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Properties

Label 16.8.38037936847...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 5^{10}\cdot 29^{8}\cdot 109^{2}$
Root discriminant $52.94$
Ramified primes $2, 5, 29, 109$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T984

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![297025, 0, -122625, 0, -98480, 0, 44040, 0, 2141, 0, -1572, 0, -56, 0, 17, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 17*x^14 - 56*x^12 - 1572*x^10 + 2141*x^8 + 44040*x^6 - 98480*x^4 - 122625*x^2 + 297025)
 
gp: K = bnfinit(x^16 + 17*x^14 - 56*x^12 - 1572*x^10 + 2141*x^8 + 44040*x^6 - 98480*x^4 - 122625*x^2 + 297025, 1)
 

Normalized defining polynomial

\( x^{16} + 17 x^{14} - 56 x^{12} - 1572 x^{10} + 2141 x^{8} + 44040 x^{6} - 98480 x^{4} - 122625 x^{2} + 297025 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3803793684729370240000000000=2^{16}\cdot 5^{10}\cdot 29^{8}\cdot 109^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.94$
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:  $2, 5, 29, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{6} - \frac{1}{5} a^{4}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{5}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{5}$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{8} - \frac{3}{10} a^{6} - \frac{1}{10} a^{4} - \frac{1}{2}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{9} - \frac{3}{10} a^{7} - \frac{1}{10} a^{5} - \frac{1}{2} a$, $\frac{1}{4484418180236470} a^{14} - \frac{89245070395199}{2242209090118235} a^{12} - \frac{136808812119157}{4484418180236470} a^{10} + \frac{1975555279831}{20665521567910} a^{8} + \frac{1900802958452047}{4484418180236470} a^{6} - \frac{98673519654301}{448441818023647} a^{4} + \frac{233723296913415}{896883636047294} a^{2} + \frac{683482817924}{4114145119483}$, $\frac{1}{4484418180236470} a^{15} - \frac{89245070395199}{2242209090118235} a^{13} - \frac{136808812119157}{4484418180236470} a^{11} + \frac{1975555279831}{20665521567910} a^{9} + \frac{1900802958452047}{4484418180236470} a^{7} - \frac{98673519654301}{448441818023647} a^{5} + \frac{233723296913415}{896883636047294} a^{3} + \frac{683482817924}{4114145119483} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 32035477.466 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T984:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n984
Character table for t16n984 is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109Data not computed