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Properties

Label 16.0.26585452170...7104.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 7^{8}$
Root discriminant $38.82$
Ramified primes $2, 7$
Class number $4$ (GRH)
Class group $[4]$ (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![158, -1456, 6408, -17680, 34192, -49136, 54432, -47584, 33497, -19160, 9068, -3480, 1126, -280, 60, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 60*x^14 - 280*x^13 + 1126*x^12 - 3480*x^11 + 9068*x^10 - 19160*x^9 + 33497*x^8 - 47584*x^7 + 54432*x^6 - 49136*x^5 + 34192*x^4 - 17680*x^3 + 6408*x^2 - 1456*x + 158)
 
gp: K = bnfinit(x^16 - 8*x^15 + 60*x^14 - 280*x^13 + 1126*x^12 - 3480*x^11 + 9068*x^10 - 19160*x^9 + 33497*x^8 - 47584*x^7 + 54432*x^6 - 49136*x^5 + 34192*x^4 - 17680*x^3 + 6408*x^2 - 1456*x + 158, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 60 x^{14} - 280 x^{13} + 1126 x^{12} - 3480 x^{11} + 9068 x^{10} - 19160 x^{9} + 33497 x^{8} - 47584 x^{7} + 54432 x^{6} - 49136 x^{5} + 34192 x^{4} - 17680 x^{3} + 6408 x^{2} - 1456 x + 158 \)

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:  \(26585452170716224216367104=2^{62}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.82$
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:  $2, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(224=2^{5}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{224}(1,·)$, $\chi_{224}(195,·)$, $\chi_{224}(139,·)$, $\chi_{224}(209,·)$, $\chi_{224}(211,·)$, $\chi_{224}(153,·)$, $\chi_{224}(27,·)$, $\chi_{224}(97,·)$, $\chi_{224}(99,·)$, $\chi_{224}(41,·)$, $\chi_{224}(43,·)$, $\chi_{224}(113,·)$, $\chi_{224}(83,·)$, $\chi_{224}(155,·)$, $\chi_{224}(169,·)$, $\chi_{224}(57,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{9284159} a^{14} - \frac{7}{9284159} a^{13} - \frac{151212}{9284159} a^{12} + \frac{907363}{9284159} a^{11} - \frac{70197}{299489} a^{10} + \frac{2562874}{9284159} a^{9} + \frac{1539952}{9284159} a^{8} - \frac{2271614}{9284159} a^{7} - \frac{3528087}{9284159} a^{6} - \frac{3892710}{9284159} a^{5} + \frac{214139}{9284159} a^{4} + \frac{1726896}{9284159} a^{3} + \frac{2456004}{9284159} a^{2} + \frac{2612508}{9284159} a + \frac{10079}{117521}$, $\frac{1}{4168587391} a^{15} + \frac{7}{134470561} a^{14} + \frac{92688810}{4168587391} a^{13} - \frac{636434460}{4168587391} a^{12} + \frac{888100971}{4168587391} a^{11} - \frac{22708247}{52766929} a^{10} - \frac{789147645}{4168587391} a^{9} + \frac{82721182}{4168587391} a^{8} + \frac{1409451290}{4168587391} a^{7} + \frac{960521853}{4168587391} a^{6} - \frac{101167704}{4168587391} a^{5} - \frac{1955684312}{4168587391} a^{4} + \frac{1354833244}{4168587391} a^{3} - \frac{64528591}{134470561} a^{2} - \frac{1521506060}{4168587391} a - \frac{25124697}{52766929}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 620029.352334 \) (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

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\zeta_{16})^+\), 4.0.100352.5, 8.0.10070523904.2, 8.0.2147483648.1, 8.8.5156108238848.1

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 ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.31.8$x^{8} + 24 x^{6} + 4 x^{4} + 16 x^{2} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
2.8.31.8$x^{8} + 24 x^{6} + 4 x^{4} + 16 x^{2} + 2$$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}$