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Properties

Label 16.12.1442115336...8721.1
Degree $16$
Signature $[12, 2]$
Discriminant $47^{2}\cdot 97^{14}$
Root discriminant $88.60$
Ramified primes $47, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T258)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![89731, 871865, 1402136, -3495249, -776294, 2730202, -423555, -396280, 75830, 11647, 964, -1110, -361, 224, -12, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 12*x^14 + 224*x^13 - 361*x^12 - 1110*x^11 + 964*x^10 + 11647*x^9 + 75830*x^8 - 396280*x^7 - 423555*x^6 + 2730202*x^5 - 776294*x^4 - 3495249*x^3 + 1402136*x^2 + 871865*x + 89731)
 
gp: K = bnfinit(x^16 - 8*x^15 - 12*x^14 + 224*x^13 - 361*x^12 - 1110*x^11 + 964*x^10 + 11647*x^9 + 75830*x^8 - 396280*x^7 - 423555*x^6 + 2730202*x^5 - 776294*x^4 - 3495249*x^3 + 1402136*x^2 + 871865*x + 89731, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 12 x^{14} + 224 x^{13} - 361 x^{12} - 1110 x^{11} + 964 x^{10} + 11647 x^{9} + 75830 x^{8} - 396280 x^{7} - 423555 x^{6} + 2730202 x^{5} - 776294 x^{4} - 3495249 x^{3} + 1402136 x^{2} + 871865 x + 89731 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14421153369104822203810274058721=47^{2}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.60$
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:  $47, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{12} a^{12} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{12} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{5}{12} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{12} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{5}{12}$, $\frac{1}{34404} a^{13} + \frac{1427}{34404} a^{12} + \frac{26}{183} a^{11} + \frac{1097}{11468} a^{10} + \frac{5579}{34404} a^{9} + \frac{497}{17202} a^{8} + \frac{8831}{34404} a^{7} + \frac{4357}{34404} a^{6} - \frac{919}{2867} a^{5} - \frac{1627}{34404} a^{4} - \frac{13703}{34404} a^{3} - \frac{797}{5734} a^{2} + \frac{11959}{34404} a - \frac{67}{188}$, $\frac{1}{56105324376580542612} a^{14} - \frac{7}{56105324376580542612} a^{13} - \frac{194411194121833227}{4675443698048378551} a^{12} - \frac{1568056271807173923}{18701774792193514204} a^{11} - \frac{259509892196878521}{18701774792193514204} a^{10} + \frac{3246841903948923211}{28052662188290271306} a^{9} - \frac{3614691208062751889}{56105324376580542612} a^{8} + \frac{17259678976194760871}{56105324376580542612} a^{7} - \frac{88127725649251750}{229939854002379273} a^{6} + \frac{277379545242283195}{18701774792193514204} a^{5} - \frac{20853411911519946913}{56105324376580542612} a^{4} - \frac{12579257422687741199}{28052662188290271306} a^{3} + \frac{13068786977354817895}{56105324376580542612} a^{2} + \frac{3887577863288461081}{56105324376580542612} a + \frac{22007610651551920}{76646618000793091}$, $\frac{1}{3422424786971413099332} a^{15} + \frac{23}{3422424786971413099332} a^{14} - \frac{44951243183005175}{3422424786971413099332} a^{13} + \frac{71738718822574117087}{3422424786971413099332} a^{12} - \frac{19248535858121824905}{1140808262323804366444} a^{11} - \frac{23559849260040923107}{1140808262323804366444} a^{10} - \frac{504748945301903026097}{3422424786971413099332} a^{9} - \frac{123239376212927562733}{1140808262323804366444} a^{8} + \frac{223022441705321231479}{3422424786971413099332} a^{7} + \frac{473864573395199250505}{3422424786971413099332} a^{6} - \frac{513371340440728613029}{3422424786971413099332} a^{5} + \frac{2704564197379578301}{26530424705204752708} a^{4} + \frac{517300904222479259603}{3422424786971413099332} a^{3} + \frac{303217709605437430781}{1140808262323804366444} a^{2} + \frac{956908205583479546275}{3422424786971413099332} a - \frac{5276993720475821645}{28052662188290271306}$

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:  $13$
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:  \( 11390078611.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T258):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.6.39149684231663.1, 8.8.80798284478113.1, 8.6.3797519370471311.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$47$47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97Data not computed