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Properties

Label 16.4.32667140577...5625.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{4}\cdot 13^{6}\cdot 101^{8}$
Root discriminant $39.32$
Ramified primes $5, 13, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1276

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-225, 1200, -2195, 945, 712, 802, -1086, -773, 297, 349, 123, -57, -13, 11, -9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 9*x^14 + 11*x^13 - 13*x^12 - 57*x^11 + 123*x^10 + 349*x^9 + 297*x^8 - 773*x^7 - 1086*x^6 + 802*x^5 + 712*x^4 + 945*x^3 - 2195*x^2 + 1200*x - 225)
 
gp: K = bnfinit(x^16 - x^15 - 9*x^14 + 11*x^13 - 13*x^12 - 57*x^11 + 123*x^10 + 349*x^9 + 297*x^8 - 773*x^7 - 1086*x^6 + 802*x^5 + 712*x^4 + 945*x^3 - 2195*x^2 + 1200*x - 225, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 9 x^{14} + 11 x^{13} - 13 x^{12} - 57 x^{11} + 123 x^{10} + 349 x^{9} + 297 x^{8} - 773 x^{7} - 1086 x^{6} + 802 x^{5} + 712 x^{4} + 945 x^{3} - 2195 x^{2} + 1200 x - 225 \)

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:  \(32667140577724797996255625=5^{4}\cdot 13^{6}\cdot 101^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.32$
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:  $5, 13, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} + \frac{4}{13} a^{11} - \frac{1}{13} a^{8} + \frac{4}{13} a^{7} - \frac{2}{13} a^{6} + \frac{2}{13} a^{5} - \frac{4}{13} a^{4} + \frac{4}{13} a^{3} - \frac{1}{13} a^{2} + \frac{4}{13} a + \frac{3}{13}$, $\frac{1}{65} a^{13} + \frac{1}{65} a^{12} - \frac{12}{65} a^{11} - \frac{1}{5} a^{10} - \frac{14}{65} a^{9} + \frac{4}{13} a^{8} - \frac{27}{65} a^{7} - \frac{1}{13} a^{6} - \frac{23}{65} a^{5} + \frac{16}{65} a^{4} + \frac{2}{5} a^{3} - \frac{6}{65} a^{2} + \frac{6}{13} a + \frac{6}{13}$, $\frac{1}{3315} a^{14} - \frac{2}{1105} a^{13} + \frac{12}{1105} a^{12} + \frac{1526}{3315} a^{11} + \frac{1312}{3315} a^{10} + \frac{1483}{3315} a^{9} - \frac{872}{3315} a^{8} - \frac{1351}{3315} a^{7} - \frac{163}{3315} a^{6} + \frac{399}{1105} a^{5} + \frac{483}{1105} a^{4} + \frac{292}{3315} a^{3} - \frac{178}{3315} a^{2} - \frac{83}{663} a - \frac{16}{221}$, $\frac{1}{164909648073054105} a^{15} - \frac{1072914546277}{54969882691018035} a^{14} + \frac{94658969347763}{18323294230339345} a^{13} - \frac{5560233127401778}{164909648073054105} a^{12} + \frac{4005119321212322}{32981929614610821} a^{11} - \frac{14598902311544842}{32981929614610821} a^{10} + \frac{13385750430367969}{164909648073054105} a^{9} - \frac{34180083729787711}{164909648073054105} a^{8} + \frac{10529046480688483}{32981929614610821} a^{7} + \frac{22671478638793669}{54969882691018035} a^{6} + \frac{9423239200901722}{54969882691018035} a^{5} - \frac{61778931809836937}{164909648073054105} a^{4} + \frac{26920438714009388}{164909648073054105} a^{3} + \frac{77586899213349044}{164909648073054105} a^{2} - \frac{359476695161697}{3664658846067869} a + \frac{258982578796777}{3664658846067869}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 4230626.67125 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1276:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1276
Character table for t16n1276 is not computed

Intermediate fields

\(\Q(\sqrt{101}) \), 4.4.132613.1, 8.4.87931038845.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.3.4$x^{4} + 104$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$101$101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$