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Properties

Label 16.0.35963452480...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{8}\cdot 11^{8}$
Root discriminant $29.66$
Ramified primes $2, 5, 11$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $C_2^4$ (as 16T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14436, -9048, 12652, -11256, 19586, -28952, 34622, -31532, 22557, -13112, 6314, -2492, 840, -224, 52, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 52*x^14 - 224*x^13 + 840*x^12 - 2492*x^11 + 6314*x^10 - 13112*x^9 + 22557*x^8 - 31532*x^7 + 34622*x^6 - 28952*x^5 + 19586*x^4 - 11256*x^3 + 12652*x^2 - 9048*x + 14436)
 
gp: K = bnfinit(x^16 - 8*x^15 + 52*x^14 - 224*x^13 + 840*x^12 - 2492*x^11 + 6314*x^10 - 13112*x^9 + 22557*x^8 - 31532*x^7 + 34622*x^6 - 28952*x^5 + 19586*x^4 - 11256*x^3 + 12652*x^2 - 9048*x + 14436, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 840 x^{12} - 2492 x^{11} + 6314 x^{10} - 13112 x^{9} + 22557 x^{8} - 31532 x^{7} + 34622 x^{6} - 28952 x^{5} + 19586 x^{4} - 11256 x^{3} + 12652 x^{2} - 9048 x + 14436 \)

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:  \(359634524805529600000000=2^{32}\cdot 5^{8}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.66$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(440=2^{3}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{440}(1,·)$, $\chi_{440}(131,·)$, $\chi_{440}(199,·)$, $\chi_{440}(329,·)$, $\chi_{440}(331,·)$, $\chi_{440}(21,·)$, $\chi_{440}(89,·)$, $\chi_{440}(219,·)$, $\chi_{440}(221,·)$, $\chi_{440}(351,·)$, $\chi_{440}(419,·)$, $\chi_{440}(109,·)$, $\chi_{440}(111,·)$, $\chi_{440}(241,·)$, $\chi_{440}(309,·)$, $\chi_{440}(439,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} + \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{4}$, $\frac{1}{174} a^{13} + \frac{4}{87} a^{12} - \frac{13}{174} a^{11} - \frac{1}{87} a^{10} - \frac{1}{87} a^{9} + \frac{11}{174} a^{8} - \frac{1}{174} a^{7} + \frac{14}{87} a^{6} - \frac{67}{174} a^{5} - \frac{49}{174} a^{4} - \frac{5}{87} a^{3} + \frac{38}{87} a^{2} + \frac{7}{87} a - \frac{14}{29}$, $\frac{1}{1126359516222} a^{14} - \frac{7}{1126359516222} a^{13} + \frac{286976158}{13097203677} a^{12} + \frac{19823444300}{563179758111} a^{11} - \frac{89042033725}{1126359516222} a^{10} - \frac{37329232703}{563179758111} a^{9} + \frac{1022338463}{26194407354} a^{8} - \frac{21340996231}{563179758111} a^{7} - \frac{149770918322}{563179758111} a^{6} - \frac{546438483223}{1126359516222} a^{5} + \frac{7026165109}{38839983318} a^{4} - \frac{184071327730}{563179758111} a^{3} + \frac{97819543172}{563179758111} a^{2} - \frac{35589885983}{187726586037} a + \frac{16387471885}{62575528679}$, $\frac{1}{368319561804594} a^{15} + \frac{26}{61386593634099} a^{14} + \frac{477813087157}{368319561804594} a^{13} + \frac{51632593544}{20462197878033} a^{12} - \frac{1139777808433}{61386593634099} a^{11} - \frac{7278071654908}{184159780902297} a^{10} + \frac{5592547007042}{184159780902297} a^{9} - \frac{8854774634369}{122773187268198} a^{8} - \frac{2921641307147}{40924395756066} a^{7} - \frac{59360274035242}{184159780902297} a^{6} + \frac{128761185884119}{368319561804594} a^{5} + \frac{4423062740837}{122773187268198} a^{4} - \frac{71030875880027}{184159780902297} a^{3} + \frac{68537199911060}{184159780902297} a^{2} + \frac{3252413799649}{61386593634099} a - \frac{6889951309526}{20462197878033}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{193400867}{2522736724689} a^{15} + \frac{967004335}{1681824483126} a^{14} - \frac{18184151065}{5045473449378} a^{13} + \frac{12366380780}{840912241563} a^{12} - \frac{88525693919}{1681824483126} a^{11} + \frac{12152294297}{82712679498} a^{10} - \frac{864214273445}{2522736724689} a^{9} + \frac{59909316435}{93434693507} a^{8} - \frac{1542949783609}{1681824483126} a^{7} + \frac{2475441894967}{2522736724689} a^{6} - \frac{2584186818433}{5045473449378} a^{5} - \frac{178390017920}{840912241563} a^{4} + \frac{1113001250573}{2522736724689} a^{3} - \frac{571096818284}{2522736724689} a^{2} - \frac{12639110983}{19556098641} a + \frac{96351625001}{280304080521} \) (order $8$)
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:  \( 136789.59927 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4$ (as 16T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_2^4$
Character table for $C_2^4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-110}) \), \(\Q(\sqrt{110}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-55}) \), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{11})\), \(\Q(i, \sqrt{22})\), \(\Q(\sqrt{-2}, \sqrt{-11})\), \(\Q(\sqrt{-2}, \sqrt{11})\), \(\Q(\sqrt{2}, \sqrt{-11})\), \(\Q(\sqrt{2}, \sqrt{11})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{110})\), \(\Q(i, \sqrt{55})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{55})\), \(\Q(\sqrt{-2}, \sqrt{-55})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{-55})\), \(\Q(\sqrt{2}, \sqrt{55})\), \(\Q(\sqrt{10}, \sqrt{-11})\), \(\Q(\sqrt{-10}, \sqrt{-11})\), \(\Q(\sqrt{-5}, \sqrt{-11})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{10}, \sqrt{11})\), \(\Q(\sqrt{-10}, \sqrt{11})\), \(\Q(\sqrt{-5}, \sqrt{11})\), \(\Q(\sqrt{5}, \sqrt{11})\), \(\Q(\sqrt{10}, \sqrt{22})\), \(\Q(\sqrt{-10}, \sqrt{22})\), \(\Q(\sqrt{-5}, \sqrt{22})\), \(\Q(\sqrt{5}, \sqrt{22})\), \(\Q(\sqrt{10}, \sqrt{-22})\), \(\Q(\sqrt{-10}, \sqrt{-22})\), \(\Q(\sqrt{-5}, \sqrt{-22})\), \(\Q(\sqrt{5}, \sqrt{-22})\), 8.0.959512576.1, 8.0.40960000.1, 8.0.599695360000.9, 8.0.599695360000.6, 8.0.2342560000.1, 8.0.599695360000.3, 8.0.599695360000.8, 8.0.599695360000.5, 8.0.37480960000.2, 8.0.599695360000.4, 8.0.599695360000.7, 8.0.37480960000.9, 8.0.599695360000.1, 8.8.599695360000.1, 8.0.599695360000.2

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$