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Properties

Label 16.0.41107867812...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 17^{14}$
Root discriminant $39.89$
Ramified primes $5, 17$
Class number $146$ (GRH)
Class group $[146]$ (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![4336, 7816, 12432, 11662, 8927, 5225, 3168, -121, 470, -399, -142, -75, 2, -17, 12, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 12*x^14 - 17*x^13 + 2*x^12 - 75*x^11 - 142*x^10 - 399*x^9 + 470*x^8 - 121*x^7 + 3168*x^6 + 5225*x^5 + 8927*x^4 + 11662*x^3 + 12432*x^2 + 7816*x + 4336)
 
gp: K = bnfinit(x^16 - x^15 + 12*x^14 - 17*x^13 + 2*x^12 - 75*x^11 - 142*x^10 - 399*x^9 + 470*x^8 - 121*x^7 + 3168*x^6 + 5225*x^5 + 8927*x^4 + 11662*x^3 + 12432*x^2 + 7816*x + 4336, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 12 x^{14} - 17 x^{13} + 2 x^{12} - 75 x^{11} - 142 x^{10} - 399 x^{9} + 470 x^{8} - 121 x^{7} + 3168 x^{6} + 5225 x^{5} + 8927 x^{4} + 11662 x^{3} + 12432 x^{2} + 7816 x + 4336 \)

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:  \(41107867812353742431640625=5^{12}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.89$
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:  $5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(85=5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{85}(64,·)$, $\chi_{85}(1,·)$, $\chi_{85}(2,·)$, $\chi_{85}(4,·)$, $\chi_{85}(69,·)$, $\chi_{85}(8,·)$, $\chi_{85}(77,·)$, $\chi_{85}(16,·)$, $\chi_{85}(81,·)$, $\chi_{85}(83,·)$, $\chi_{85}(84,·)$, $\chi_{85}(21,·)$, $\chi_{85}(32,·)$, $\chi_{85}(42,·)$, $\chi_{85}(43,·)$, $\chi_{85}(53,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{32} a^{13} - \frac{1}{16} a^{12} - \frac{1}{32} a^{11} - \frac{1}{8} a^{10} + \frac{1}{32} a^{9} + \frac{1}{16} a^{8} + \frac{1}{32} a^{7} - \frac{1}{4} a^{6} + \frac{3}{32} a^{5} + \frac{1}{16} a^{4} + \frac{1}{32} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{287648} a^{14} - \frac{59}{35956} a^{13} + \frac{17107}{287648} a^{12} - \frac{3155}{143824} a^{11} - \frac{23583}{287648} a^{10} + \frac{8239}{71912} a^{9} - \frac{34515}{287648} a^{8} - \frac{9767}{143824} a^{7} + \frac{16651}{287648} a^{6} - \frac{37}{356} a^{5} + \frac{35581}{287648} a^{4} - \frac{1389}{143824} a^{3} - \frac{4649}{17978} a^{2} - \frac{9005}{35956} a - \frac{7115}{17978}$, $\frac{1}{765438287643647648} a^{15} + \frac{21037299445}{95679785955455956} a^{14} - \frac{3018935244648289}{382719143821823824} a^{13} + \frac{585556165806002}{23919946488863989} a^{12} + \frac{20594261425782973}{382719143821823824} a^{11} + \frac{4048834567557887}{191359571910911912} a^{10} + \frac{435936975185113}{95679785955455956} a^{9} + \frac{5613929370750201}{47839892977727978} a^{8} + \frac{24505262305291299}{382719143821823824} a^{7} - \frac{35030872440772747}{191359571910911912} a^{6} - \frac{47757350764564027}{382719143821823824} a^{5} - \frac{595290168700045}{23919946488863989} a^{4} + \frac{273564998943326139}{765438287643647648} a^{3} - \frac{23789804694035911}{191359571910911912} a^{2} - \frac{10220102721128446}{23919946488863989} a + \frac{18051254894262569}{95679785955455956}$

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

Class group and class number

$C_{146}$, which has order $146$ (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:  \( 55081.0821685 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}, \sqrt{17})\), 4.4.122825.1, 4.4.4913.1, 8.8.15085980625.1, 8.0.6411541765625.2, 8.0.6411541765625.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
17Data not computed