Properties

Label 16.0.6298009600000000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{8}\cdot 31^{2}$
Root discriminant $9.72$
Ramified primes $2, 5, 31$
Class number $1$
Class group Trivial
Galois Group $D_4^2.C_2$ (as 16T396)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 8, -10, 11, -12, 10, -4, 1, -4, 10, -12, 11, -10, 8, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 10*x^13 + 11*x^12 - 12*x^11 + 10*x^10 - 4*x^9 + x^8 - 4*x^7 + 10*x^6 - 12*x^5 + 11*x^4 - 10*x^3 + 8*x^2 - 4*x + 1)
gp: K = bnfinit(x^16 - 4*x^15 + 8*x^14 - 10*x^13 + 11*x^12 - 12*x^11 + 10*x^10 - 4*x^9 + x^8 - 4*x^7 + 10*x^6 - 12*x^5 + 11*x^4 - 10*x^3 + 8*x^2 - 4*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 4 x^{15} \) \(\mathstrut +\mathstrut 8 x^{14} \) \(\mathstrut -\mathstrut 10 x^{13} \) \(\mathstrut +\mathstrut 11 x^{12} \) \(\mathstrut -\mathstrut 12 x^{11} \) \(\mathstrut +\mathstrut 10 x^{10} \) \(\mathstrut -\mathstrut 4 x^{9} \) \(\mathstrut +\mathstrut x^{8} \) \(\mathstrut -\mathstrut 4 x^{7} \) \(\mathstrut +\mathstrut 10 x^{6} \) \(\mathstrut -\mathstrut 12 x^{5} \) \(\mathstrut +\mathstrut 11 x^{4} \) \(\mathstrut -\mathstrut 10 x^{3} \) \(\mathstrut +\mathstrut 8 x^{2} \) \(\mathstrut -\mathstrut 4 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(6298009600000000=2^{24}\cdot 5^{8}\cdot 31^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.72$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 5, 31$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} + \frac{4}{13} a^{13} - \frac{1}{13} a^{11} + \frac{3}{13} a^{10} - \frac{6}{13} a^{8} - \frac{6}{13} a^{6} + \frac{3}{13} a^{4} - \frac{1}{13} a^{3} + \frac{4}{13} a + \frac{1}{13}$, $\frac{1}{143} a^{15} - \frac{5}{143} a^{14} + \frac{68}{143} a^{13} - \frac{1}{143} a^{12} + \frac{12}{143} a^{11} + \frac{64}{143} a^{10} - \frac{32}{143} a^{9} + \frac{28}{143} a^{8} - \frac{71}{143} a^{7} + \frac{67}{143} a^{6} + \frac{42}{143} a^{5} - \frac{54}{143} a^{4} - \frac{56}{143} a^{3} - \frac{9}{143} a^{2} + \frac{17}{143} a + \frac{56}{143}$

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:  \( \frac{21}{143} a^{15} - \frac{25}{11} a^{14} + \frac{834}{143} a^{13} - \frac{1165}{143} a^{12} + \frac{1044}{143} a^{11} - \frac{1318}{143} a^{10} + \frac{1330}{143} a^{9} - \frac{523}{143} a^{8} - \frac{204}{143} a^{7} - \frac{419}{143} a^{6} + \frac{1025}{143} a^{5} - \frac{116}{11} a^{4} + \frac{1046}{143} a^{3} - \frac{1190}{143} a^{2} + \frac{1050}{143} a - \frac{474}{143} \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( a \),  \( \frac{112}{143} a^{15} - \frac{16}{11} a^{14} + \frac{158}{143} a^{13} + \frac{31}{143} a^{12} + \frac{134}{143} a^{11} - \frac{70}{143} a^{10} - \frac{295}{143} a^{9} + \frac{309}{143} a^{8} + \frac{199}{143} a^{7} - \frac{328}{143} a^{6} + \frac{271}{143} a^{5} + \frac{12}{11} a^{4} + \frac{97}{143} a^{3} - \frac{150}{143} a^{2} - \frac{263}{143} a + \frac{189}{143} \),  \( \frac{54}{143} a^{15} - \frac{347}{143} a^{14} + \frac{790}{143} a^{13} - \frac{1055}{143} a^{12} + \frac{1011}{143} a^{11} - \frac{1208}{143} a^{10} + \frac{1132}{143} a^{9} - \frac{457}{143} a^{8} - \frac{116}{143} a^{7} - \frac{353}{143} a^{6} + \frac{981}{143} a^{5} - \frac{1288}{143} a^{4} + \frac{1057}{143} a^{3} - \frac{1058}{143} a^{2} + \frac{896}{143} a - \frac{485}{143} \),  \( \frac{19}{143} a^{15} - \frac{51}{143} a^{14} + \frac{38}{143} a^{13} - \frac{19}{143} a^{12} + \frac{41}{143} a^{11} - \frac{82}{143} a^{10} - \frac{36}{143} a^{9} - \frac{18}{143} a^{8} + \frac{81}{143} a^{7} - \frac{135}{143} a^{6} + \frac{83}{143} a^{5} - \frac{36}{143} a^{4} - \frac{107}{143} a^{3} - \frac{171}{143} a^{2} - \frac{73}{143} a - \frac{36}{143} \),  \( \frac{304}{143} a^{15} - \frac{1179}{143} a^{14} + \frac{2159}{143} a^{13} - \frac{2449}{143} a^{12} + \frac{2592}{143} a^{11} - \frac{2973}{143} a^{10} + \frac{2141}{143} a^{9} - \frac{398}{143} a^{8} + \frac{9}{143} a^{7} - \frac{1269}{143} a^{6} + \frac{2615}{143} a^{5} - \frac{2809}{143} a^{4} + \frac{2369}{143} a^{3} - \frac{2450}{143} a^{2} + \frac{1670}{143} a - \frac{510}{143} \),  \( \frac{710}{143} a^{15} - \frac{2615}{143} a^{14} + \frac{4687}{143} a^{13} - \frac{5143}{143} a^{12} + \frac{5440}{143} a^{11} - \frac{6095}{143} a^{10} + \frac{4307}{143} a^{9} - \frac{602}{143} a^{8} + \frac{69}{143} a^{7} - \frac{2942}{143} a^{6} + \frac{6082}{143} a^{5} - \frac{6077}{143} a^{4} + \frac{4922}{143} a^{3} - \frac{4817}{143} a^{2} + \frac{3369}{143} a - \frac{1061}{143} \),  \( \frac{25}{11} a^{15} - \frac{1020}{143} a^{14} + \frac{1640}{143} a^{13} - \frac{124}{11} a^{12} + \frac{1865}{143} a^{11} - \frac{1981}{143} a^{10} + \frac{80}{11} a^{9} + \frac{179}{143} a^{8} + \frac{18}{11} a^{7} - \frac{1160}{143} a^{6} + \frac{159}{11} a^{5} - \frac{1864}{143} a^{4} + \frac{1787}{143} a^{3} - \frac{115}{11} a^{2} + \frac{795}{143} a - \frac{71}{143} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 22.3523537457 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_4^2.C_2$ (as 16T396):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 128
The 20 conjugacy class representatives for $D_4^2.C_2$
Character table for $D_4^2.C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), 4.0.320.1 x2, 4.2.400.1 x2, \(\Q(i, \sqrt{5})\), 8.2.4960000.1, 8.2.79360000.2, 8.0.2560000.1

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
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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$