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Properties

Label 16.0.37682348496...4208.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 17^{11}$
Root discriminant $39.67$
Ramified primes $2, 17$
Class number $8$
Class group $[8]$
Galois group 16T1230

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5508, 0, -2448, 0, -68, 0, -680, 0, 865, 0, -432, 0, 118, 0, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 16*x^14 + 118*x^12 - 432*x^10 + 865*x^8 - 680*x^6 - 68*x^4 - 2448*x^2 + 5508)
 
gp: K = bnfinit(x^16 - 16*x^14 + 118*x^12 - 432*x^10 + 865*x^8 - 680*x^6 - 68*x^4 - 2448*x^2 + 5508, 1)
 

Normalized defining polynomial

\( x^{16} - 16 x^{14} + 118 x^{12} - 432 x^{10} + 865 x^{8} - 680 x^{6} - 68 x^{4} - 2448 x^{2} + 5508 \)

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:  \(37682348496175843883614208=2^{40}\cdot 17^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.67$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{8} a^{6} - \frac{3}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{8} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{104} a^{12} + \frac{3}{104} a^{10} + \frac{11}{104} a^{8} - \frac{1}{8} a^{6} + \frac{9}{26} a^{4} + \frac{15}{52} a^{2} - \frac{5}{26}$, $\frac{1}{312} a^{13} - \frac{5}{156} a^{11} - \frac{1}{156} a^{9} - \frac{1}{4} a^{7} - \frac{29}{312} a^{5} + \frac{1}{78} a^{3} + \frac{29}{156} a$, $\frac{1}{432549000} a^{14} - \frac{661121}{216274500} a^{12} - \frac{1247713}{86509800} a^{10} - \frac{1}{8} a^{9} - \frac{343361}{6007625} a^{8} - \frac{1}{4} a^{7} - \frac{22907659}{216274500} a^{6} - \frac{3}{8} a^{5} + \frac{37439047}{108137250} a^{4} - \frac{1}{2} a^{3} + \frac{13370243}{54068625} a^{2} + \frac{1}{4} a + \frac{1220364}{6007625}$, $\frac{1}{2595294000} a^{15} - \frac{1}{865098000} a^{14} + \frac{2836883}{2595294000} a^{13} - \frac{2836883}{865098000} a^{12} - \frac{31193413}{519058800} a^{11} + \frac{9565963}{173019600} a^{10} - \frac{21694013}{288366000} a^{9} + \frac{9678763}{96122000} a^{8} - \frac{19244071}{324411750} a^{7} + \frac{22907659}{432549000} a^{6} - \frac{553149781}{1297647000} a^{5} + \frac{16633039}{108137250} a^{4} + \frac{247174111}{648823500} a^{3} - \frac{21242059}{54068625} a^{2} - \frac{4473193}{18022875} a + \frac{11885147}{24030500}$

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

Class group and class number

$C_{8}$, which has order $8$

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:  \( \frac{3097}{3460392} a^{14} - \frac{42091}{3460392} a^{12} + \frac{262927}{3460392} a^{10} - \frac{74831}{384488} a^{8} + \frac{184705}{865098} a^{6} + \frac{584885}{1730196} a^{4} - \frac{113849}{865098} a^{2} - \frac{97793}{48061} \) (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:  \( 2336931.55822 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1230:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 58 conjugacy class representatives for t16n1230 are not computed
Character table for t16n1230 is not computed

Intermediate fields

\(\Q(\sqrt{-17}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{17})\), 8.0.23262937088.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.24.78$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 2$$8$$1$$24$$Z_8 : Z_8^\times$$[2, 2, 3, 4]^{2}$
2.8.16.65$x^{8} + 4 x^{6} + 28 x^{4} + 20$$8$$1$$16$$QD_{16}$$[2, 2, 5/2]^{2}$
$17$17.8.7.2$x^{8} - 153$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$