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Properties

Label 16.8.12010916375...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 5^{8}\cdot 7^{4}\cdot 29^{8}$
Root discriminant $27.70$
Ramified primes $2, 5, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4:C_2^2$ (as 16T119)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4519, -11092, -2816, 10232, 5581, -2982, -2690, -471, 390, 452, 153, -88, -54, 27, 2, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 2*x^14 + 27*x^13 - 54*x^12 - 88*x^11 + 153*x^10 + 452*x^9 + 390*x^8 - 471*x^7 - 2690*x^6 - 2982*x^5 + 5581*x^4 + 10232*x^3 - 2816*x^2 - 11092*x - 4519)
 
gp: K = bnfinit(x^16 - 5*x^15 + 2*x^14 + 27*x^13 - 54*x^12 - 88*x^11 + 153*x^10 + 452*x^9 + 390*x^8 - 471*x^7 - 2690*x^6 - 2982*x^5 + 5581*x^4 + 10232*x^3 - 2816*x^2 - 11092*x - 4519, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 2 x^{14} + 27 x^{13} - 54 x^{12} - 88 x^{11} + 153 x^{10} + 452 x^{9} + 390 x^{8} - 471 x^{7} - 2690 x^{6} - 2982 x^{5} + 5581 x^{4} + 10232 x^{3} - 2816 x^{2} - 11092 x - 4519 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(120109163751936100000000=2^{8}\cdot 5^{8}\cdot 7^{4}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.70$
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:  $2, 5, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{13} a^{13} + \frac{3}{13} a^{11} + \frac{2}{13} a^{10} - \frac{2}{13} a^{9} + \frac{5}{13} a^{8} + \frac{5}{13} a^{7} + \frac{3}{13} a^{6} + \frac{2}{13} a^{5} + \frac{5}{13} a^{4} - \frac{1}{13} a^{3} + \frac{6}{13} a^{2} + \frac{2}{13} a + \frac{5}{13}$, $\frac{1}{3705} a^{14} - \frac{44}{1235} a^{13} + \frac{21}{247} a^{12} + \frac{393}{1235} a^{11} - \frac{119}{1235} a^{10} + \frac{737}{3705} a^{9} - \frac{239}{3705} a^{8} - \frac{154}{1235} a^{7} + \frac{568}{3705} a^{6} - \frac{161}{741} a^{5} - \frac{216}{1235} a^{4} - \frac{811}{3705} a^{3} - \frac{1609}{3705} a^{2} - \frac{259}{3705} a + \frac{1576}{3705}$, $\frac{1}{57621079174829894857185} a^{15} - \frac{479880909061102057}{11524215834965978971437} a^{14} - \frac{519694493658489651108}{19207026391609964952395} a^{13} + \frac{1204155275899207262223}{19207026391609964952395} a^{12} + \frac{1125598902411233216622}{19207026391609964952395} a^{11} + \frac{12369116249598413381873}{57621079174829894857185} a^{10} - \frac{910160483448353639618}{3841405278321992990479} a^{9} + \frac{1306808207125308112742}{11524215834965978971437} a^{8} - \frac{24469941706944021247826}{57621079174829894857185} a^{7} - \frac{7036896030849787296138}{19207026391609964952395} a^{6} + \frac{4012733651768271020402}{57621079174829894857185} a^{5} + \frac{5905666704754941227138}{57621079174829894857185} a^{4} + \frac{19984425239692451637409}{57621079174829894857185} a^{3} + \frac{488569901372210417371}{4432390705756145758245} a^{2} + \frac{2827443300830542921811}{19207026391609964952395} a - \frac{1070395991448443296337}{3032688377622626045115}$

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:  $11$
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:  \( 372761.232576 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_2^2$ (as 16T119):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^4:C_2^2$
Character table for $C_2^4:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1, 8.4.346567690000.1, 8.4.346567690000.2

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$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}$