↑

Properties

Label 16.0.66283426716...3249.6
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 89^{8}$
Root discriminant $112.55$
Ramified primes $17, 89$
Class number $200$ (GRH)
Class group $[2, 10, 10]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6029561, -30043348, 55061061, -49841930, 42254283, -23360988, 11524894, -4306308, 1604631, -406704, 128672, -20302, 5424, -472, 118, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 118*x^14 - 472*x^13 + 5424*x^12 - 20302*x^11 + 128672*x^10 - 406704*x^9 + 1604631*x^8 - 4306308*x^7 + 11524894*x^6 - 23360988*x^5 + 42254283*x^4 - 49841930*x^3 + 55061061*x^2 - 30043348*x + 6029561)
 
gp: K = bnfinit(x^16 - 4*x^15 + 118*x^14 - 472*x^13 + 5424*x^12 - 20302*x^11 + 128672*x^10 - 406704*x^9 + 1604631*x^8 - 4306308*x^7 + 11524894*x^6 - 23360988*x^5 + 42254283*x^4 - 49841930*x^3 + 55061061*x^2 - 30043348*x + 6029561, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 118 x^{14} - 472 x^{13} + 5424 x^{12} - 20302 x^{11} + 128672 x^{10} - 406704 x^{9} + 1604631 x^{8} - 4306308 x^{7} + 11524894 x^{6} - 23360988 x^{5} + 42254283 x^{4} - 49841930 x^{3} + 55061061 x^{2} - 30043348 x + 6029561 \)

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:  \(662834267162184237456650608633249=17^{14}\cdot 89^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.55$
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:  $17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{34} a^{8} - \frac{1}{17} a^{7} + \frac{3}{17} a^{6} + \frac{7}{17} a^{5} - \frac{15}{34} a^{4} - \frac{7}{17} a^{3} - \frac{11}{34} a^{2} - \frac{15}{34} a + \frac{1}{34}$, $\frac{1}{34} a^{9} + \frac{1}{17} a^{7} - \frac{4}{17} a^{6} + \frac{13}{34} a^{5} - \frac{5}{17} a^{4} - \frac{5}{34} a^{3} - \frac{3}{34} a^{2} + \frac{5}{34} a + \frac{1}{17}$, $\frac{1}{34} a^{10} - \frac{2}{17} a^{7} + \frac{1}{34} a^{6} - \frac{2}{17} a^{5} - \frac{9}{34} a^{4} - \frac{9}{34} a^{3} - \frac{7}{34} a^{2} - \frac{1}{17} a - \frac{1}{17}$, $\frac{1}{34} a^{11} - \frac{7}{34} a^{7} - \frac{7}{17} a^{6} + \frac{13}{34} a^{5} - \frac{1}{34} a^{4} + \frac{5}{34} a^{3} - \frac{6}{17} a^{2} + \frac{3}{17} a + \frac{2}{17}$, $\frac{1}{48586} a^{12} - \frac{383}{48586} a^{11} - \frac{277}{48586} a^{10} - \frac{343}{48586} a^{9} - \frac{665}{48586} a^{8} + \frac{6445}{48586} a^{7} + \frac{3324}{24293} a^{6} + \frac{1361}{48586} a^{5} + \frac{10129}{48586} a^{4} - \frac{6561}{48586} a^{3} + \frac{11263}{24293} a^{2} + \frac{19369}{48586} a + \frac{11147}{24293}$, $\frac{1}{48586} a^{13} + \frac{13}{2858} a^{11} - \frac{344}{24293} a^{10} - \frac{283}{24293} a^{9} + \frac{198}{24293} a^{8} + \frac{5029}{24293} a^{7} + \frac{12499}{48586} a^{6} + \frac{21239}{48586} a^{5} + \frac{13117}{48586} a^{4} + \frac{3063}{24293} a^{3} - \frac{1481}{48586} a^{2} - \frac{13043}{48586} a + \frac{878}{24293}$, $\frac{1}{2283542} a^{14} + \frac{3}{1141771} a^{13} + \frac{15}{2283542} a^{12} - \frac{13105}{1141771} a^{11} - \frac{23369}{2283542} a^{10} + \frac{31933}{2283542} a^{9} + \frac{16527}{2283542} a^{8} - \frac{211653}{2283542} a^{7} + \frac{441553}{1141771} a^{6} - \frac{116350}{1141771} a^{5} - \frac{354965}{1141771} a^{4} + \frac{955283}{2283542} a^{3} - \frac{112620}{1141771} a^{2} + \frac{25314}{67163} a - \frac{976661}{2283542}$, $\frac{1}{3560617928841156834962505241441764394} a^{15} + \frac{6570593991708039464622836768}{104724056730622259851838389454169541} a^{14} - \frac{21209216486426480657859109149755}{3560617928841156834962505241441764394} a^{13} - \frac{428796946869437979571485132889}{209448113461244519703676778908339082} a^{12} - \frac{11435673802468136280660525556712481}{3560617928841156834962505241441764394} a^{11} + \frac{35181636007125008027935812246681401}{3560617928841156834962505241441764394} a^{10} + \frac{20702124056813853883952809565933841}{3560617928841156834962505241441764394} a^{9} - \frac{20214721388256825857813471460804377}{1780308964420578417481252620720882197} a^{8} - \frac{29896586212520240904659394462715020}{1780308964420578417481252620720882197} a^{7} - \frac{566841966816848944360448464836415779}{3560617928841156834962505241441764394} a^{6} - \frac{1550914320508297687367549150618754997}{3560617928841156834962505241441764394} a^{5} - \frac{832459831365455188678655998647102900}{1780308964420578417481252620720882197} a^{4} + \frac{252559519828898541808739499459061440}{1780308964420578417481252620720882197} a^{3} - \frac{571380949822486573777976272306082343}{1780308964420578417481252620720882197} a^{2} - \frac{393458103787724298663360970825121223}{3560617928841156834962505241441764394} a - \frac{354825911274067914105192432324226162}{1780308964420578417481252620720882197}$

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

Class group and class number

$C_{2}\times C_{10}\times C_{10}$, which has order $200$ (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:  \( 22977967.6331 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{1513}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{89}) \), \(\Q(\sqrt{17}, \sqrt{89})\), 4.4.4913.1, 4.4.38915873.1, 8.8.1514445171352129.1, 8.0.3250292628833.2 x2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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}$
$89$89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$