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Properties

Label 16.2.13849203506...4891.1
Degree $16$
Signature $[2, 7]$
Discriminant $-\,3^{7}\cdot 97^{15}$
Root discriminant $117.85$
Ramified primes $3, 97$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-235224, 1299672, -2517036, 1806831, -309548, -235268, 31703, 103959, -90470, 41759, -15581, 4640, -755, 105, 14, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 14*x^14 + 105*x^13 - 755*x^12 + 4640*x^11 - 15581*x^10 + 41759*x^9 - 90470*x^8 + 103959*x^7 + 31703*x^6 - 235268*x^5 - 309548*x^4 + 1806831*x^3 - 2517036*x^2 + 1299672*x - 235224)
 
gp: K = bnfinit(x^16 - 2*x^15 + 14*x^14 + 105*x^13 - 755*x^12 + 4640*x^11 - 15581*x^10 + 41759*x^9 - 90470*x^8 + 103959*x^7 + 31703*x^6 - 235268*x^5 - 309548*x^4 + 1806831*x^3 - 2517036*x^2 + 1299672*x - 235224, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 14 x^{14} + 105 x^{13} - 755 x^{12} + 4640 x^{11} - 15581 x^{10} + 41759 x^{9} - 90470 x^{8} + 103959 x^{7} + 31703 x^{6} - 235268 x^{5} - 309548 x^{4} + 1806831 x^{3} - 2517036 x^{2} + 1299672 x - 235224 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1384920350642158387276644512694891=-\,3^{7}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $117.85$
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:  $3, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{8} - \frac{1}{12} a^{2}$, $\frac{1}{180} a^{9} + \frac{1}{60} a^{8} - \frac{1}{30} a^{7} + \frac{1}{15} a^{6} + \frac{1}{10} a^{5} - \frac{2}{15} a^{4} - \frac{37}{180} a^{3} - \frac{9}{20} a^{2} + \frac{13}{30} a + \frac{1}{5}$, $\frac{1}{360} a^{10} - \frac{1}{360} a^{9} + \frac{1}{30} a^{8} - \frac{1}{15} a^{7} - \frac{1}{12} a^{6} - \frac{11}{60} a^{5} + \frac{59}{360} a^{4} - \frac{173}{360} a^{3} - \frac{13}{60} a^{2} + \frac{7}{30} a - \frac{2}{5}$, $\frac{1}{2160} a^{11} + \frac{1}{432} a^{9} - \frac{1}{24} a^{8} + \frac{17}{360} a^{7} - \frac{1}{45} a^{6} + \frac{37}{432} a^{5} + \frac{5}{72} a^{4} + \frac{91}{2160} a^{3} - \frac{49}{180} a^{2} + \frac{23}{90} a - \frac{1}{2}$, $\frac{1}{2160} a^{12} - \frac{1}{2160} a^{10} - \frac{13}{360} a^{8} - \frac{1}{45} a^{7} - \frac{67}{2160} a^{6} + \frac{43}{360} a^{5} - \frac{479}{2160} a^{4} + \frac{157}{360} a^{3} + \frac{29}{90} a^{2} + \frac{3}{10} a - \frac{1}{5}$, $\frac{1}{38880} a^{13} + \frac{1}{38880} a^{12} + \frac{1}{4860} a^{11} - \frac{1}{38880} a^{10} + \frac{5}{2592} a^{9} + \frac{1}{135} a^{8} - \frac{257}{7776} a^{7} - \frac{293}{7776} a^{6} + \frac{1567}{9720} a^{5} - \frac{7619}{38880} a^{4} + \frac{509}{4320} a^{3} + \frac{1087}{3240} a^{2} + \frac{2}{45} a + \frac{9}{20}$, $\frac{1}{59486400} a^{14} - \frac{19}{14871600} a^{13} - \frac{401}{19828800} a^{12} + \frac{3163}{59486400} a^{11} + \frac{4153}{29743200} a^{10} - \frac{2077}{1166400} a^{9} - \frac{2015089}{59486400} a^{8} - \frac{67742}{929475} a^{7} + \frac{85841}{1166400} a^{6} + \frac{3736433}{59486400} a^{5} + \frac{1071457}{5948640} a^{4} + \frac{9058453}{19828800} a^{3} + \frac{1327573}{4957200} a^{2} + \frac{37103}{275400} a - \frac{11281}{30600}$, $\frac{1}{711497547920202595200} a^{15} - \frac{374034185777}{79055283102244732800} a^{14} + \frac{3809983031411549}{711497547920202595200} a^{13} + \frac{36231661903820507}{355748773960101297600} a^{12} - \frac{8606692284054611}{47433169861346839680} a^{11} - \frac{773514092089315909}{711497547920202595200} a^{10} - \frac{153364581411748537}{71149754792020259520} a^{9} + \frac{302522149570186417}{15811056620448946560} a^{8} - \frac{53620504712852292353}{711497547920202595200} a^{7} - \frac{19370309081954842397}{355748773960101297600} a^{6} - \frac{57322340528861675857}{237165849306734198400} a^{5} - \frac{2717687528513208061}{64681595265472963200} a^{4} - \frac{10432027914973735729}{237165849306734198400} a^{3} - \frac{1403726210457671191}{3487733078040208800} a^{2} + \frac{15391209898356451}{41174626615752465} a - \frac{4152771104166773}{33272425548082800}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  $8$
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:  \( 1403138189870 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{97}) \), 4.2.2738019.1, 8.2.2181553680909051.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
97Data not computed