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Properties

Label 16.0.74247530256...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 29^{4}$
Root discriminant $26.88$
Ramified primes $2, 3, 5, 29$
Class number $16$
Class group $[2, 8]$
Galois group $C_2 \times (C_2^2:C_4)$ (as 16T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![961, 0, 2250, 0, 2049, 0, 840, 0, 206, 0, 90, 0, 24, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 24*x^12 + 90*x^10 + 206*x^8 + 840*x^6 + 2049*x^4 + 2250*x^2 + 961)
 
gp: K = bnfinit(x^16 + 24*x^12 + 90*x^10 + 206*x^8 + 840*x^6 + 2049*x^4 + 2250*x^2 + 961, 1)
 

Normalized defining polynomial

\( x^{16} + 24 x^{12} + 90 x^{10} + 206 x^{8} + 840 x^{6} + 2049 x^{4} + 2250 x^{2} + 961 \)

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:  \(74247530256000000000000=2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.88$
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:  $2, 3, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3}$, $\frac{1}{236} a^{12} - \frac{7}{59} a^{10} + \frac{11}{118} a^{8} + \frac{65}{236} a^{6} + \frac{33}{236} a^{4} - \frac{20}{59} a^{2} + \frac{1}{59}$, $\frac{1}{236} a^{13} - \frac{7}{59} a^{11} + \frac{11}{118} a^{9} + \frac{65}{236} a^{7} + \frac{33}{236} a^{5} - \frac{20}{59} a^{3} + \frac{1}{59} a$, $\frac{1}{988604} a^{14} - \frac{125}{494302} a^{12} + \frac{15631}{247151} a^{10} - \frac{60397}{988604} a^{8} - \frac{471057}{988604} a^{6} - \frac{93272}{247151} a^{4} - \frac{1395}{3481} a^{2} + \frac{93425}{494302}$, $\frac{1}{30646724} a^{15} - \frac{2035}{988604} a^{13} + \frac{1652405}{15323362} a^{11} - \frac{1937069}{30646724} a^{9} - \frac{2868890}{7661681} a^{7} - \frac{8872569}{30646724} a^{5} + \frac{33710}{107911} a^{3} - \frac{3352661}{7661681} a$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3121.7160225 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:C_4$ (as 16T21):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2 \times (C_2^2:C_4)$
Character table for $C_2 \times (C_2^2:C_4)$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{15}) \), 4.0.3625.1, \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{15})^+\), 4.0.522000.3, 4.0.11600.1, 4.0.6525.1, \(\Q(\sqrt{3}, \sqrt{5})\), 8.0.10899360000.14, 8.0.3364000000.2, 8.0.272484000000.14, 8.0.1064390625.1, 8.0.272484000000.12, 8.0.272484000000.11, \(\Q(\zeta_{60})^+\)

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

Sibling fields

Galois closure: data not computed
Degree 16 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{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}$