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Properties

Label 16.0.11400342036...000.25
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 5^{12}\cdot 61^{12}$
Root discriminant $490.97$
Ramified primes $2, 5, 61$
Class number $1098923680$ (GRH)
Class group $[2, 2, 2, 74, 1856290]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![908425628010000, 0, 201872361780000, 0, 15315089013000, 0, 543392514000, 0, 10034792800, 0, 98978600, 0, 508130, 0, 1220, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 1220*x^14 + 508130*x^12 + 98978600*x^10 + 10034792800*x^8 + 543392514000*x^6 + 15315089013000*x^4 + 201872361780000*x^2 + 908425628010000)
 
gp: K = bnfinit(x^16 + 1220*x^14 + 508130*x^12 + 98978600*x^10 + 10034792800*x^8 + 543392514000*x^6 + 15315089013000*x^4 + 201872361780000*x^2 + 908425628010000, 1)
 

Normalized defining polynomial

\( x^{16} + 1220 x^{14} + 508130 x^{12} + 98978600 x^{10} + 10034792800 x^{8} + 543392514000 x^{6} + 15315089013000 x^{4} + 201872361780000 x^{2} + 908425628010000 \)

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:  \(11400342036779277121730973270016000000000000=2^{44}\cdot 5^{12}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $490.97$
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:  $2, 5, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4880=2^{4}\cdot 5\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{4880}(1,·)$, $\chi_{4880}(2307,·)$, $\chi_{4880}(2561,·)$, $\chi_{4880}(2441,·)$, $\chi_{4880}(843,·)$, $\chi_{4880}(3283,·)$, $\chi_{4880}(987,·)$, $\chi_{4880}(609,·)$, $\chi_{4880}(3427,·)$, $\chi_{4880}(4747,·)$, $\chi_{4880}(3049,·)$, $\chi_{4880}(1963,·)$, $\chi_{4880}(2929,·)$, $\chi_{4880}(4403,·)$, $\chi_{4880}(489,·)$, $\chi_{4880}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{610} a^{4}$, $\frac{1}{610} a^{5}$, $\frac{1}{610} a^{6}$, $\frac{1}{1830} a^{7} + \frac{1}{1830} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{5581500} a^{8} + \frac{1}{9150} a^{6} + \frac{2}{4575} a^{4} + \frac{4}{15} a^{2} + \frac{2}{5}$, $\frac{1}{5581500} a^{9} + \frac{1}{9150} a^{7} + \frac{2}{4575} a^{5} + \frac{4}{15} a^{3} + \frac{2}{5} a$, $\frac{1}{16744500} a^{10} - \frac{1}{16744500} a^{8} + \frac{4}{13725} a^{6} + \frac{11}{27450} a^{4} + \frac{7}{45} a^{2} + \frac{1}{5}$, $\frac{1}{50233500} a^{11} - \frac{1}{12558375} a^{9} + \frac{1}{16470} a^{7} - \frac{1}{82350} a^{5} - \frac{10}{27} a^{3} + \frac{4}{15} a$, $\frac{1}{32358411360000} a^{12} + \frac{169}{6630822000} a^{10} - \frac{419}{6630822000} a^{8} + \frac{1321}{43480800} a^{6} - \frac{5971}{10870200} a^{4} + \frac{421}{1980} a^{2} - \frac{871}{1760}$, $\frac{1}{97075234080000} a^{13} + \frac{169}{19892466000} a^{11} - \frac{419}{19892466000} a^{9} + \frac{1321}{130442400} a^{7} - \frac{5971}{32610600} a^{5} - \frac{1559}{5940} a^{3} - \frac{871}{5280} a$, $\frac{1}{392281020917280000} a^{14} + \frac{5639}{392281020917280000} a^{12} - \frac{46169}{20096363776500} a^{10} - \frac{7959121}{160770910212000} a^{8} + \frac{412697459}{527117738400} a^{6} + \frac{8627237}{14642159400} a^{4} + \frac{7419409}{21336480} a^{2} + \frac{152315}{474144}$, $\frac{1}{1176843062751840000} a^{15} + \frac{5639}{1176843062751840000} a^{13} - \frac{46169}{60289091329500} a^{11} - \frac{36763369}{482312730636000} a^{9} + \frac{355088963}{1581353215200} a^{7} - \frac{21777247}{43926478200} a^{5} - \frac{19606799}{64009440} a^{3} - \frac{2557433}{7112160} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{74}\times C_{1856290}$, which has order $1098923680$ (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:  \( -1 \) (order $2$)
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:  \( 2927643.5650334265 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{61}) \), \(\Q(\sqrt{122}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{610}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{305}) \), \(\Q(\sqrt{2}, \sqrt{61})\), \(\Q(\sqrt{10}, \sqrt{61})\), \(\Q(\sqrt{5}, \sqrt{61})\), \(\Q(\sqrt{5}, \sqrt{122})\), \(\Q(\sqrt{10}, \sqrt{122})\), \(\Q(\sqrt{2}, \sqrt{305})\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.58107136000.4, 4.0.58107136000.7, 4.0.58107136000.6, 4.0.58107136000.8, 8.8.35445352960000.2, 8.0.3376439254122496000000.5, 8.0.3376439254122496000000.3, 8.0.3376439254122496000000.6, 8.0.3376439254122496000000.1, 8.0.3376439254122496000000.4, 8.0.3376439254122496000000.2

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$61$61.8.6.2$x^{8} + 183 x^{4} + 14884$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
61.8.6.2$x^{8} + 183 x^{4} + 14884$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$