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Properties

Label 16.0.46335197011...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{4}\cdot 109^{6}$
Root discriminant $30.14$
Ramified primes $5, 29, 109$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T392)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![475, 1700, 2865, 2225, -689, -2461, -447, -314, 1134, -336, 387, -218, 125, -63, 21, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 21*x^14 - 63*x^13 + 125*x^12 - 218*x^11 + 387*x^10 - 336*x^9 + 1134*x^8 - 314*x^7 - 447*x^6 - 2461*x^5 - 689*x^4 + 2225*x^3 + 2865*x^2 + 1700*x + 475)
 
gp: K = bnfinit(x^16 - 5*x^15 + 21*x^14 - 63*x^13 + 125*x^12 - 218*x^11 + 387*x^10 - 336*x^9 + 1134*x^8 - 314*x^7 - 447*x^6 - 2461*x^5 - 689*x^4 + 2225*x^3 + 2865*x^2 + 1700*x + 475, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 21 x^{14} - 63 x^{13} + 125 x^{12} - 218 x^{11} + 387 x^{10} - 336 x^{9} + 1134 x^{8} - 314 x^{7} - 447 x^{6} - 2461 x^{5} - 689 x^{4} + 2225 x^{3} + 2865 x^{2} + 1700 x + 475 \)

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:  \(463351970115520828515625=5^{8}\cdot 29^{4}\cdot 109^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.14$
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:  $5, 29, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{12} + \frac{1}{5} a^{11} - \frac{1}{10} a^{10} - \frac{1}{5} a^{9} - \frac{3}{10} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{3}{10} a^{5} + \frac{2}{5} a^{4} + \frac{1}{10} a^{3} + \frac{1}{10} a^{2}$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{9} - \frac{1}{10} a^{8} - \frac{1}{2} a^{7} + \frac{1}{10} a^{6} + \frac{3}{10} a^{5} + \frac{3}{10} a^{4} + \frac{2}{5} a^{3} + \frac{3}{10} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{20} a^{14} - \frac{1}{20} a^{13} + \frac{1}{20} a^{10} - \frac{1}{10} a^{9} - \frac{1}{5} a^{8} + \frac{3}{10} a^{7} - \frac{2}{5} a^{6} - \frac{1}{2} a^{5} + \frac{1}{20} a^{4} + \frac{9}{20} a^{3} + \frac{1}{10} a^{2} + \frac{1}{4}$, $\frac{1}{159377522388060362101160} a^{15} - \frac{28008374539510428791}{2570605199807425195180} a^{14} - \frac{1680974464973394782529}{159377522388060362101160} a^{13} + \frac{2233388684196729609257}{79688761194030181050580} a^{12} - \frac{20590819100532543457241}{159377522388060362101160} a^{11} - \frac{1195781789086389103647}{5141210399614850390360} a^{10} - \frac{973031289123440870367}{3984438059701509052529} a^{9} - \frac{9432353470936912385123}{39844380597015090525290} a^{8} - \frac{23595247018043736826251}{79688761194030181050580} a^{7} + \frac{10954810018112694797303}{39844380597015090525290} a^{6} - \frac{52735238348082647999387}{159377522388060362101160} a^{5} + \frac{39414558548728645139747}{79688761194030181050580} a^{4} - \frac{9884467227146913891003}{159377522388060362101160} a^{3} - \frac{251060837460660328627}{1285302599903712597590} a^{2} + \frac{751319331280110286101}{31875504477612072420232} a - \frac{7707973509390749929009}{31875504477612072420232}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( 18311.5213263 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T392):

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 $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.2725.1, 4.0.79025.2, 4.4.725.1, 8.0.6244950625.4, 8.8.6244950625.1, 8.0.6244950625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.4.2.1$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}$
$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}$
109Data not computed