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Properties

Label 16.0.80985213602...2016.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 7^{8}\cdot 11^{8}$
Root discriminant $17.55$
Ramified primes $2, 7, 11$
Class number $1$
Class group Trivial
Galois group $D_4\times C_2$ (as 16T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 0, -52, 0, -3, 0, 67, 0, 231, 0, 86, 0, 43, 0, 11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 11*x^14 + 43*x^12 + 86*x^10 + 231*x^8 + 67*x^6 - 3*x^4 - 52*x^2 + 16)
 
gp: K = bnfinit(x^16 + 11*x^14 + 43*x^12 + 86*x^10 + 231*x^8 + 67*x^6 - 3*x^4 - 52*x^2 + 16, 1)
 

Normalized defining polynomial

\( x^{16} + 11 x^{14} + 43 x^{12} + 86 x^{10} + 231 x^{8} + 67 x^{6} - 3 x^{4} - 52 x^{2} + 16 \)

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:  \(80985213602868822016=2^{16}\cdot 7^{8}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.55$
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:  $2, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} + \frac{1}{8} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{120} a^{12} + \frac{3}{40} a^{10} - \frac{1}{4} a^{9} + \frac{1}{15} a^{8} - \frac{23}{120} a^{6} - \frac{1}{2} a^{5} - \frac{13}{40} a^{4} + \frac{1}{4} a^{3} + \frac{7}{15} a^{2} - \frac{1}{2} a + \frac{1}{15}$, $\frac{1}{120} a^{13} - \frac{1}{20} a^{11} - \frac{7}{120} a^{9} - \frac{23}{120} a^{7} - \frac{9}{20} a^{5} + \frac{41}{120} a^{3} + \frac{1}{15} a$, $\frac{1}{2576400} a^{14} - \frac{801}{214700} a^{12} + \frac{304139}{2576400} a^{10} - \frac{67931}{2576400} a^{8} + \frac{5193}{11300} a^{6} - \frac{1}{2} a^{5} - \frac{157337}{515280} a^{4} - \frac{1}{2} a^{3} - \frac{88207}{644100} a^{2} - \frac{1}{2} a - \frac{1831}{53675}$, $\frac{1}{2576400} a^{15} - \frac{801}{214700} a^{13} - \frac{17911}{2576400} a^{11} - \frac{389981}{2576400} a^{9} - \frac{1}{4} a^{8} - \frac{457}{11300} a^{7} - \frac{221747}{515280} a^{5} - \frac{337439}{1288200} a^{3} - \frac{1}{4} a^{2} + \frac{50013}{107350} a$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{1259}{33900} a^{15} + \frac{29269}{67800} a^{13} + \frac{31523}{16950} a^{11} + \frac{96649}{22600} a^{9} + \frac{739717}{67800} a^{7} + \frac{28939}{3390} a^{5} + \frac{216181}{67800} a^{3} - \frac{15038}{8475} a \) (order $4$)
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:  \( 8995.10035275 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4$ (as 16T9):

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 $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{77}) \), \(\Q(\sqrt{-77}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{11}) \), \(\Q(i, \sqrt{77})\), \(\Q(i, \sqrt{7})\), \(\Q(i, \sqrt{11})\), \(\Q(\sqrt{-7}, \sqrt{-11})\), \(\Q(\sqrt{7}, \sqrt{11})\), \(\Q(\sqrt{-7}, \sqrt{11})\), \(\Q(\sqrt{7}, \sqrt{-11})\), 4.0.2156.1 x2, 4.0.2156.2 x2, 4.2.13552.2 x2, 4.2.13552.1 x2, 8.0.8999178496.1, 8.0.74373376.3 x2, 8.0.183656704.1 x2, 8.0.562448656.1 x2, 8.4.8999178496.1 x2, 8.0.8999178496.2, 8.0.8999178496.3

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ R R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$