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Properties

Label 16.0.11127404190...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 11^{4}\cdot 41^{6}$
Root discriminant $49.02$
Ramified primes $2, 5, 11, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T646)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![615288025, 0, 313783250, 0, 70680940, 0, 9245500, 0, 785991, 0, 47378, 0, 2149, 0, 67, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 67*x^14 + 2149*x^12 + 47378*x^10 + 785991*x^8 + 9245500*x^6 + 70680940*x^4 + 313783250*x^2 + 615288025)
 
gp: K = bnfinit(x^16 + 67*x^14 + 2149*x^12 + 47378*x^10 + 785991*x^8 + 9245500*x^6 + 70680940*x^4 + 313783250*x^2 + 615288025, 1)
 

Normalized defining polynomial

\( x^{16} + 67 x^{14} + 2149 x^{12} + 47378 x^{10} + 785991 x^{8} + 9245500 x^{6} + 70680940 x^{4} + 313783250 x^{2} + 615288025 \)

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:  \(1112740419079696000000000000=2^{16}\cdot 5^{12}\cdot 11^{4}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.02$
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:  $2, 5, 11, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{33} a^{10} + \frac{1}{33} a^{8} + \frac{4}{33} a^{6} + \frac{1}{33} a^{4} - \frac{1}{11} a^{2} + \frac{1}{3}$, $\frac{1}{33} a^{11} + \frac{1}{33} a^{9} + \frac{4}{33} a^{7} + \frac{1}{33} a^{5} - \frac{1}{11} a^{3} + \frac{1}{3} a$, $\frac{1}{1815} a^{12} + \frac{4}{605} a^{10} + \frac{93}{605} a^{8} + \frac{191}{605} a^{6} + \frac{41}{1815} a^{4} - \frac{10}{33} a^{2} - \frac{1}{3}$, $\frac{1}{1815} a^{13} + \frac{4}{605} a^{11} + \frac{93}{605} a^{9} + \frac{191}{605} a^{7} + \frac{41}{1815} a^{5} - \frac{10}{33} a^{3} - \frac{1}{3} a$, $\frac{1}{1179564571864324580385} a^{14} - \frac{42125443328243527}{1179564571864324580385} a^{12} + \frac{13666093497603995221}{1179564571864324580385} a^{10} - \frac{147020858981395289758}{1179564571864324580385} a^{8} + \frac{523270880614686242354}{1179564571864324580385} a^{6} - \frac{51017609377063139}{2615442509676994635} a^{4} - \frac{286026145027501909}{649897835737919879} a^{2} - \frac{1613321144943257}{4323045470540487}$, $\frac{1}{5897822859321622901925} a^{15} + \frac{607772392409676352}{5897822859321622901925} a^{13} - \frac{14279513439126559576}{5897822859321622901925} a^{11} + \frac{784932637466781816728}{5897822859321622901925} a^{9} - \frac{1213256136477035674334}{5897822859321622901925} a^{7} + \frac{68795909303747607}{174362833978466309} a^{5} - \frac{1290413709349741722}{3249489178689599395} a^{3} + \frac{541944865119446}{4323045470540487} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{1833464705186}{1917991173763129399} a^{14} + \frac{1586123141802116}{28769867606446940985} a^{12} + \frac{14812724905319894}{9589955868815646995} a^{10} + \frac{894223017675761944}{28769867606446940985} a^{8} + \frac{13420900005548642933}{28769867606446940985} a^{6} + \frac{11980557994298832596}{2615442509676994635} a^{4} + \frac{1237282581182031214}{47553500175945357} a^{2} + \frac{95993360798452224}{1441015156846829} \) (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:  \( 8483670.4147 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T646):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.5125.1, 4.0.1025.1, 8.0.26265625.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.8.9$x^{8} + 6 x^{6} + 4 x^{5} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
2.8.8.10$x^{8} + 2 x^{6} + 8 x^{3} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 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}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$