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Properties

Label 16.0.18492726723...3125.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{9}\cdot 79^{10}$
Root discriminant $37.95$
Ramified primes $5, 79$
Class number $10$ (GRH)
Class group $[10]$ (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![104, -1172, 5710, -9331, 8032, -4001, 4446, 302, -1405, -37, 381, -221, 182, -64, 21, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 21*x^14 - 64*x^13 + 182*x^12 - 221*x^11 + 381*x^10 - 37*x^9 - 1405*x^8 + 302*x^7 + 4446*x^6 - 4001*x^5 + 8032*x^4 - 9331*x^3 + 5710*x^2 - 1172*x + 104)
 
gp: K = bnfinit(x^16 - 4*x^15 + 21*x^14 - 64*x^13 + 182*x^12 - 221*x^11 + 381*x^10 - 37*x^9 - 1405*x^8 + 302*x^7 + 4446*x^6 - 4001*x^5 + 8032*x^4 - 9331*x^3 + 5710*x^2 - 1172*x + 104, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 21 x^{14} - 64 x^{13} + 182 x^{12} - 221 x^{11} + 381 x^{10} - 37 x^{9} - 1405 x^{8} + 302 x^{7} + 4446 x^{6} - 4001 x^{5} + 8032 x^{4} - 9331 x^{3} + 5710 x^{2} - 1172 x + 104 \)

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:  \(18492726723880560939453125=5^{9}\cdot 79^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.95$
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:  $5, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{40} a^{12} + \frac{1}{20} a^{11} - \frac{1}{5} a^{10} + \frac{1}{20} a^{9} - \frac{1}{20} a^{8} + \frac{1}{8} a^{7} + \frac{13}{40} a^{6} + \frac{3}{10} a^{5} - \frac{7}{20} a^{4} + \frac{7}{20} a^{3} - \frac{3}{10} a^{2} + \frac{17}{40} a - \frac{3}{20}$, $\frac{1}{40} a^{13} - \frac{1}{20} a^{11} + \frac{1}{5} a^{10} + \frac{1}{10} a^{9} - \frac{1}{40} a^{8} + \frac{13}{40} a^{7} - \frac{7}{20} a^{6} + \frac{3}{10} a^{5} - \frac{1}{5} a^{4} + \frac{1}{4} a^{3} - \frac{9}{40} a^{2} + \frac{1}{4} a - \frac{1}{5}$, $\frac{1}{800} a^{14} + \frac{9}{800} a^{13} - \frac{1}{100} a^{12} - \frac{41}{400} a^{11} - \frac{29}{200} a^{10} - \frac{157}{800} a^{9} + \frac{19}{200} a^{8} - \frac{127}{800} a^{7} + \frac{3}{50} a^{6} - \frac{83}{200} a^{5} - \frac{39}{400} a^{4} + \frac{257}{800} a^{3} + \frac{361}{800} a^{2} + \frac{3}{10} a + \frac{81}{200}$, $\frac{1}{1788984712052682007115200} a^{15} - \frac{239570631740978203437}{447246178013170501778800} a^{14} - \frac{14916126094257776957501}{1788984712052682007115200} a^{13} + \frac{10796656448216485645847}{894492356026341003557600} a^{12} - \frac{89966750163698183684221}{894492356026341003557600} a^{11} + \frac{74987162887554012745451}{357796942410536401423040} a^{10} + \frac{63260912802380461993089}{357796942410536401423040} a^{9} - \frac{113964058423190728111419}{1788984712052682007115200} a^{8} - \frac{458341875075681325095253}{1788984712052682007115200} a^{7} - \frac{119058092042420657194897}{447246178013170501778800} a^{6} - \frac{54095434086850359813537}{894492356026341003557600} a^{5} + \frac{858063069311615570703463}{1788984712052682007115200} a^{4} + \frac{119368973452239529401023}{447246178013170501778800} a^{3} + \frac{597799510697140271171243}{1788984712052682007115200} a^{2} - \frac{168071518286600424555629}{447246178013170501778800} a - \frac{14038995733059948274749}{34403552154859269367600}$

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

Class group and class number

$C_{10}$, which has order $10$ (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:  \( 4985395.89387 \) (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

\(\Q(\sqrt{-79}) \), 4.0.31205.1, 8.0.4868760125.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{4}$ $16$ R $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ $16$ $16$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$79$79.8.4.1$x^{8} + 37446 x^{4} - 493039 x^{2} + 350550729$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
79.8.6.2$x^{8} + 395 x^{4} + 56169$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$