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Properties

Label 16.0.10483151353...4369.7
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 53^{8}$
Root discriminant $86.85$
Ramified primes $17, 53$
Class number $484$ (GRH)
Class group $[11, 44]$ (GRH)
Galois group $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22193008, -48013336, 76467296, -64659186, 52839433, -27873848, 15012390, -5486248, 2130764, -555422, 161878, -29640, 6578, -784, 132, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 132*x^14 - 784*x^13 + 6578*x^12 - 29640*x^11 + 161878*x^10 - 555422*x^9 + 2130764*x^8 - 5486248*x^7 + 15012390*x^6 - 27873848*x^5 + 52839433*x^4 - 64659186*x^3 + 76467296*x^2 - 48013336*x + 22193008)
 
gp: K = bnfinit(x^16 - 8*x^15 + 132*x^14 - 784*x^13 + 6578*x^12 - 29640*x^11 + 161878*x^10 - 555422*x^9 + 2130764*x^8 - 5486248*x^7 + 15012390*x^6 - 27873848*x^5 + 52839433*x^4 - 64659186*x^3 + 76467296*x^2 - 48013336*x + 22193008, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 132 x^{14} - 784 x^{13} + 6578 x^{12} - 29640 x^{11} + 161878 x^{10} - 555422 x^{9} + 2130764 x^{8} - 5486248 x^{7} + 15012390 x^{6} - 27873848 x^{5} + 52839433 x^{4} - 64659186 x^{3} + 76467296 x^{2} - 48013336 x + 22193008 \)

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:  \(10483151353726139536553735554369=17^{14}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.85$
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:  $17, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{204} a^{8} - \frac{1}{51} a^{7} - \frac{11}{51} a^{6} + \frac{11}{51} a^{5} - \frac{1}{102} a^{4} - \frac{10}{51} a^{3} - \frac{41}{204} a^{2} + \frac{43}{102} a - \frac{4}{51}$, $\frac{1}{204} a^{9} + \frac{7}{34} a^{7} - \frac{5}{34} a^{6} - \frac{5}{34} a^{5} - \frac{4}{17} a^{4} - \frac{33}{68} a^{3} - \frac{13}{34} a^{2} + \frac{11}{102} a - \frac{16}{51}$, $\frac{1}{204} a^{10} + \frac{3}{17} a^{7} - \frac{3}{34} a^{6} + \frac{7}{34} a^{5} - \frac{5}{68} a^{4} - \frac{5}{34} a^{3} + \frac{5}{102} a^{2} + \frac{49}{102} a + \frac{5}{17}$, $\frac{1}{204} a^{11} + \frac{2}{17} a^{7} - \frac{1}{34} a^{6} + \frac{11}{68} a^{5} + \frac{7}{34} a^{4} + \frac{11}{102} a^{3} - \frac{29}{102} a^{2} - \frac{13}{34} a - \frac{3}{17}$, $\frac{1}{12758160} a^{12} - \frac{1}{2126360} a^{11} + \frac{551}{425272} a^{10} - \frac{1337}{850544} a^{9} - \frac{6877}{3189540} a^{8} - \frac{196679}{797385} a^{7} + \frac{39223}{2551632} a^{6} + \frac{1099403}{6379080} a^{5} - \frac{3139}{2126360} a^{4} + \frac{1061969}{4252720} a^{3} - \frac{219193}{3189540} a^{2} - \frac{1183}{6018} a + \frac{8491}{30090}$, $\frac{1}{12758160} a^{13} + \frac{2749}{2126360} a^{11} + \frac{3317}{2551632} a^{10} - \frac{3793}{2126360} a^{9} + \frac{677}{3189540} a^{8} + \frac{2953771}{12758160} a^{7} + \frac{203047}{1594770} a^{6} - \frac{229859}{6379080} a^{5} - \frac{131941}{750480} a^{4} - \frac{6187}{21624} a^{3} - \frac{79311}{1063180} a^{2} + \frac{1817}{10030} a - \frac{341}{885}$, $\frac{1}{12758160} a^{14} - \frac{3177}{4252720} a^{11} + \frac{2911}{6379080} a^{10} + \frac{7909}{6379080} a^{9} + \frac{3643}{12758160} a^{8} - \frac{82828}{797385} a^{7} + \frac{8672}{797385} a^{6} - \frac{2027921}{12758160} a^{5} + \frac{776153}{6379080} a^{4} - \frac{363157}{1275816} a^{3} + \frac{163133}{531590} a^{2} - \frac{13459}{30090} a + \frac{1591}{5015}$, $\frac{1}{661395772560} a^{15} + \frac{25913}{661395772560} a^{14} - \frac{802}{41337235785} a^{13} - \frac{30}{2755815719} a^{12} + \frac{44192923}{44093051504} a^{11} + \frac{66472007}{330697886280} a^{10} + \frac{168796571}{82674471570} a^{9} + \frac{504559149}{220465257520} a^{8} + \frac{9565133767}{41337235785} a^{7} + \frac{15913674461}{82674471570} a^{6} + \frac{8867416457}{44093051504} a^{5} - \frac{67518294871}{330697886280} a^{4} - \frac{6712283867}{220465257520} a^{3} + \frac{75437128543}{165348943140} a^{2} + \frac{146491089}{519965230} a - \frac{45562199}{91758570}$

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

Class group and class number

$C_{11}\times C_{44}$, which has order $484$ (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:  \( 6878292.5579 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{53}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{901}) \), \(\Q(\sqrt{17}, \sqrt{53})\), 4.4.260389.1 x2, 4.4.13800617.1 x2, 8.8.190457029580689.1, 8.0.61089990620221.2 x4

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$17.8.7.4$x^{8} - 12393$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.4$x^{8} - 12393$$8$$1$$7$$C_8$$[\ ]_{8}$
$53$53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$