Properties

Label 16.0.3143861048180736.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 13^{4}$
Root discriminant $9.30$
Ramified primes $2, 3, 13$
Class number $1$
Class group Trivial
Galois Group $C_2\wr C_2^2$ (as 16T149)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 18, -40, 73, -102, 98, -48, -18, 58, -56, 30, -4, -8, 8, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 8*x^13 - 4*x^12 + 30*x^11 - 56*x^10 + 58*x^9 - 18*x^8 - 48*x^7 + 98*x^6 - 102*x^5 + 73*x^4 - 40*x^3 + 18*x^2 - 6*x + 1)
gp: K = bnfinit(x^16 - 4*x^15 + 8*x^14 - 8*x^13 - 4*x^12 + 30*x^11 - 56*x^10 + 58*x^9 - 18*x^8 - 48*x^7 + 98*x^6 - 102*x^5 + 73*x^4 - 40*x^3 + 18*x^2 - 6*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 4 x^{15} \) \(\mathstrut +\mathstrut 8 x^{14} \) \(\mathstrut -\mathstrut 8 x^{13} \) \(\mathstrut -\mathstrut 4 x^{12} \) \(\mathstrut +\mathstrut 30 x^{11} \) \(\mathstrut -\mathstrut 56 x^{10} \) \(\mathstrut +\mathstrut 58 x^{9} \) \(\mathstrut -\mathstrut 18 x^{8} \) \(\mathstrut -\mathstrut 48 x^{7} \) \(\mathstrut +\mathstrut 98 x^{6} \) \(\mathstrut -\mathstrut 102 x^{5} \) \(\mathstrut +\mathstrut 73 x^{4} \) \(\mathstrut -\mathstrut 40 x^{3} \) \(\mathstrut +\mathstrut 18 x^{2} \) \(\mathstrut -\mathstrut 6 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(3143861048180736=2^{24}\cdot 3^{8}\cdot 13^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.30$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{23711} a^{15} + \frac{7757}{23711} a^{14} - \frac{144}{23711} a^{13} - \frac{3175}{23711} a^{12} - \frac{5450}{23711} a^{11} + \frac{3004}{23711} a^{10} + \frac{6075}{23711} a^{9} + \frac{10665}{23711} a^{8} - \frac{4054}{23711} a^{7} + \frac{1355}{23711} a^{6} - \frac{11431}{23711} a^{5} + \frac{10469}{23711} a^{4} - \frac{7615}{23711} a^{3} + \frac{11468}{23711} a^{2} - \frac{7928}{23711} a + \frac{831}{23711}$

