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Properties

Label 16.0.15638154465...9601.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 61^{8}$
Root discriminant $28.16$
Ramified primes $13, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2.SD_{16}$ (as 16T163)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13, 117, 478, 997, 1274, 1382, 1434, 1057, 596, 350, 128, 20, -3, -7, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - x^14 - 7*x^13 - 3*x^12 + 20*x^11 + 128*x^10 + 350*x^9 + 596*x^8 + 1057*x^7 + 1434*x^6 + 1382*x^5 + 1274*x^4 + 997*x^3 + 478*x^2 + 117*x + 13)
 
gp: K = bnfinit(x^16 - x^15 - x^14 - 7*x^13 - 3*x^12 + 20*x^11 + 128*x^10 + 350*x^9 + 596*x^8 + 1057*x^7 + 1434*x^6 + 1382*x^5 + 1274*x^4 + 997*x^3 + 478*x^2 + 117*x + 13, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - x^{14} - 7 x^{13} - 3 x^{12} + 20 x^{11} + 128 x^{10} + 350 x^{9} + 596 x^{8} + 1057 x^{7} + 1434 x^{6} + 1382 x^{5} + 1274 x^{4} + 997 x^{3} + 478 x^{2} + 117 x + 13 \)

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:  \(156381544652244701169601=13^{8}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.16$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{13} a^{13} - \frac{6}{13} a^{12} - \frac{1}{13} a^{11} + \frac{1}{13} a^{10} + \frac{5}{13} a^{9} - \frac{1}{13} a^{8} + \frac{1}{13} a^{7} - \frac{3}{13} a^{6} + \frac{4}{13} a^{5} + \frac{1}{13} a^{4} - \frac{6}{13} a^{3} - \frac{2}{13} a^{2}$, $\frac{1}{39} a^{14} + \frac{1}{39} a^{13} - \frac{4}{39} a^{12} + \frac{7}{39} a^{11} - \frac{14}{39} a^{10} + \frac{8}{39} a^{9} + \frac{7}{39} a^{8} - \frac{3}{13} a^{7} + \frac{3}{13} a^{6} + \frac{16}{39} a^{5} + \frac{14}{39} a^{4} - \frac{5}{39} a^{3} + \frac{4}{13} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{1217361053706135} a^{15} - \frac{10711573534214}{1217361053706135} a^{14} + \frac{15141823248086}{1217361053706135} a^{13} - \frac{524996061820}{243472210741227} a^{12} + \frac{371154532501162}{1217361053706135} a^{11} + \frac{586217762863334}{1217361053706135} a^{10} + \frac{34930888604707}{93643157977395} a^{9} + \frac{173216125615979}{405787017902045} a^{8} + \frac{34846899166289}{81157403580409} a^{7} - \frac{190893295923818}{1217361053706135} a^{6} - \frac{344958375719587}{1217361053706135} a^{5} + \frac{10938026512648}{1217361053706135} a^{4} - \frac{33283749002091}{81157403580409} a^{3} - \frac{340929762885823}{1217361053706135} a^{2} + \frac{42657200445929}{93643157977395} a - \frac{10874553048676}{31214385992465}$

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:  \( 240179.93942 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.SD_{16}$ (as 16T163):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $C_2^2.SD_{16}$
Character table for $C_2^2.SD_{16}$

Intermediate fields

\(\Q(\sqrt{61}) \), 4.4.48373.1, 8.0.30419312677.1, 8.4.30419312677.1, 8.4.395451064801.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$61$61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$