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Properties

Label 16.0.18545366580...0864.8
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 7^{8}\cdot 17^{8}$
Root discriminant $160.05$
Ramified primes $2, 7, 17$
Class number $19763200$ (GRH)
Class group $[4, 4, 4, 40, 7720]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1082924229118, -239749602736, 263791224520, -49508817424, 28373018608, -4496812400, 1760597328, -232736704, 68918345, -7411672, 1741116, -145272, 27670, -1624, 252, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 252*x^14 - 1624*x^13 + 27670*x^12 - 145272*x^11 + 1741116*x^10 - 7411672*x^9 + 68918345*x^8 - 232736704*x^7 + 1760597328*x^6 - 4496812400*x^5 + 28373018608*x^4 - 49508817424*x^3 + 263791224520*x^2 - 239749602736*x + 1082924229118)
 
gp: K = bnfinit(x^16 - 8*x^15 + 252*x^14 - 1624*x^13 + 27670*x^12 - 145272*x^11 + 1741116*x^10 - 7411672*x^9 + 68918345*x^8 - 232736704*x^7 + 1760597328*x^6 - 4496812400*x^5 + 28373018608*x^4 - 49508817424*x^3 + 263791224520*x^2 - 239749602736*x + 1082924229118, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 252 x^{14} - 1624 x^{13} + 27670 x^{12} - 145272 x^{11} + 1741116 x^{10} - 7411672 x^{9} + 68918345 x^{8} - 232736704 x^{7} + 1760597328 x^{6} - 4496812400 x^{5} + 28373018608 x^{4} - 49508817424 x^{3} + 263791224520 x^{2} - 239749602736 x + 1082924229118 \)

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:  \(185453665802223303376747219715620864=2^{62}\cdot 7^{8}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $160.05$
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:  $2, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3808=2^{5}\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{3808}(1,·)$, $\chi_{3808}(3333,·)$, $\chi_{3808}(1665,·)$, $\chi_{3808}(713,·)$, $\chi_{3808}(2381,·)$, $\chi_{3808}(3093,·)$, $\chi_{3808}(2617,·)$, $\chi_{3808}(477,·)$, $\chi_{3808}(1189,·)$, $\chi_{3808}(1905,·)$, $\chi_{3808}(2857,·)$, $\chi_{3808}(237,·)$, $\chi_{3808}(2141,·)$, $\chi_{3808}(3569,·)$, $\chi_{3808}(953,·)$, $\chi_{3808}(1429,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11916846145500052417919521} a^{14} - \frac{7}{11916846145500052417919521} a^{13} - \frac{3946614775289306062233198}{11916846145500052417919521} a^{12} - \frac{154003639264268462439763}{11916846145500052417919521} a^{11} + \frac{5668510689393913116678295}{11916846145500052417919521} a^{10} + \frac{4847402967619701770091575}{11916846145500052417919521} a^{9} - \frac{1902777039756319245737095}{11916846145500052417919521} a^{8} + \frac{666342612400218111400865}{11916846145500052417919521} a^{7} + \frac{2365947729569047936044108}{11916846145500052417919521} a^{6} - \frac{1677125210798269578317069}{11916846145500052417919521} a^{5} + \frac{3052857356128134700769957}{11916846145500052417919521} a^{4} - \frac{5130368167318410779365804}{11916846145500052417919521} a^{3} - \frac{1627154299733551835803219}{11916846145500052417919521} a^{2} - \frac{2163018222950889671088646}{11916846145500052417919521} a + \frac{2397684802555783526854831}{11916846145500052417919521}$, $\frac{1}{3087808527863802095658981799162337} a^{15} + \frac{129556441}{3087808527863802095658981799162337} a^{14} - \frac{32823907971933865669853512804986}{3087808527863802095658981799162337} a^{13} - \frac{297953992066537926598411173263187}{3087808527863802095658981799162337} a^{12} + \frac{805228100755126475252927805911884}{3087808527863802095658981799162337} a^{11} + \frac{128839213130646181955094206543}{16166536795098440291408281671007} a^{10} - \frac{160413716158972305606416116830683}{3087808527863802095658981799162337} a^{9} + \frac{1679112528378865589694019417415}{9162636581198225803142379226001} a^{8} + \frac{128983428461994900128968367867768}{3087808527863802095658981799162337} a^{7} + \frac{598968859148841754898032200523311}{3087808527863802095658981799162337} a^{6} - \frac{637259174935512845811922099430272}{3087808527863802095658981799162337} a^{5} - \frac{1128151388027386158691804554125216}{3087808527863802095658981799162337} a^{4} - \frac{1268060570908453449401148143795031}{3087808527863802095658981799162337} a^{3} + \frac{1235031601615426223508350104641600}{3087808527863802095658981799162337} a^{2} + \frac{34532541795331368395378422662974}{99606726705283938569644574166527} a + \frac{1402276167903703911330726355047676}{3087808527863802095658981799162337}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{4}\times C_{40}\times C_{7720}$, which has order $19763200$ (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:  \( 15753.94986242651 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

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 $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-238}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-119}) \), \(\Q(\sqrt{2}, \sqrt{-119})\), \(\Q(\zeta_{16})^+\), 4.0.29001728.14, 8.0.841100226985984.129, \(\Q(\zeta_{32})^+\), 8.0.430643316216823808.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
2.8.31.6$x^{8} + 16 x^{7} + 28 x^{4} + 16 x^{3} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$