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Properties

Label 16.8.16726428348...0000.5
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{8}\cdot 761^{5}$
Root discriminant $50.29$
Ramified primes $2, 5, 761$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2.D_4^2.C_2$ (as 16T659)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![263009, 71066, -506387, 57532, 194160, -27434, -15494, 3314, -5584, -2954, 685, 838, 181, -62, -32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 32*x^14 - 62*x^13 + 181*x^12 + 838*x^11 + 685*x^10 - 2954*x^9 - 5584*x^8 + 3314*x^7 - 15494*x^6 - 27434*x^5 + 194160*x^4 + 57532*x^3 - 506387*x^2 + 71066*x + 263009)
 
gp: K = bnfinit(x^16 - 32*x^14 - 62*x^13 + 181*x^12 + 838*x^11 + 685*x^10 - 2954*x^9 - 5584*x^8 + 3314*x^7 - 15494*x^6 - 27434*x^5 + 194160*x^4 + 57532*x^3 - 506387*x^2 + 71066*x + 263009, 1)
 

Normalized defining polynomial

\( x^{16} - 32 x^{14} - 62 x^{13} + 181 x^{12} + 838 x^{11} + 685 x^{10} - 2954 x^{9} - 5584 x^{8} + 3314 x^{7} - 15494 x^{6} - 27434 x^{5} + 194160 x^{4} + 57532 x^{3} - 506387 x^{2} + 71066 x + 263009 \)

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:  \(1672642834856679833600000000=2^{24}\cdot 5^{8}\cdot 761^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.29$
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:  $2, 5, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{195139189869553582010858520852176968049369} a^{15} + \frac{67241896828446830963687089648505635976014}{195139189869553582010858520852176968049369} a^{14} + \frac{34728752935219708761130914589732423493422}{195139189869553582010858520852176968049369} a^{13} + \frac{95592169449113283307597969854247742479224}{195139189869553582010858520852176968049369} a^{12} - \frac{17246323754042125970571282389112648104564}{195139189869553582010858520852176968049369} a^{11} + \frac{10680322444560641988471184701950878877934}{195139189869553582010858520852176968049369} a^{10} + \frac{39076973648973669040074830718309994603847}{195139189869553582010858520852176968049369} a^{9} + \frac{89719956250313168986163350216238733931875}{195139189869553582010858520852176968049369} a^{8} + \frac{34157450224381952656445999301650446826675}{195139189869553582010858520852176968049369} a^{7} - \frac{88537186471215844419371021998593808287200}{195139189869553582010858520852176968049369} a^{6} + \frac{60805712647547825804652230755614184091268}{195139189869553582010858520852176968049369} a^{5} - \frac{25528894240533763256137205410635342371035}{195139189869553582010858520852176968049369} a^{4} - \frac{70907196601982851491365952707513671952589}{195139189869553582010858520852176968049369} a^{3} - \frac{82215299900925973472553965828186544130011}{195139189869553582010858520852176968049369} a^{2} - \frac{36866638328135968695276881044873237691113}{195139189869553582010858520852176968049369} a - \frac{65362255456400459339891028360048293641349}{195139189869553582010858520852176968049369}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 18606574.5705 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2.D_4^2.C_2$ (as 16T659):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 25 conjugacy class representatives for $C_2.D_4^2.C_2$
Character table for $C_2.D_4^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.1948160000.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$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}$
761Data not computed