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Properties

Label 16.0.96108125139...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 89^{8}$
Root discriminant $31.54$
Ramified primes $5, 89$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -72, 260, -866, 2793, -3876, 5534, -7039, 7060, -18, 793, 44, 142, -31, 12, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 12*x^14 - 31*x^13 + 142*x^12 + 44*x^11 + 793*x^10 - 18*x^9 + 7060*x^8 - 7039*x^7 + 5534*x^6 - 3876*x^5 + 2793*x^4 - 866*x^3 + 260*x^2 - 72*x + 16)
 
gp: K = bnfinit(x^16 - 2*x^15 + 12*x^14 - 31*x^13 + 142*x^12 + 44*x^11 + 793*x^10 - 18*x^9 + 7060*x^8 - 7039*x^7 + 5534*x^6 - 3876*x^5 + 2793*x^4 - 866*x^3 + 260*x^2 - 72*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 12 x^{14} - 31 x^{13} + 142 x^{12} + 44 x^{11} + 793 x^{10} - 18 x^{9} + 7060 x^{8} - 7039 x^{7} + 5534 x^{6} - 3876 x^{5} + 2793 x^{4} - 866 x^{3} + 260 x^{2} - 72 x + 16 \)

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:  \(961081251392109619140625=5^{12}\cdot 89^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.54$
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:  $5, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{5} - \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{6} - \frac{1}{3} a$, $\frac{1}{15} a^{12} + \frac{1}{15} a^{11} + \frac{1}{15} a^{10} - \frac{2}{5} a^{8} + \frac{2}{15} a^{7} + \frac{2}{15} a^{6} - \frac{7}{15} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{4}{15} a^{2} + \frac{1}{3} a - \frac{4}{15}$, $\frac{1}{1493089530} a^{13} + \frac{6237616}{746544765} a^{12} + \frac{29876752}{248848255} a^{11} + \frac{79659117}{497696510} a^{10} + \frac{95118329}{248848255} a^{9} - \frac{189700652}{746544765} a^{8} - \frac{379224851}{1493089530} a^{7} - \frac{8300429}{49769651} a^{6} + \frac{110949459}{248848255} a^{5} + \frac{31188095}{99539302} a^{4} - \frac{6390346}{149308953} a^{3} + \frac{342789383}{746544765} a^{2} - \frac{50487943}{497696510} a - \frac{53664189}{248848255}$, $\frac{1}{2986179060} a^{14} + \frac{2855626}{149308953} a^{12} - \frac{160816937}{995393020} a^{11} + \frac{106602191}{746544765} a^{10} + \frac{244159397}{746544765} a^{9} + \frac{40833959}{199078604} a^{8} + \frac{140798599}{746544765} a^{7} + \frac{123082274}{248848255} a^{6} + \frac{1054296073}{2986179060} a^{5} - \frac{220446607}{746544765} a^{4} + \frac{67118682}{248848255} a^{3} - \frac{941367479}{2986179060} a^{2} + \frac{80897157}{248848255} a - \frac{282736}{149308953}$, $\frac{1}{29861790600} a^{15} - \frac{1}{3732723825} a^{13} + \frac{845821513}{29861790600} a^{12} + \frac{218896696}{3732723825} a^{11} - \frac{12550247}{149308953} a^{10} + \frac{1713134051}{9953930200} a^{9} - \frac{1262997179}{3732723825} a^{8} - \frac{1064095028}{3732723825} a^{7} + \frac{6122983993}{29861790600} a^{6} - \frac{216777247}{746544765} a^{5} - \frac{603867334}{1244241275} a^{4} + \frac{9021838001}{29861790600} a^{3} - \frac{38573303}{3732723825} a^{2} + \frac{1395705104}{3732723825} a - \frac{306183472}{1244241275}$

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:  \( -\frac{38580599}{1990786040} a^{15} + \frac{115854013}{2986179060} a^{14} - \frac{115741797}{497696510} a^{13} + \frac{1195998569}{1990786040} a^{12} - \frac{38580599}{14019620} a^{11} - \frac{424386589}{497696510} a^{10} - \frac{91677516293}{5972358120} a^{9} + \frac{347225391}{995393020} a^{8} - \frac{13618951447}{99539302} a^{7} + \frac{271568836361}{1990786040} a^{6} - \frac{106752517433}{995393020} a^{5} + \frac{108388426813}{1493089530} a^{4} - \frac{107755613007}{1990786040} a^{3} + \frac{16705399367}{995393020} a^{2} - \frac{501547787}{99539302} a + \frac{347225391}{248848255} \) (order $10$)
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:  \( 95443.616698 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{445}) \), \(\Q(\sqrt{89}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{89})\), 4.4.39605.1 x2, 4.4.2225.1 x2, 4.0.990125.1 x2, 4.0.11125.1 x2, \(\Q(\zeta_{5})\), 4.0.990125.2, 8.8.39213900625.1, 8.0.980347515625.4, 8.0.980347515625.5, 8.0.123765625.1 x2, 8.0.980347515625.3 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$89$89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$