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Properties

Label 16.0.63456228123...000.18
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}$
Root discriminant $40.99$
Ramified primes $2, 3, 5, 7$
Class number $256$ (GRH)
Class group $[2, 4, 4, 8]$ (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![37809, -47772, 78354, -73380, 70504, -50244, 34410, -18388, 8969, -3340, 998, -76, -48, 0, 20, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 20*x^14 - 48*x^12 - 76*x^11 + 998*x^10 - 3340*x^9 + 8969*x^8 - 18388*x^7 + 34410*x^6 - 50244*x^5 + 70504*x^4 - 73380*x^3 + 78354*x^2 - 47772*x + 37809)
 
gp: K = bnfinit(x^16 - 8*x^15 + 20*x^14 - 48*x^12 - 76*x^11 + 998*x^10 - 3340*x^9 + 8969*x^8 - 18388*x^7 + 34410*x^6 - 50244*x^5 + 70504*x^4 - 73380*x^3 + 78354*x^2 - 47772*x + 37809, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 20 x^{14} - 48 x^{12} - 76 x^{11} + 998 x^{10} - 3340 x^{9} + 8969 x^{8} - 18388 x^{7} + 34410 x^{6} - 50244 x^{5} + 70504 x^{4} - 73380 x^{3} + 78354 x^{2} - 47772 x + 37809 \)

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:  \(63456228123711897600000000=2^{32}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.99$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(840=2^{3}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(491,·)$, $\chi_{840}(769,·)$, $\chi_{840}(589,·)$, $\chi_{840}(659,·)$, $\chi_{840}(601,·)$, $\chi_{840}(349,·)$, $\chi_{840}(671,·)$, $\chi_{840}(419,·)$, $\chi_{840}(421,·)$, $\chi_{840}(169,·)$, $\chi_{840}(71,·)$, $\chi_{840}(239,·)$, $\chi_{840}(839,·)$, $\chi_{840}(181,·)$, $\chi_{840}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{210} a^{12} - \frac{1}{35} a^{11} + \frac{1}{210} a^{10} + \frac{5}{21} a^{9} + \frac{9}{35} a^{7} - \frac{73}{210} a^{6} + \frac{16}{35} a^{5} - \frac{2}{5} a^{4} + \frac{4}{105} a^{3} - \frac{83}{210} a^{2} + \frac{6}{35} a - \frac{23}{70}$, $\frac{1}{210} a^{13} - \frac{1}{6} a^{11} - \frac{7}{30} a^{10} - \frac{1}{14} a^{9} - \frac{17}{70} a^{8} + \frac{41}{210} a^{7} - \frac{9}{70} a^{6} - \frac{11}{70} a^{5} + \frac{29}{210} a^{4} - \frac{1}{6} a^{3} + \frac{3}{10} a^{2} + \frac{1}{5} a - \frac{33}{70}$, $\frac{1}{434466595290} a^{14} - \frac{1}{62066656470} a^{13} + \frac{49390097}{43446659529} a^{12} - \frac{2963405729}{434466595290} a^{11} - \frac{83297158117}{434466595290} a^{10} + \frac{4591873822}{217233297645} a^{9} - \frac{6759758179}{33420507330} a^{8} - \frac{170659118579}{434466595290} a^{7} - \frac{27731684257}{62066656470} a^{6} - \frac{18289827721}{43446659529} a^{5} + \frac{6966363791}{62066656470} a^{4} + \frac{190766134933}{434466595290} a^{3} - \frac{20203316317}{217233297645} a^{2} + \frac{26182771009}{144822198430} a + \frac{47738734191}{144822198430}$, $\frac{1}{18477429831088410} a^{15} + \frac{21257}{18477429831088410} a^{14} + \frac{5721938389038}{3079571638514735} a^{13} - \frac{18338759381248}{9238714915544205} a^{12} + \frac{239182448807404}{3079571638514735} a^{11} + \frac{134137927364201}{3695485966217682} a^{10} - \frac{1175622207638618}{9238714915544205} a^{9} + \frac{7472820227632}{174315375764985} a^{8} - \frac{701951373521176}{9238714915544205} a^{7} + \frac{7613142147869813}{18477429831088410} a^{6} - \frac{1133913017532899}{9238714915544205} a^{5} - \frac{1480201361514102}{3079571638514735} a^{4} - \frac{7033285022649553}{18477429831088410} a^{3} - \frac{13535337192743}{615914327702947} a^{2} + \frac{313461579418961}{3079571638514735} a - \frac{686643557446519}{3079571638514735}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{8}$, which has order $256$ (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:  \( 15197.4244561 \) (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

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-210}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{2}, \sqrt{-35})\), \(\Q(\sqrt{3}, \sqrt{-35})\), \(\Q(\sqrt{6}, \sqrt{-35})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-105})\), \(\Q(\sqrt{3}, \sqrt{-70})\), \(\Q(\sqrt{6}, \sqrt{-70})\), \(\Q(\sqrt{10}, \sqrt{-14})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{30}, \sqrt{-35})\), \(\Q(\sqrt{15}, \sqrt{-21})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{2}, \sqrt{-21})\), \(\Q(\sqrt{-7}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{-14})\), \(\Q(\sqrt{-21}, \sqrt{30})\), \(\Q(\sqrt{15}, \sqrt{-42})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{-14})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{10}, \sqrt{-42})\), \(\Q(\sqrt{-14}, \sqrt{30})\), \(\Q(\sqrt{5}, \sqrt{-21})\), \(\Q(\sqrt{-7}, \sqrt{15})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{6}, \sqrt{-14})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{-7})\), \(\Q(\sqrt{10}, \sqrt{-21})\), \(\Q(\sqrt{-14}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{-42})\), \(\Q(\sqrt{-7}, \sqrt{30})\), 8.0.7965941760000.58, 8.0.6146560000.2, 8.0.7965941760000.16, 8.0.7965941760000.7, 8.0.31116960000.6, 8.0.7965941760000.52, 8.0.497871360000.15, 8.8.3317760000.1, 8.0.12745506816.6, 8.0.7965941760000.62, 8.0.7965941760000.47, 8.0.7965941760000.43, 8.0.7965941760000.53, 8.0.7965941760000.14, 8.0.7965941760000.27

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 R R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{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}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$