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Properties

Label 16.0.18182894681...3125.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{10}\cdot 11^{6}\cdot 101^{5}$
Root discriminant $28.43$
Ramified primes $5, 11, 101$
Class number $2$
Class group $[2]$
Galois group 16T1574

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![67519, 24712, 27951, 4914, 21648, 136, 5754, 789, 1529, -92, 615, -183, 169, -41, 21, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 21*x^14 - 41*x^13 + 169*x^12 - 183*x^11 + 615*x^10 - 92*x^9 + 1529*x^8 + 789*x^7 + 5754*x^6 + 136*x^5 + 21648*x^4 + 4914*x^3 + 27951*x^2 + 24712*x + 67519)
 
gp: K = bnfinit(x^16 - 3*x^15 + 21*x^14 - 41*x^13 + 169*x^12 - 183*x^11 + 615*x^10 - 92*x^9 + 1529*x^8 + 789*x^7 + 5754*x^6 + 136*x^5 + 21648*x^4 + 4914*x^3 + 27951*x^2 + 24712*x + 67519, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 21 x^{14} - 41 x^{13} + 169 x^{12} - 183 x^{11} + 615 x^{10} - 92 x^{9} + 1529 x^{8} + 789 x^{7} + 5754 x^{6} + 136 x^{5} + 21648 x^{4} + 4914 x^{3} + 27951 x^{2} + 24712 x + 67519 \)

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:  \(181828946813008408203125=5^{10}\cdot 11^{6}\cdot 101^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.43$
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:  $5, 11, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{45} a^{14} + \frac{1}{9} a^{13} + \frac{14}{45} a^{11} + \frac{11}{45} a^{10} - \frac{4}{45} a^{9} + \frac{2}{45} a^{8} - \frac{4}{15} a^{7} + \frac{2}{15} a^{6} - \frac{2}{15} a^{5} - \frac{1}{3} a^{4} + \frac{22}{45} a^{3} + \frac{14}{45} a^{2} - \frac{2}{15} a + \frac{4}{45}$, $\frac{1}{2338453817553155069962322811945} a^{15} + \frac{2268638225069738750975783461}{2338453817553155069962322811945} a^{14} + \frac{61188055027210847740761258245}{467690763510631013992464562389} a^{13} - \frac{127013871824783003783485733476}{2338453817553155069962322811945} a^{12} + \frac{10272365834323119311075528293}{155896921170210337997488187463} a^{11} + \frac{6488330744661827259711570551}{13211603489000876101482049785} a^{10} - \frac{281281384145043987766566718474}{779484605851051689987440937315} a^{9} + \frac{69841445846883346296990638909}{467690763510631013992464562389} a^{8} + \frac{9504735772442876207087598442}{86609400650116854443048993035} a^{7} + \frac{22243923511208854880063615068}{155896921170210337997488187463} a^{6} - \frac{929778874530302578957108316}{4403867829666958700494016595} a^{5} - \frac{626514430926991148902836352898}{2338453817553155069962322811945} a^{4} + \frac{938438123526743027358678083956}{2338453817553155069962322811945} a^{3} + \frac{984840252063069207245321757643}{2338453817553155069962322811945} a^{2} - \frac{751815433863056267799327030527}{2338453817553155069962322811945} a + \frac{862905227895954158514174240224}{2338453817553155069962322811945}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 71186.6489359 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1574:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 88 conjugacy class representatives for t16n1574 are not computed
Character table for t16n1574 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 8.0.7638125.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R $16$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.8.0.1$x^{8} + x^{2} - 2 x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
$101$101.4.3.3$x^{4} + 202$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.2.2$x^{4} - 101 x^{2} + 30603$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
101.4.0.1$x^{4} - x + 12$$1$$4$$0$$C_4$$[\ ]^{4}$
101.4.0.1$x^{4} - x + 12$$1$$4$$0$$C_4$$[\ ]^{4}$