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Properties

Label 16.12.5729212928...8192.1
Degree $16$
Signature $[12, 2]$
Discriminant $2^{24}\cdot 41^{6}\cdot 193^{3}$
Root discriminant $30.54$
Ramified primes $2, 41, 193$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1722

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1297, 2626, -11481, -9136, 11594, 11310, 1448, -3908, -3184, 70, 649, 140, 28, -14, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 16*x^14 - 14*x^13 + 28*x^12 + 140*x^11 + 649*x^10 + 70*x^9 - 3184*x^8 - 3908*x^7 + 1448*x^6 + 11310*x^5 + 11594*x^4 - 9136*x^3 - 11481*x^2 + 2626*x + 1297)
 
gp: K = bnfinit(x^16 - 16*x^14 - 14*x^13 + 28*x^12 + 140*x^11 + 649*x^10 + 70*x^9 - 3184*x^8 - 3908*x^7 + 1448*x^6 + 11310*x^5 + 11594*x^4 - 9136*x^3 - 11481*x^2 + 2626*x + 1297, 1)
 

Normalized defining polynomial

\( x^{16} - 16 x^{14} - 14 x^{13} + 28 x^{12} + 140 x^{11} + 649 x^{10} + 70 x^{9} - 3184 x^{8} - 3908 x^{7} + 1448 x^{6} + 11310 x^{5} + 11594 x^{4} - 9136 x^{3} - 11481 x^{2} + 2626 x + 1297 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(572921292848472304648192=2^{24}\cdot 41^{6}\cdot 193^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.54$
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:  $2, 41, 193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{13} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{296} a^{14} - \frac{1}{37} a^{13} - \frac{23}{296} a^{12} - \frac{1}{148} a^{11} + \frac{49}{296} a^{10} - \frac{55}{148} a^{9} + \frac{61}{148} a^{8} - \frac{13}{74} a^{7} - \frac{17}{148} a^{6} - \frac{23}{74} a^{5} + \frac{5}{148} a^{4} - \frac{73}{148} a^{3} + \frac{8}{37} a^{2} - \frac{11}{148} a - \frac{13}{296}$, $\frac{1}{1905086013306836240347448} a^{15} + \frac{953444590103967391483}{1905086013306836240347448} a^{14} - \frac{150440008155134324769917}{1905086013306836240347448} a^{13} + \frac{212613808339503553865487}{1905086013306836240347448} a^{12} + \frac{295477555330865800226611}{1905086013306836240347448} a^{11} + \frac{62042441531268632235917}{1905086013306836240347448} a^{10} - \frac{429054650996391221823825}{952543006653418120173724} a^{9} - \frac{37119892501711898973895}{476271503326709060086862} a^{8} + \frac{196657831281520843345593}{476271503326709060086862} a^{7} - \frac{209840650942017698921423}{476271503326709060086862} a^{6} - \frac{2502212689123556713476}{6436101396306879190363} a^{5} - \frac{311552586571569210385541}{952543006653418120173724} a^{4} + \frac{217132464020081355538715}{476271503326709060086862} a^{3} + \frac{146985209722486992201577}{476271503326709060086862} a^{2} - \frac{653460699192352004411317}{1905086013306836240347448} a - \frac{120169046794330129885745}{1905086013306836240347448}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 1577899.16521 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1722:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 104 conjugacy class representatives for t16n1722 are not computed
Character table for t16n1722 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 8.8.282300416.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$41$41.8.6.2$x^{8} + 943 x^{4} + 242064$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41.8.0.1$x^{8} - x + 12$$1$$8$$0$$C_8$$[\ ]^{8}$
$193$193.2.1.1$x^{2} - 193$$2$$1$$1$$C_2$$[\ ]_{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.1.1$x^{2} - 193$$2$$1$$1$$C_2$$[\ ]_{2}$
193.2.1.1$x^{2} - 193$$2$$1$$1$$C_2$$[\ ]_{2}$
193.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$