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Properties

Label 16.4.98340377973...249.36
Degree $16$
Signature $[4, 6]$
Discriminant $13^{10}\cdot 61^{10}$
Root discriminant $64.87$
Ramified primes $13, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T875

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![168464, -151560, 156920, 19720, -93281, 116978, -56404, -5942, 26238, -14232, 224, 1862, -644, 126, 2, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 2*x^14 + 126*x^13 - 644*x^12 + 1862*x^11 + 224*x^10 - 14232*x^9 + 26238*x^8 - 5942*x^7 - 56404*x^6 + 116978*x^5 - 93281*x^4 + 19720*x^3 + 156920*x^2 - 151560*x + 168464)
 
gp: K = bnfinit(x^16 - 8*x^15 + 2*x^14 + 126*x^13 - 644*x^12 + 1862*x^11 + 224*x^10 - 14232*x^9 + 26238*x^8 - 5942*x^7 - 56404*x^6 + 116978*x^5 - 93281*x^4 + 19720*x^3 + 156920*x^2 - 151560*x + 168464, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 2 x^{14} + 126 x^{13} - 644 x^{12} + 1862 x^{11} + 224 x^{10} - 14232 x^{9} + 26238 x^{8} - 5942 x^{7} - 56404 x^{6} + 116978 x^{5} - 93281 x^{4} + 19720 x^{3} + 156920 x^{2} - 151560 x + 168464 \)

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:  \(98340377973019428085802419249=13^{10}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.87$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{12} a^{9} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{4} a^{3} - \frac{1}{3}$, $\frac{1}{12} a^{10} + \frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{3} a$, $\frac{1}{12} a^{11} + \frac{1}{12} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{4} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} + \frac{5}{12} a^{2} + \frac{1}{3}$, $\frac{1}{19032} a^{12} - \frac{1}{3172} a^{11} - \frac{17}{793} a^{10} + \frac{509}{19032} a^{9} + \frac{1111}{9516} a^{8} + \frac{85}{2379} a^{7} - \frac{113}{1464} a^{6} + \frac{1481}{9516} a^{5} - \frac{7}{4758} a^{4} - \frac{4081}{19032} a^{3} + \frac{701}{3172} a^{2} + \frac{439}{4758} a - \frac{890}{2379}$, $\frac{1}{19032} a^{13} - \frac{37}{1586} a^{11} - \frac{353}{19032} a^{10} + \frac{259}{9516} a^{9} + \frac{331}{4758} a^{8} - \frac{187}{6344} a^{7} + \frac{458}{2379} a^{6} + \frac{157}{1586} a^{5} + \frac{509}{19032} a^{4} - \frac{77}{244} a^{3} - \frac{389}{4758} a^{2} + \frac{9}{26} a - \frac{194}{793}$, $\frac{1}{9913239386856} a^{14} - \frac{7}{9913239386856} a^{13} - \frac{9425363}{413051641119} a^{12} + \frac{1357252363}{9913239386856} a^{11} + \frac{122328025793}{9913239386856} a^{10} + \frac{2805856571}{137683880373} a^{9} + \frac{107586193979}{3304413128952} a^{8} + \frac{272059512937}{3304413128952} a^{7} + \frac{2317457845}{21182135442} a^{6} + \frac{340727869777}{9913239386856} a^{5} + \frac{388368310789}{3304413128952} a^{4} - \frac{1136350474115}{2478309846714} a^{3} - \frac{644143823735}{4956619693428} a^{2} - \frac{381386130481}{2478309846714} a - \frac{616481482274}{1239154923357}$, $\frac{1}{386616336087384} a^{15} + \frac{1}{32218028007282} a^{14} - \frac{2309697577}{386616336087384} a^{13} - \frac{93385111}{3717464770071} a^{12} - \frac{167594642401}{21478685338188} a^{11} + \frac{11807154719873}{386616336087384} a^{10} - \frac{1712987902313}{64436056014564} a^{9} - \frac{1199733557377}{21478685338188} a^{8} + \frac{27516321765931}{128872112029128} a^{7} - \frac{1016396041603}{48327042010923} a^{6} - \frac{8584582936897}{193308168043692} a^{5} - \frac{56802498590645}{386616336087384} a^{4} + \frac{28775024430479}{128872112029128} a^{3} + \frac{23532470003693}{193308168043692} a^{2} - \frac{46688399299835}{96654084021846} a - \frac{5300169846110}{48327042010923}$

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:  \( 255676212.326 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T875:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n875
Character table for t16n875 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.10309.1, 8.4.395451064801.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$
61.8.7.2$x^{8} - 244$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$