Properties

Label 16.0.8084777718513664.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 7^{6}$
Root discriminant $9.87$
Ramified primes $2, 7$
Class number $1$
Class group Trivial
Galois Group $D_4:D_4$ (as 16T141)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 12, -20, 24, -24, 22, -20, 15, 0, -18, 24, -14, 0, 6, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 - 14*x^12 + 24*x^11 - 18*x^10 + 15*x^8 - 20*x^7 + 22*x^6 - 24*x^5 + 24*x^4 - 20*x^3 + 12*x^2 - 4*x + 1)
gp: K = bnfinit(x^16 - 4*x^15 + 6*x^14 - 14*x^12 + 24*x^11 - 18*x^10 + 15*x^8 - 20*x^7 + 22*x^6 - 24*x^5 + 24*x^4 - 20*x^3 + 12*x^2 - 4*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 4 x^{15} \) \(\mathstrut +\mathstrut 6 x^{14} \) \(\mathstrut -\mathstrut 14 x^{12} \) \(\mathstrut +\mathstrut 24 x^{11} \) \(\mathstrut -\mathstrut 18 x^{10} \) \(\mathstrut +\mathstrut 15 x^{8} \) \(\mathstrut -\mathstrut 20 x^{7} \) \(\mathstrut +\mathstrut 22 x^{6} \) \(\mathstrut -\mathstrut 24 x^{5} \) \(\mathstrut +\mathstrut 24 x^{4} \) \(\mathstrut -\mathstrut 20 x^{3} \) \(\mathstrut +\mathstrut 12 x^{2} \) \(\mathstrut -\mathstrut 4 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(8084777718513664=2^{36}\cdot 7^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.87$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5549} a^{15} + \frac{2615}{5549} a^{14} + \frac{1225}{5549} a^{13} + \frac{953}{5549} a^{12} - \frac{1157}{5549} a^{11} - \frac{405}{5549} a^{10} - \frac{854}{5549} a^{9} - \frac{379}{5549} a^{8} + \frac{685}{5549} a^{7} + \frac{1668}{5549} a^{6} + \frac{1451}{5549} a^{5} - \frac{920}{5549} a^{4} - \frac{1190}{5549} a^{3} + \frac{1908}{5549} a^{2} - \frac{2585}{5549} a - \frac{339}{5549}$

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:  \( -\frac{6779}{5549} a^{15} + \frac{24166}{5549} a^{14} - \frac{30716}{5549} a^{13} - \frac{12449}{5549} a^{12} + \frac{91350}{5549} a^{11} - \frac{128887}{5549} a^{10} + \frac{68247}{5549} a^{9} + \frac{38897}{5549} a^{8} - \frac{98984}{5549} a^{7} + \frac{95823}{5549} a^{6} - \frac{97834}{5549} a^{5} + \frac{110584}{5549} a^{4} - \frac{112216}{5549} a^{3} + \frac{83622}{5549} a^{2} - \frac{38870}{5549} a + \frac{6344}{5549} \) (order $8$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{2266}{5549} a^{15} - \frac{6291}{5549} a^{14} + \frac{6899}{5549} a^{13} + \frac{6486}{5549} a^{12} - \frac{24830}{5549} a^{11} + \frac{31149}{5549} a^{10} - \frac{9661}{5549} a^{9} - \frac{15366}{5549} a^{8} + \frac{26235}{5549} a^{7} - \frac{21377}{5549} a^{6} + \frac{25154}{5549} a^{5} - \frac{26041}{5549} a^{4} + \frac{22470}{5549} a^{3} - \frac{10241}{5549} a^{2} + \frac{7683}{5549} a + \frac{3137}{5549} \),  \( \frac{2568}{5549} a^{15} - \frac{10068}{5549} a^{14} + \frac{10615}{5549} a^{13} + \frac{11293}{5549} a^{12} - \frac{41304}{5549} a^{11} + \frac{42015}{5549} a^{10} - \frac{6766}{5549} a^{9} - \frac{29942}{5549} a^{8} + \frac{38890}{5549} a^{7} - \frac{28149}{5549} a^{6} + \frac{30534}{5549} a^{5} - \frac{31980}{5549} a^{4} + \frac{23775}{5549} a^{3} - \frac{16670}{5549} a^{2} + \frac{3873}{5549} a + \frac{6190}{5549} \),  \( \frac{5171}{5549} a^{15} - \frac{17395}{5549} a^{14} + \frac{19713}{5549} a^{13} + \frac{11549}{5549} a^{12} - \frac{62064}{5549} a^{11} + \frac{80953}{5549} a^{10} - \frac{43422}{5549} a^{9} - \frac{17659}{5549} a^{8} + \frac{51814}{5549} a^{7} - \frac{64506}{5549} a^{6} + \frac{78559}{5549} a^{5} - \frac{85062}{5549} a^{4} + \frac{78037}{5549} a^{3} - \frac{60893}{5549} a^{2} + \frac{28251}{5549} a - \frac{10583}{5549} \),  \( \frac{1528}{5549} a^{15} - \frac{5109}{5549} a^{14} + \frac{1787}{5549} a^{13} + \frac{13444}{5549} a^{12} - \frac{25510}{5549} a^{11} + \frac{8197}{5549} a^{10} + \frac{26848}{5549} a^{9} - \frac{40859}{5549} a^{8} + \frac{14566}{5549} a^{7} + \frac{12811}{5549} a^{6} - \frac{8021}{5549} a^{5} - \frac{1863}{5549} a^{4} - \frac{3797}{5549} a^{3} + \frac{13297}{5549} a^{2} - \frac{21188}{5549} a + \frac{9163}{5549} \),  \( \frac{2721}{5549} a^{15} - \frac{3952}{5549} a^{14} - \frac{1724}{5549} a^{13} + \frac{12828}{5549} a^{12} - \frac{13012}{5549} a^{11} + \frac{2246}{5549} a^{10} + \frac{12395}{5549} a^{9} - \frac{15792}{5549} a^{8} + \frac{10519}{5549} a^{7} - \frac{6003}{5549} a^{6} + \frac{2832}{5549} a^{5} + \frac{4828}{5549} a^{4} + \frac{2626}{5549} a^{3} + \frac{3353}{5549} a^{2} - \frac{8751}{5549} a - \frac{1285}{5549} \),  \( \frac{754}{5549} a^{15} - \frac{3734}{5549} a^{14} + \frac{2516}{5549} a^{13} + \frac{8290}{5549} a^{12} - \frac{17832}{5549} a^{11} + \frac{5374}{5549} a^{10} + \frac{16415}{5549} a^{9} - \frac{19414}{5549} a^{8} + \frac{433}{5549} a^{7} + \frac{9147}{5549} a^{6} + \frac{6450}{5549} a^{5} - \frac{16702}{5549} a^{4} + \frac{1678}{5549} a^{3} + \frac{6990}{5549} a^{2} - \frac{6940}{5549} a + \frac{5197}{5549} \),  \( \frac{2868}{5549} a^{15} - \frac{7977}{5549} a^{14} + \frac{6332}{5549} a^{13} + \frac{14194}{5549} a^{12} - \frac{33268}{5549} a^{11} + \frac{25946}{5549} a^{10} + \frac{8935}{5549} a^{9} - \frac{27113}{5549} a^{8} + \frac{22430}{5549} a^{7} - \frac{10512}{5549} a^{6} + \frac{21914}{5549} a^{5} - \frac{24981}{5549} a^{4} + \frac{16362}{5549} a^{3} - \frac{4719}{5549} a^{2} + \frac{5233}{5549} a + \frac{4372}{5549} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 60.0447869363 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_4:D_4$ (as 16T141):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 64
The 16 conjugacy class representatives for $D_4:D_4$
Character table for $D_4:D_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.2.1792.1, 4.2.448.1, \(\Q(\zeta_{8})\), 8.2.89915392.1 x2, 8.2.22478848.1 x2, 8.0.3211264.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$