Properties

Label 16.0.6634204312890625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 19^{8}$
Root discriminant $9.75$
Ramified primes $5, 19$
Class number $1$
Class group Trivial
Galois Group $D_{8}$ (as 16T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 34, -98, 221, -416, 654, -839, 869, -730, 509, -304, 160, -73, 27, -7, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 27*x^14 - 73*x^13 + 160*x^12 - 304*x^11 + 509*x^10 - 730*x^9 + 869*x^8 - 839*x^7 + 654*x^6 - 416*x^5 + 221*x^4 - 98*x^3 + 34*x^2 - 8*x + 1)
gp: K = bnfinit(x^16 - 7*x^15 + 27*x^14 - 73*x^13 + 160*x^12 - 304*x^11 + 509*x^10 - 730*x^9 + 869*x^8 - 839*x^7 + 654*x^6 - 416*x^5 + 221*x^4 - 98*x^3 + 34*x^2 - 8*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 7 x^{15} \) \(\mathstrut +\mathstrut 27 x^{14} \) \(\mathstrut -\mathstrut 73 x^{13} \) \(\mathstrut +\mathstrut 160 x^{12} \) \(\mathstrut -\mathstrut 304 x^{11} \) \(\mathstrut +\mathstrut 509 x^{10} \) \(\mathstrut -\mathstrut 730 x^{9} \) \(\mathstrut +\mathstrut 869 x^{8} \) \(\mathstrut -\mathstrut 839 x^{7} \) \(\mathstrut +\mathstrut 654 x^{6} \) \(\mathstrut -\mathstrut 416 x^{5} \) \(\mathstrut +\mathstrut 221 x^{4} \) \(\mathstrut -\mathstrut 98 x^{3} \) \(\mathstrut +\mathstrut 34 x^{2} \) \(\mathstrut -\mathstrut 8 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(6634204312890625=5^{8}\cdot 19^{8}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.75$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 19$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{13} + \frac{1}{5} a^{12} - \frac{2}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{85} a^{15} - \frac{1}{17} a^{14} + \frac{1}{5} a^{13} - \frac{39}{85} a^{12} - \frac{3}{85} a^{11} + \frac{6}{17} a^{10} - \frac{26}{85} a^{9} - \frac{1}{5} a^{8} - \frac{3}{17} a^{7} - \frac{19}{85} a^{6} + \frac{21}{85} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{38}{85} a^{2} + \frac{5}{17} a + \frac{42}{85}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( a^{15} - \frac{21}{5} a^{14} + \frac{48}{5} a^{13} - \frac{61}{5} a^{12} + \frac{52}{5} a^{11} + \frac{12}{5} a^{10} - \frac{216}{5} a^{9} + \frac{719}{5} a^{8} - 278 a^{7} + 364 a^{6} - \frac{1631}{5} a^{5} + \frac{1072}{5} a^{4} - \frac{552}{5} a^{3} + \frac{258}{5} a^{2} - \frac{82}{5} a + \frac{11}{5} \),  \( \frac{331}{85} a^{15} - \frac{2029}{85} a^{14} + \frac{417}{5} a^{13} - \frac{17533}{85} a^{12} + \frac{7227}{17} a^{11} - \frac{65397}{85} a^{10} + \frac{20763}{17} a^{9} - \frac{8083}{5} a^{8} + \frac{29250}{17} a^{7} - \frac{122059}{85} a^{6} + \frac{80442}{85} a^{5} - \frac{2538}{5} a^{4} + 230 a^{3} - \frac{1408}{17} a^{2} + \frac{1577}{85} a - \frac{174}{85} \),  \( \frac{5}{17} a^{15} - \frac{346}{85} a^{14} + \frac{104}{5} a^{13} - \frac{5701}{85} a^{12} + \frac{13627}{85} a^{11} - \frac{27368}{85} a^{10} + \frac{48429}{85} a^{9} - \frac{4408}{5} a^{8} + \frac{19305}{17} a^{7} - \frac{19917}{17} a^{6} + \frac{80119}{85} a^{5} - \frac{2979}{5} a^{4} + \frac{1534}{5} a^{3} - \frac{11392}{85} a^{2} + \frac{3668}{85} a - \frac{599}{85} \),  \( a^{15} - 6 a^{14} + 20 a^{13} - 47 a^{12} + 93 a^{11} - 164 a^{10} + 252 a^{9} - 314 a^{8} + 303 a^{7} - 222 a^{6} + 129 a^{5} - 65 a^{4} + 27 a^{3} - 6 a^{2} + a - 1 \),  \( \frac{32}{17} a^{15} - \frac{868}{85} a^{14} + \frac{162}{5} a^{13} - \frac{6138}{85} a^{12} + \frac{11726}{85} a^{11} - \frac{19799}{85} a^{10} + \frac{28922}{85} a^{9} - \frac{1929}{5} a^{8} + \frac{5300}{17} a^{7} - \frac{2461}{17} a^{6} + \frac{572}{85} a^{5} + \frac{223}{5} a^{4} - \frac{188}{5} a^{3} + \frac{2204}{85} a^{2} - \frac{1236}{85} a + \frac{328}{85} \),  \( \frac{67}{17} a^{15} - \frac{2202}{85} a^{14} + \frac{473}{5} a^{13} - \frac{20647}{85} a^{12} + \frac{43484}{85} a^{11} - \frac{80101}{85} a^{10} + \frac{129823}{85} a^{9} - \frac{10441}{5} a^{8} + \frac{39404}{17} a^{7} - \frac{34610}{17} a^{6} + \frac{120323}{85} a^{5} - \frac{3998}{5} a^{4} + \frac{1908}{5} a^{3} - \frac{12549}{85} a^{2} + \frac{3156}{85} a - \frac{363}{85} \),  \( \frac{462}{85} a^{15} - \frac{3024}{85} a^{14} + \frac{648}{5} a^{13} - \frac{28252}{85} a^{12} + \frac{59542}{85} a^{11} - \frac{109747}{85} a^{10} + \frac{177929}{85} a^{9} - \frac{14314}{5} a^{8} + \frac{54136}{17} a^{7} - \frac{238873}{85} a^{6} + \frac{167343}{85} a^{5} - 1120 a^{4} + \frac{2674}{5} a^{3} - \frac{17702}{85} a^{2} + \frac{4682}{85} a - \frac{622}{85} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 11.4915268613 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_8$ (as 16T13):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-95}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-19})\), 4.0.1805.1 x2, 4.2.475.1 x2, 8.0.81450625.1, 8.0.16290125.1 x4, 8.2.4286875.1 x4

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$