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Properties

Label 16.4.18149808491...641.12
Degree $16$
Signature $[4, 6]$
Discriminant $17^{14}\cdot 47^{6}$
Root discriminant $50.55$
Ramified primes $17, 47$
Class number $2$ (GRH)
Class group $[2]$ (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![16967, 23332, 41492, 73843, -22203, 16726, -5845, -12449, 6250, 1353, -923, -52, 1, -61, 20, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 20*x^14 - 61*x^13 + x^12 - 52*x^11 - 923*x^10 + 1353*x^9 + 6250*x^8 - 12449*x^7 - 5845*x^6 + 16726*x^5 - 22203*x^4 + 73843*x^3 + 41492*x^2 + 23332*x + 16967)
 
gp: K = bnfinit(x^16 - 4*x^15 + 20*x^14 - 61*x^13 + x^12 - 52*x^11 - 923*x^10 + 1353*x^9 + 6250*x^8 - 12449*x^7 - 5845*x^6 + 16726*x^5 - 22203*x^4 + 73843*x^3 + 41492*x^2 + 23332*x + 16967, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 20 x^{14} - 61 x^{13} + x^{12} - 52 x^{11} - 923 x^{10} + 1353 x^{9} + 6250 x^{8} - 12449 x^{7} - 5845 x^{6} + 16726 x^{5} - 22203 x^{4} + 73843 x^{3} + 41492 x^{2} + 23332 x + 16967 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1814980849112797822933640641=17^{14}\cdot 47^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.55$
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:  $17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{188} a^{14} + \frac{15}{94} a^{13} - \frac{1}{188} a^{12} + \frac{71}{188} a^{11} + \frac{18}{47} a^{10} - \frac{75}{188} a^{9} - \frac{29}{188} a^{8} + \frac{23}{94} a^{7} + \frac{27}{188} a^{6} - \frac{9}{188} a^{5} - \frac{21}{94} a^{4} + \frac{39}{188} a^{3} - \frac{91}{188} a^{2} + \frac{35}{94} a + \frac{1}{4}$, $\frac{1}{1376157142235904502894020938667036581368} a^{15} + \frac{110828482601498826037453964015794797}{1376157142235904502894020938667036581368} a^{14} + \frac{293336835616909587136573049769903919717}{1376157142235904502894020938667036581368} a^{13} + \frac{59020059534083546858119325691367732798}{172019642779488062861752617333379572671} a^{12} - \frac{142173058594476071762352973314174300563}{1376157142235904502894020938667036581368} a^{11} - \frac{587937395006528680550520258949025136011}{1376157142235904502894020938667036581368} a^{10} - \frac{129371657253159910916915232616551347999}{688078571117952251447010469333518290684} a^{9} + \frac{4602248573910403426636758248884979643}{105858241710454192530309302974387429336} a^{8} + \frac{480032199234913536791416820178312799861}{1376157142235904502894020938667036581368} a^{7} - \frac{130425333019946062369032055280776944073}{344039285558976125723505234666759145342} a^{6} - \frac{445952887575522437382891023192137793453}{1376157142235904502894020938667036581368} a^{5} + \frac{290720425673173575574947896256186879285}{1376157142235904502894020938667036581368} a^{4} - \frac{4268674710145528738367652970634596311}{688078571117952251447010469333518290684} a^{3} - \frac{463088958508482882239050067409700002503}{1376157142235904502894020938667036581368} a^{2} - \frac{418616350250196170586324178955038743615}{1376157142235904502894020938667036581368} a + \frac{708698704837769831597815728087145809}{1541049431395189812871244052258719576}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 15727953.1114 \) (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

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.2.230911.1, 4.2.13583.1, 8.4.906438128657.2 x2, 8.4.53319889921.1

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 ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$