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Properties

Label 16.0.62459261715...2809.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 173^{6}$
Root discriminant $47.29$
Ramified primes $13, 173$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729, 0, 4203, 0, 10738, 0, 6681, 0, -660, 0, -525, 0, 104, 0, 32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 32*x^14 + 104*x^12 - 525*x^10 - 660*x^8 + 6681*x^6 + 10738*x^4 + 4203*x^2 + 729)
 
gp: K = bnfinit(x^16 + 32*x^14 + 104*x^12 - 525*x^10 - 660*x^8 + 6681*x^6 + 10738*x^4 + 4203*x^2 + 729, 1)
 

Normalized defining polynomial

\( x^{16} + 32 x^{14} + 104 x^{12} - 525 x^{10} - 660 x^{8} + 6681 x^{6} + 10738 x^{4} + 4203 x^{2} + 729 \)

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:  \(624592617158605666432592809=13^{12}\cdot 173^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.29$
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:  $13, 173$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{11} + \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{9} - \frac{1}{6} a^{7} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2}$, $\frac{1}{169194967575642} a^{14} - \frac{1405928027023}{169194967575642} a^{12} + \frac{3775604658559}{84597483787821} a^{10} - \frac{3746442512087}{18799440841738} a^{8} - \frac{3952470566942}{9399720420869} a^{6} - \frac{1}{2} a^{5} - \frac{11107777852736}{28199161262607} a^{4} - \frac{1}{2} a^{3} + \frac{6524114312495}{84597483787821} a^{2} - \frac{3752847461781}{9399720420869}$, $\frac{1}{1522754708180778} a^{15} - \frac{14802544644815}{761377354090389} a^{13} - \frac{77046274470703}{1522754708180778} a^{11} - \frac{29118964820303}{253792451363463} a^{9} - \frac{127111748031211}{507584902726926} a^{7} + \frac{7691662989002}{253792451363463} a^{5} - \frac{1}{2} a^{4} + \frac{316714888201172}{761377354090389} a^{3} + \frac{1894025497307}{169194967575642} a$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( 9490787.05287 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.29237.1, 4.0.2197.1, 4.0.380081.1, 8.4.24991851015053.1, 8.4.147880775237.1, 8.0.144461566561.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
13Data not computed
$173$173.4.0.1$x^{4} - x + 26$$1$$4$$0$$C_4$$[\ ]^{4}$
173.4.0.1$x^{4} - x + 26$$1$$4$$0$$C_4$$[\ ]^{4}$
173.8.6.2$x^{8} + 1557 x^{4} + 748225$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$