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Properties

Label 16.0.17415183620...4081.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 61^{12}$
Root discriminant $37.81$
Ramified primes $3, 61$
Class number $18$ (GRH)
Class group $[3, 6]$ (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![729, 3159, 9315, 13284, 13716, 14526, 11097, 5301, 2370, 501, -320, -113, 41, -32, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - 32*x^13 + 41*x^12 - 113*x^11 - 320*x^10 + 501*x^9 + 2370*x^8 + 5301*x^7 + 11097*x^6 + 14526*x^5 + 13716*x^4 + 13284*x^3 + 9315*x^2 + 3159*x + 729)
 
gp: K = bnfinit(x^16 - 2*x^15 + 2*x^14 - 32*x^13 + 41*x^12 - 113*x^11 - 320*x^10 + 501*x^9 + 2370*x^8 + 5301*x^7 + 11097*x^6 + 14526*x^5 + 13716*x^4 + 13284*x^3 + 9315*x^2 + 3159*x + 729, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 2 x^{14} - 32 x^{13} + 41 x^{12} - 113 x^{11} - 320 x^{10} + 501 x^{9} + 2370 x^{8} + 5301 x^{7} + 11097 x^{6} + 14526 x^{5} + 13716 x^{4} + 13284 x^{3} + 9315 x^{2} + 3159 x + 729 \)

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:  \(17415183620366462784744081=3^{8}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.81$
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:  $3, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{4}{9} a^{5} + \frac{1}{9} a^{4} - \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{4}$, $\frac{1}{27} a^{11} + \frac{1}{27} a^{10} - \frac{1}{27} a^{9} + \frac{1}{27} a^{8} - \frac{1}{27} a^{7} + \frac{1}{27} a^{6} + \frac{7}{27} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{405} a^{12} + \frac{1}{405} a^{11} - \frac{2}{81} a^{10} + \frac{4}{405} a^{9} - \frac{4}{405} a^{8} - \frac{23}{405} a^{7} - \frac{1}{81} a^{6} - \frac{13}{27} a^{5} + \frac{8}{135} a^{4} - \frac{2}{45} a^{3} + \frac{7}{45} a^{2} + \frac{7}{15} a - \frac{2}{5}$, $\frac{1}{405} a^{13} + \frac{4}{405} a^{11} - \frac{16}{405} a^{10} + \frac{22}{405} a^{9} - \frac{4}{405} a^{8} + \frac{1}{135} a^{7} - \frac{8}{81} a^{6} + \frac{7}{15} a^{5} + \frac{1}{135} a^{4} - \frac{11}{45} a^{3} - \frac{1}{45} a^{2} + \frac{7}{15} a + \frac{2}{5}$, $\frac{1}{397305} a^{14} - \frac{56}{397305} a^{13} - \frac{217}{397305} a^{12} + \frac{1999}{397305} a^{11} - \frac{10432}{397305} a^{10} - \frac{1411}{79461} a^{9} - \frac{15464}{397305} a^{8} + \frac{2762}{26487} a^{7} - \frac{2183}{14715} a^{6} + \frac{4147}{14715} a^{5} + \frac{43}{1635} a^{4} - \frac{4856}{14715} a^{3} - \frac{496}{2943} a^{2} + \frac{664}{4905} a + \frac{155}{327}$, $\frac{1}{3857120801152875} a^{15} + \frac{2345760614}{3857120801152875} a^{14} + \frac{25797844289}{428568977905875} a^{13} - \frac{1223363400472}{1285706933717625} a^{12} - \frac{301692671461}{257141386743525} a^{11} - \frac{17810200289326}{1285706933717625} a^{10} - \frac{69475657740793}{3857120801152875} a^{9} + \frac{137323076254888}{3857120801152875} a^{8} + \frac{88340107619776}{1285706933717625} a^{7} - \frac{4311660772289}{428568977905875} a^{6} - \frac{38387285922847}{142856325968625} a^{5} + \frac{4163869430237}{47618775322875} a^{4} - \frac{61105709244691}{142856325968625} a^{3} - \frac{41100184322789}{142856325968625} a^{2} - \frac{4999551551843}{47618775322875} a + \frac{6539745510517}{15872925107625}$

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

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (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:  \( -\frac{34529650088}{35386429368375} a^{15} + \frac{88405322068}{35386429368375} a^{14} - \frac{124215397663}{35386429368375} a^{13} + \frac{1191386976583}{35386429368375} a^{12} - \frac{422481201821}{7077285873675} a^{11} + \frac{5297549546164}{35386429368375} a^{10} + \frac{7636268460259}{35386429368375} a^{9} - \frac{6815990014598}{11795476456125} a^{8} - \frac{7750305281671}{3931825485375} a^{7} - \frac{1816696502927}{436869498375} a^{6} - \frac{11515100244364}{1310608495125} a^{5} - \frac{12892658031593}{1310608495125} a^{4} - \frac{12039532228142}{1310608495125} a^{3} - \frac{3971055818381}{436869498375} a^{2} - \frac{833390383222}{145623166125} a - \frac{44362097007}{48541055375} \) (order $6$)
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:  \( 2820654.85259 \) (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

\(\Q(\sqrt{-183}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{61})\), 4.2.11163.1 x2, 4.0.549.1 x2, 8.0.1121513121.2, 8.2.1391050107747.2 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$61$61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$