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Properties

Label 16.0.13185833497...6049.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 23^{8}$
Root discriminant $57.21$
Ramified primes $17, 23$
Class number $5145$ (GRH)
Class group $[7, 7, 105]$ (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![10693577, -4933595, 8317312, -3529051, 3154322, -1200952, 749631, -251532, 121104, -35071, 13568, -3286, 1027, -194, 48, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 48*x^14 - 194*x^13 + 1027*x^12 - 3286*x^11 + 13568*x^10 - 35071*x^9 + 121104*x^8 - 251532*x^7 + 749631*x^6 - 1200952*x^5 + 3154322*x^4 - 3529051*x^3 + 8317312*x^2 - 4933595*x + 10693577)
 
gp: K = bnfinit(x^16 - 6*x^15 + 48*x^14 - 194*x^13 + 1027*x^12 - 3286*x^11 + 13568*x^10 - 35071*x^9 + 121104*x^8 - 251532*x^7 + 749631*x^6 - 1200952*x^5 + 3154322*x^4 - 3529051*x^3 + 8317312*x^2 - 4933595*x + 10693577, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 48 x^{14} - 194 x^{13} + 1027 x^{12} - 3286 x^{11} + 13568 x^{10} - 35071 x^{9} + 121104 x^{8} - 251532 x^{7} + 749631 x^{6} - 1200952 x^{5} + 3154322 x^{4} - 3529051 x^{3} + 8317312 x^{2} - 4933595 x + 10693577 \)

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:  \(13185833497340017023096726049=17^{14}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.21$
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:  $17, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(391=17\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{391}(1,·)$, $\chi_{391}(390,·)$, $\chi_{391}(321,·)$, $\chi_{391}(137,·)$, $\chi_{391}(206,·)$, $\chi_{391}(208,·)$, $\chi_{391}(344,·)$, $\chi_{391}(93,·)$, $\chi_{391}(229,·)$, $\chi_{391}(162,·)$, $\chi_{391}(70,·)$, $\chi_{391}(298,·)$, $\chi_{391}(47,·)$, $\chi_{391}(183,·)$, $\chi_{391}(185,·)$, $\chi_{391}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{1420066336313268079323697673207535485638} a^{15} - \frac{114508356691323773943734489100514292461}{1420066336313268079323697673207535485638} a^{14} + \frac{20523632831455624706662478881202812220}{710033168156634039661848836603767742819} a^{13} + \frac{38035165719175662361289484016968843915}{710033168156634039661848836603767742819} a^{12} + \frac{330904446064102077864297969967425768213}{1420066336313268079323697673207535485638} a^{11} + \frac{277127639976811437785439716171371083472}{710033168156634039661848836603767742819} a^{10} - \frac{136429160006479670981291122941659161291}{710033168156634039661848836603767742819} a^{9} - \frac{657265325910297897800127996150551025787}{1420066336313268079323697673207535485638} a^{8} - \frac{122906967058605139483885621840606311809}{710033168156634039661848836603767742819} a^{7} + \frac{314377857511426365372102590035057842995}{710033168156634039661848836603767742819} a^{6} + \frac{298190279239011529305752345971495669389}{1420066336313268079323697673207535485638} a^{5} + \frac{73863679348527786631330619892595726238}{710033168156634039661848836603767742819} a^{4} - \frac{289542179745691055983663972767660857959}{710033168156634039661848836603767742819} a^{3} + \frac{394162451673052402545626156846961423291}{1420066336313268079323697673207535485638} a^{2} - \frac{341948660049627717557572459287834681100}{710033168156634039661848836603767742819} a + \frac{648001352876436462739859740953679118655}{1420066336313268079323697673207535485638}$

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

Class group and class number

$C_{7}\times C_{7}\times C_{105}$, which has order $5145$ (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

\(\Q(\sqrt{-23}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-391}) \), \(\Q(\sqrt{17}, \sqrt{-23})\), 4.4.4913.1, 4.0.2598977.1, 8.0.6754681446529.1, \(\Q(\zeta_{17})^+\), 8.0.114829584590993.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
17Data not computed
23Data not computed