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Properties

Label 16.0.83680629612...2769.4
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 89^{6}$
Root discriminant $64.22$
Ramified primes $17, 89$
Class number $56$ (GRH)
Class group $[2, 28]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![554608, -1788400, 3526208, -4672450, 4676785, -3596503, 2206158, -1087932, 441642, -148549, 42053, -10072, 2151, -407, 65, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 65*x^14 - 407*x^13 + 2151*x^12 - 10072*x^11 + 42053*x^10 - 148549*x^9 + 441642*x^8 - 1087932*x^7 + 2206158*x^6 - 3596503*x^5 + 4676785*x^4 - 4672450*x^3 + 3526208*x^2 - 1788400*x + 554608)
 
gp: K = bnfinit(x^16 - 7*x^15 + 65*x^14 - 407*x^13 + 2151*x^12 - 10072*x^11 + 42053*x^10 - 148549*x^9 + 441642*x^8 - 1087932*x^7 + 2206158*x^6 - 3596503*x^5 + 4676785*x^4 - 4672450*x^3 + 3526208*x^2 - 1788400*x + 554608, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 65 x^{14} - 407 x^{13} + 2151 x^{12} - 10072 x^{11} + 42053 x^{10} - 148549 x^{9} + 441642 x^{8} - 1087932 x^{7} + 2206158 x^{6} - 3596503 x^{5} + 4676785 x^{4} - 4672450 x^{3} + 3526208 x^{2} - 1788400 x + 554608 \)

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:  \(83680629612698426645202702769=17^{14}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.22$
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:  $17, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8164} a^{14} - \frac{329}{8164} a^{13} + \frac{1279}{8164} a^{12} - \frac{1929}{8164} a^{11} - \frac{587}{8164} a^{10} + \frac{957}{4082} a^{9} + \frac{3457}{8164} a^{8} - \frac{3655}{8164} a^{7} + \frac{948}{2041} a^{6} - \frac{366}{2041} a^{5} - \frac{173}{4082} a^{4} - \frac{3047}{8164} a^{3} + \frac{2939}{8164} a^{2} + \frac{528}{2041} a + \frac{1009}{2041}$, $\frac{1}{6675435632359660659691871969456994392} a^{15} - \frac{18344745214024541651906069738539}{513495048643050819976297843804384184} a^{14} + \frac{713821291680891733458469201611349761}{6675435632359660659691871969456994392} a^{13} - \frac{287581047960796741146210408964452459}{6675435632359660659691871969456994392} a^{12} + \frac{1101755756540632730985715661676219643}{6675435632359660659691871969456994392} a^{11} - \frac{365683411333127572548226500719126751}{1668858908089915164922967992364248598} a^{10} - \frac{1485651606007649374714277143163808803}{6675435632359660659691871969456994392} a^{9} + \frac{1389399568827765124948423159647389375}{6675435632359660659691871969456994392} a^{8} + \frac{308207506860723764527132425776494277}{3337717816179830329845935984728497196} a^{7} - \frac{373735870906768124722480849187517711}{1668858908089915164922967992364248598} a^{6} - \frac{228477361753217301463170135543104233}{3337717816179830329845935984728497196} a^{5} - \frac{2094475168202388054450758789699358627}{6675435632359660659691871969456994392} a^{4} - \frac{1375735317744905572883053781905492775}{6675435632359660659691871969456994392} a^{3} - \frac{1337096545877511726584276672741663317}{3337717816179830329845935984728497196} a^{2} - \frac{326336134839797124008575359902934893}{834429454044957582461483996182124299} a + \frac{184095900044551560584004599035226294}{834429454044957582461483996182124299}$

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

Class group and class number

$C_{2}\times C_{28}$, which has order $56$ (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:  \( 1325060.89169 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.0.437257.1, 4.4.4913.1, 4.0.25721.1, 8.4.3250292628833.1 x2, 8.0.191193684049.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$