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Properties

Label 16.0.97066466292...0129.1
Degree $16$
Signature $[0, 8]$
Discriminant $7^{8}\cdot 17^{14}$
Root discriminant $31.56$
Ramified primes $7, 17$
Class number $45$ (GRH)
Class group $[3, 15]$ (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![256, 512, 3968, -960, 10480, -1072, 10340, -1206, 5217, -512, 1478, -128, 247, -16, 23, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 23*x^14 - 16*x^13 + 247*x^12 - 128*x^11 + 1478*x^10 - 512*x^9 + 5217*x^8 - 1206*x^7 + 10340*x^6 - 1072*x^5 + 10480*x^4 - 960*x^3 + 3968*x^2 + 512*x + 256)
 
gp: K = bnfinit(x^16 - x^15 + 23*x^14 - 16*x^13 + 247*x^12 - 128*x^11 + 1478*x^10 - 512*x^9 + 5217*x^8 - 1206*x^7 + 10340*x^6 - 1072*x^5 + 10480*x^4 - 960*x^3 + 3968*x^2 + 512*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 23 x^{14} - 16 x^{13} + 247 x^{12} - 128 x^{11} + 1478 x^{10} - 512 x^{9} + 5217 x^{8} - 1206 x^{7} + 10340 x^{6} - 1072 x^{5} + 10480 x^{4} - 960 x^{3} + 3968 x^{2} + 512 x + 256 \)

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:  \(970664662927461034900129=7^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.56$
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:  $7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(119=7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{119}(64,·)$, $\chi_{119}(1,·)$, $\chi_{119}(69,·)$, $\chi_{119}(8,·)$, $\chi_{119}(76,·)$, $\chi_{119}(13,·)$, $\chi_{119}(15,·)$, $\chi_{119}(83,·)$, $\chi_{119}(36,·)$, $\chi_{119}(104,·)$, $\chi_{119}(106,·)$, $\chi_{119}(43,·)$, $\chi_{119}(111,·)$, $\chi_{119}(50,·)$, $\chi_{119}(118,·)$, $\chi_{119}(55,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{4} a^{5} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} + \frac{7}{16} a^{8} - \frac{1}{2} a^{7} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{16} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{12} - \frac{1}{32} a^{11} + \frac{7}{32} a^{9} - \frac{1}{4} a^{8} + \frac{7}{16} a^{7} + \frac{1}{4} a^{6} + \frac{1}{32} a^{5} - \frac{3}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{13} - \frac{1}{64} a^{12} + \frac{7}{64} a^{10} - \frac{1}{8} a^{9} + \frac{7}{32} a^{8} - \frac{3}{8} a^{7} - \frac{31}{64} a^{6} - \frac{3}{32} a^{5} - \frac{5}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{233166472395434373248} a^{15} + \frac{1744111799362360603}{233166472395434373248} a^{14} + \frac{2059134807952905427}{233166472395434373248} a^{13} + \frac{820749644659564679}{58291618098858593312} a^{12} - \frac{14445605618290027281}{233166472395434373248} a^{11} - \frac{167366262782780091}{58291618098858593312} a^{10} - \frac{17787808042702612481}{116583236197717186624} a^{9} - \frac{1554317510717170887}{29145809049429296656} a^{8} + \frac{31767959797613682337}{233166472395434373248} a^{7} - \frac{22633920275511305981}{116583236197717186624} a^{6} - \frac{11047626730501128435}{58291618098858593312} a^{5} - \frac{4596926853070110713}{14572904524714648328} a^{4} + \frac{1674656964873030791}{7286452262357324164} a^{3} - \frac{409451417165024872}{1821613065589331041} a^{2} - \frac{396373735951031421}{1821613065589331041} a + \frac{869706355501611714}{1821613065589331041}$

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

Class group and class number

$C_{3}\times C_{15}$, which has order $45$ (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:  \( 3640.01221338 \) (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

\(\Q(\sqrt{-119}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-7}, \sqrt{17})\), 4.4.4913.1, 4.0.240737.1, 8.0.57954303169.1, 8.0.985223153873.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
7Data not computed
17Data not computed