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:  \( \frac{43703}{23711} a^{15} - \frac{134262}{23711} a^{14} + \frac{227293}{23711} a^{13} - \frac{142519}{23711} a^{12} - \frac{312598}{23711} a^{11} + \frac{1039289}{23711} a^{10} - \frac{1513846}{23711} a^{9} + \frac{1143496}{23711} a^{8} + \frac{328584}{23711} a^{7} - \frac{1909393}{23711} a^{6} + \frac{2629987}{23711} a^{5} - \frac{2039895}{23711} a^{4} + \frac{1147379}{23711} a^{3} - \frac{491334}{23711} a^{2} + \frac{201147}{23711} a - \frac{31770}{23711} \) (order $12$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{48145}{23711} a^{15} - \frac{177173}{23711} a^{14} + \frac{322686}{23711} a^{13} - \frac{256379}{23711} a^{12} - \frac{312567}{23711} a^{11} + \frac{1365718}{23711} a^{10} - \frac{2199433}{23711} a^{9} + \frac{1901600}{23711} a^{8} - \frac{14589}{23711} a^{7} - \frac{2458430}{23711} a^{6} + \frac{3827997}{23711} a^{5} - \frac{3314262}{23711} a^{4} + \frac{2081875}{23711} a^{3} - \frac{1027059}{23711} a^{2} + \frac{432916}{23711} a - \frac{86806}{23711} \),  \( \frac{7262}{23711} a^{15} + \frac{17709}{23711} a^{14} - \frac{73577}{23711} a^{13} + \frac{132508}{23711} a^{12} - \frac{99085}{23711} a^{11} - \frac{165049}{23711} a^{10} + \frac{583254}{23711} a^{9} - \frac{797070}{23711} a^{8} + \frac{554267}{23711} a^{7} + \frac{331899}{23711} a^{6} - \frac{1114128}{23711} a^{5} + \frac{1265095}{23711} a^{4} - \frac{788541}{23711} a^{3} + \frac{410671}{23711} a^{2} - \frac{168805}{23711} a + \frac{59550}{23711} \),  \( \frac{45487}{23711} a^{15} - \frac{142998}{23711} a^{14} + \frac{231218}{23711} a^{13} - \frac{139790}{23711} a^{12} - \frac{337599}{23711} a^{11} + \frac{1087161}{23711} a^{10} - \frac{1535684}{23711} a^{9} + \frac{1129923}{23711} a^{8} + \frac{375525}{23711} a^{7} - \frac{1981728}{23711} a^{6} + \frac{2604832}{23711} a^{5} - \frac{2047467}{23711} a^{4} + \frac{1196044}{23711} a^{3} - \frac{566148}{23711} a^{2} + \frac{236773}{23711} a - \frac{43059}{23711} \),  \( \frac{89192}{23711} a^{15} - \frac{332879}{23711} a^{14} + \frac{600489}{23711} a^{13} - \frac{478347}{23711} a^{12} - \frac{589964}{23711} a^{11} + \frac{2559256}{23711} a^{10} - \frac{4104375}{23711} a^{9} + \frac{3527721}{23711} a^{8} + \frac{8382}{23711} a^{7} - \frac{4647163}{23711} a^{6} + \frac{7132548}{23711} a^{5} - \frac{6176703}{23711} a^{4} + \frac{3798275}{23711} a^{3} - \frac{1864431}{23711} a^{2} + \frac{730307}{23711} a - \frac{144300}{23711} \),  \( \frac{468}{131} a^{15} - \frac{1568}{131} a^{14} + \frac{2824}{131} a^{13} - \frac{2194}{131} a^{12} - \frac{2912}{131} a^{11} + \frac{12032}{131} a^{10} - \frac{19250}{131} a^{9} + \frac{16757}{131} a^{8} + \frac{1}{131} a^{7} - \frac{21253}{131} a^{6} + \frac{33737}{131} a^{5} - \frac{29776}{131} a^{4} + \frac{18768}{131} a^{3} - \frac{8954}{131} a^{2} + \frac{3808}{131} a - \frac{948}{131} \),  \( \frac{67493}{23711} a^{15} - \frac{232789}{23711} a^{14} + \frac{405605}{23711} a^{13} - \frac{298500}{23711} a^{12} - \frac{458616}{23711} a^{11} + \frac{1774536}{23711} a^{10} - \frac{2741113}{23711} a^{9} + \frac{2270563}{23711} a^{8} + \frac{174295}{23711} a^{7} - \frac{3177586}{23711} a^{6} + \frac{4714524}{23711} a^{5} - \frac{4034453}{23711} a^{4} + \frac{2513807}{23711} a^{3} - \frac{1268843}{23711} a^{2} + \frac{499564}{23711} a - \frac{108387}{23711} \),  \( \frac{920}{23711} a^{15} - \frac{571}{23711} a^{14} - \frac{13925}{23711} a^{13} + \frac{42875}{23711} a^{12} - \frac{58401}{23711} a^{11} + \frac{13204}{23711} a^{10} + \frac{135470}{23711} a^{9} - \frac{312797}{23711} a^{8} + \frac{348612}{23711} a^{7} - \frac{128638}{23711} a^{6} - \frac{297079}{23711} a^{5} + \frac{597589}{23711} a^{4} - \frac{532697}{23711} a^{3} + \frac{283697}{23711} a^{2} - \frac{109327}{23711} a + \frac{53190}{23711} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 44.6663869612 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2\wr C_2^2$ (as 16T149):

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 $C_2\wr C_2^2$
Character table for $C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), 4.0.117.1, \(\Q(\zeta_{12})\), 4.0.1872.1, 8.0.3504384.1 x2, 8.0.3504384.2, 8.0.4313088.1 x2

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$