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Properties

Label 16.0.12370988147...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 5^{6}\cdot 13^{4}\cdot 101^{8}$
Root discriminant $49.35$
Ramified primes $2, 5, 13, 101$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1275

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![361, 0, 843, 0, 503, 0, 161, 0, 302, 0, 279, 0, 133, 0, 22, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 22*x^14 + 133*x^12 + 279*x^10 + 302*x^8 + 161*x^6 + 503*x^4 + 843*x^2 + 361)
 
gp: K = bnfinit(x^16 + 22*x^14 + 133*x^12 + 279*x^10 + 302*x^8 + 161*x^6 + 503*x^4 + 843*x^2 + 361, 1)
 

Normalized defining polynomial

\( x^{16} + 22 x^{14} + 133 x^{12} + 279 x^{10} + 302 x^{8} + 161 x^{6} + 503 x^{4} + 843 x^{2} + 361 \)

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:  \(1237098814777743829444000000=2^{8}\cdot 5^{6}\cdot 13^{4}\cdot 101^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.35$
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:  $2, 5, 13, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{22} a^{10} - \frac{1}{2} a^{9} - \frac{7}{22} a^{8} + \frac{4}{11} a^{6} - \frac{7}{22} a^{4} - \frac{1}{2} a^{3} - \frac{1}{11} a^{2} - \frac{1}{2} a - \frac{5}{22}$, $\frac{1}{22} a^{11} + \frac{2}{11} a^{9} - \frac{1}{2} a^{8} + \frac{4}{11} a^{7} - \frac{7}{22} a^{5} + \frac{9}{22} a^{3} - \frac{1}{2} a^{2} + \frac{3}{11} a - \frac{1}{2}$, $\frac{1}{484} a^{12} - \frac{1}{484} a^{10} - \frac{1}{2} a^{9} - \frac{67}{484} a^{8} + \frac{107}{484} a^{6} + \frac{9}{22} a^{4} - \frac{1}{2} a^{3} - \frac{91}{242} a^{2} - \frac{1}{2} a + \frac{223}{484}$, $\frac{1}{484} a^{13} - \frac{1}{484} a^{11} + \frac{175}{484} a^{9} - \frac{1}{2} a^{8} + \frac{107}{484} a^{7} + \frac{9}{22} a^{5} + \frac{15}{121} a^{3} - \frac{1}{2} a^{2} - \frac{19}{484} a - \frac{1}{2}$, $\frac{1}{1671736} a^{14} - \frac{1}{968} a^{13} + \frac{235}{417934} a^{12} - \frac{21}{968} a^{11} - \frac{14903}{835868} a^{10} - \frac{263}{968} a^{9} + \frac{174235}{417934} a^{8} - \frac{283}{968} a^{7} + \frac{409193}{1671736} a^{6} - \frac{1}{22} a^{5} - \frac{374113}{835868} a^{4} - \frac{129}{484} a^{3} - \frac{56667}{151976} a^{2} - \frac{113}{968} a - \frac{146865}{1671736}$, $\frac{1}{31762984} a^{15} + \frac{6121}{31762984} a^{13} - \frac{1}{968} a^{12} + \frac{8157}{1671736} a^{11} - \frac{21}{968} a^{10} + \frac{5706967}{31762984} a^{9} + \frac{221}{968} a^{8} + \frac{1982151}{3970373} a^{7} - \frac{283}{968} a^{6} - \frac{6377165}{15881492} a^{5} - \frac{1}{22} a^{4} + \frac{1342831}{2887544} a^{3} + \frac{113}{484} a^{2} - \frac{1448107}{7940746} a + \frac{371}{968}$

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:  \( -1 \) (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:  \( 7476878.56849 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1275:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1275
Character table for t16n1275 is not computed

Intermediate fields

\(\Q(\sqrt{101}) \), 4.4.663065.1, 8.4.2198275971125.4

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$101$101.8.4.1$x^{8} + 244824 x^{4} - 1030301 x^{2} + 14984697744$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
101.8.4.1$x^{8} + 244824 x^{4} - 1030301 x^{2} + 14984697744$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$