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Properties

Label 16.0.22917895064...5877.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{9}\cdot 43^{10}$
Root discriminant $44.41$
Ramified primes $13, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2^3\times C_4).D_4$ (as 16T675)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, -290, 1089, -928, -1219, 2939, -1863, -942, 2743, -2663, 1587, -440, -61, 51, -2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 2*x^14 + 51*x^13 - 61*x^12 - 440*x^11 + 1587*x^10 - 2663*x^9 + 2743*x^8 - 942*x^7 - 1863*x^6 + 2939*x^5 - 1219*x^4 - 928*x^3 + 1089*x^2 - 290*x + 31)
 
gp: K = bnfinit(x^16 - 2*x^15 - 2*x^14 + 51*x^13 - 61*x^12 - 440*x^11 + 1587*x^10 - 2663*x^9 + 2743*x^8 - 942*x^7 - 1863*x^6 + 2939*x^5 - 1219*x^4 - 928*x^3 + 1089*x^2 - 290*x + 31, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 2 x^{14} + 51 x^{13} - 61 x^{12} - 440 x^{11} + 1587 x^{10} - 2663 x^{9} + 2743 x^{8} - 942 x^{7} - 1863 x^{6} + 2939 x^{5} - 1219 x^{4} - 928 x^{3} + 1089 x^{2} - 290 x + 31 \)

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:  \(229178950640823408091275877=13^{9}\cdot 43^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.41$
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:  $13, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{1161} a^{14} + \frac{91}{1161} a^{13} + \frac{118}{387} a^{12} - \frac{424}{1161} a^{11} + \frac{95}{1161} a^{10} - \frac{85}{1161} a^{9} - \frac{548}{1161} a^{8} + \frac{401}{1161} a^{7} + \frac{146}{387} a^{6} - \frac{563}{1161} a^{5} - \frac{61}{387} a^{4} + \frac{202}{1161} a^{3} - \frac{412}{1161} a^{2} + \frac{13}{1161} a + \frac{250}{1161}$, $\frac{1}{7486747353580741243299} a^{15} - \frac{1175562218208050602}{7486747353580741243299} a^{14} + \frac{329703059636529641797}{7486747353580741243299} a^{13} + \frac{3457518446226874548881}{7486747353580741243299} a^{12} + \frac{2375310696454427798515}{7486747353580741243299} a^{11} - \frac{1051589574238782767291}{7486747353580741243299} a^{10} - \frac{138806716242215656051}{2495582451193580414433} a^{9} + \frac{714281942900967807230}{2495582451193580414433} a^{8} + \frac{287470598527045470593}{7486747353580741243299} a^{7} + \frac{2632451233383207925525}{7486747353580741243299} a^{6} + \frac{2897016045769070683102}{7486747353580741243299} a^{5} + \frac{2427313110718020701590}{7486747353580741243299} a^{4} - \frac{336867817457131627072}{2495582451193580414433} a^{3} - \frac{275939859254323167361}{2495582451193580414433} a^{2} - \frac{2882273433491725413937}{7486747353580741243299} a - \frac{1735878712915010230121}{7486747353580741243299}$

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:  $7$
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:  \( 7044960.06615 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-43}) \), 4.0.24037.1, 8.0.7511105797.1

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 $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$43$43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.4.3.1$x^{4} + 387$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
43.4.3.1$x^{4} + 387$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$