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Properties

Label 16.0.14098938471...0625.4
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 11^{8}\cdot 17^{14}$
Root discriminant $88.48$
Ramified primes $5, 11, 17$
Class number $126560$ (GRH)
Class group $[126560]$ (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![3605932049, -1059674139, 1613418232, -418732795, 330143602, -74720456, 40179631, -7799164, 3172440, -514575, 166080, -21510, 5619, -530, 112, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 112*x^14 - 530*x^13 + 5619*x^12 - 21510*x^11 + 166080*x^10 - 514575*x^9 + 3172440*x^8 - 7799164*x^7 + 40179631*x^6 - 74720456*x^5 + 330143602*x^4 - 418732795*x^3 + 1613418232*x^2 - 1059674139*x + 3605932049)
 
gp: K = bnfinit(x^16 - 6*x^15 + 112*x^14 - 530*x^13 + 5619*x^12 - 21510*x^11 + 166080*x^10 - 514575*x^9 + 3172440*x^8 - 7799164*x^7 + 40179631*x^6 - 74720456*x^5 + 330143602*x^4 - 418732795*x^3 + 1613418232*x^2 - 1059674139*x + 3605932049, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 112 x^{14} - 530 x^{13} + 5619 x^{12} - 21510 x^{11} + 166080 x^{10} - 514575 x^{9} + 3172440 x^{8} - 7799164 x^{7} + 40179631 x^{6} - 74720456 x^{5} + 330143602 x^{4} - 418732795 x^{3} + 1613418232 x^{2} - 1059674139 x + 3605932049 \)

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:  \(14098938471283305926093925390625=5^{8}\cdot 11^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.48$
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:  $5, 11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(935=5\cdot 11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{935}(1,·)$, $\chi_{935}(769,·)$, $\chi_{935}(331,·)$, $\chi_{935}(716,·)$, $\chi_{935}(274,·)$, $\chi_{935}(659,·)$, $\chi_{935}(276,·)$, $\chi_{935}(661,·)$, $\chi_{935}(219,·)$, $\chi_{935}(604,·)$, $\chi_{935}(166,·)$, $\chi_{935}(934,·)$, $\chi_{935}(494,·)$, $\chi_{935}(111,·)$, $\chi_{935}(824,·)$, $\chi_{935}(441,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{5042482081315548164431726648730537810768791902166} a^{15} + \frac{620940958171901658440360038781130769842498886318}{2521241040657774082215863324365268905384395951083} a^{14} + \frac{140618657730798303441982385924783462992783097801}{2521241040657774082215863324365268905384395951083} a^{13} - \frac{622327037337099403485919885509481416653432204065}{5042482081315548164431726648730537810768791902166} a^{12} + \frac{1005434591596546403191665861708256834655043732206}{2521241040657774082215863324365268905384395951083} a^{11} - \frac{381909574097953355404549640533108253505076516466}{2521241040657774082215863324365268905384395951083} a^{10} - \frac{1460188123004357925312019605688098927944169686051}{5042482081315548164431726648730537810768791902166} a^{9} - \frac{539459016851580398941754562311440787516547806727}{2521241040657774082215863324365268905384395951083} a^{8} + \frac{570337291487685916211476737307101769334974017418}{2521241040657774082215863324365268905384395951083} a^{7} - \frac{961304870430019316355391117514005042814755739931}{5042482081315548164431726648730537810768791902166} a^{6} - \frac{868488662089467114518622131151234686968261869040}{2521241040657774082215863324365268905384395951083} a^{5} - \frac{182370591976077119390945465679224926829682777674}{2521241040657774082215863324365268905384395951083} a^{4} + \frac{917656315601680777251702672553403079485056465927}{5042482081315548164431726648730537810768791902166} a^{3} - \frac{1187807823696541804617829633820586429113837847992}{2521241040657774082215863324365268905384395951083} a^{2} - \frac{882787294926855988585482181132926550749811022301}{2521241040657774082215863324365268905384395951083} a + \frac{720154570243011873658385606326142837999921657228}{2521241040657774082215863324365268905384395951083}$

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

Class group and class number

$C_{126560}$, which has order $126560$ (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:  \( 3640.012213375973 \) (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

\(\Q(\sqrt{-935}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{17}, \sqrt{-55})\), 4.4.4913.1, 4.0.14861825.3, 8.0.220873842330625.9, 8.0.3754855319620625.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ 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
5Data not computed
11Data not computed
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$