^{10} - \frac{1680452600922765023}{224654050240824783600} a^{9} - \frac{10222878072585380941}{1572578351685773485200} a^{8} - \frac{318954534844712407609}{1572578351685773485200} a^{7} - \frac{97298359622084344763}{786289175842886742600} a^{6} + \frac{193136430323372857201}{524192783895257828400} a^{5} + \frac{88869980584571780539}{196572293960721685650} a^{4} + \frac{146920046312659204033}{314515670337154697040} a^{3} - \frac{24320710579847557969}{174730927965085942800} a^{2} - \frac{17727767935718547631}{87365463982542971400} a - \frac{126773056036903794607}{314515670337154697040}$

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

Class group and class number

$C_{2}\times C_{96}$, which has order $192$ (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:  \( 100117.930697 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T373):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), 4.0.108025.1, 4.0.108025.2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.4.65865543125.1, 8.0.9813965925625.1, 8.4.65865543125.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$149$149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$