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Properties

Label 16.0.14149003321...0336.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 7^{4}\cdot 37^{8}$
Root discriminant $27.98$
Ramified primes $2, 7, 37$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $\GL(2,Z/4)$ (as 16T193)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![243, -990, 1120, 1780, 4286, 2368, 2132, 140, 251, -188, 96, -48, 20, -10, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 6*x^14 - 10*x^13 + 20*x^12 - 48*x^11 + 96*x^10 - 188*x^9 + 251*x^8 + 140*x^7 + 2132*x^6 + 2368*x^5 + 4286*x^4 + 1780*x^3 + 1120*x^2 - 990*x + 243)
 
gp: K = bnfinit(x^16 + 6*x^14 - 10*x^13 + 20*x^12 - 48*x^11 + 96*x^10 - 188*x^9 + 251*x^8 + 140*x^7 + 2132*x^6 + 2368*x^5 + 4286*x^4 + 1780*x^3 + 1120*x^2 - 990*x + 243, 1)
 

Normalized defining polynomial

\( x^{16} + 6 x^{14} - 10 x^{13} + 20 x^{12} - 48 x^{11} + 96 x^{10} - 188 x^{9} + 251 x^{8} + 140 x^{7} + 2132 x^{6} + 2368 x^{5} + 4286 x^{4} + 1780 x^{3} + 1120 x^{2} - 990 x + 243 \)

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:  \(141490033212081188110336=2^{24}\cdot 7^{4}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.98$
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:  $2, 7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{531} a^{14} - \frac{1}{531} a^{13} + \frac{34}{531} a^{12} - \frac{71}{531} a^{11} - \frac{53}{531} a^{10} + \frac{212}{531} a^{9} + \frac{46}{531} a^{8} + \frac{20}{59} a^{7} + \frac{251}{531} a^{6} - \frac{10}{177} a^{5} - \frac{88}{531} a^{4} + \frac{53}{531} a^{3} + \frac{88}{177} a^{2} - \frac{239}{531} a - \frac{1}{59}$, $\frac{1}{27762811151319695963853} a^{15} - \frac{843611284382767831}{3084756794591077329317} a^{14} - \frac{259853549157827105740}{9254270383773231987951} a^{13} - \frac{4260373028488604512450}{27762811151319695963853} a^{12} + \frac{2815115612517097036943}{27762811151319695963853} a^{11} + \frac{1331841817032327083449}{9254270383773231987951} a^{10} + \frac{4318460684563655730151}{9254270383773231987951} a^{9} - \frac{5527243475655010703048}{27762811151319695963853} a^{8} + \frac{11181910504664546134859}{27762811151319695963853} a^{7} - \frac{12682830586506525999709}{27762811151319695963853} a^{6} + \frac{13455414207469521096992}{27762811151319695963853} a^{5} - \frac{8161340586122708446595}{27762811151319695963853} a^{4} - \frac{11075586472258719946927}{27762811151319695963853} a^{3} + \frac{5639554787317380024688}{27762811151319695963853} a^{2} + \frac{5852556103107271437754}{27762811151319695963853} a + \frac{436054527551041766435}{3084756794591077329317}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( 80443.548319 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\GL(2,Z/4)$ (as 16T193):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 14 conjugacy class representatives for $\GL(2,Z/4)$
Character table for $\GL(2,Z/4)$

Intermediate fields

\(\Q(\sqrt{37}) \), 4.0.592.1, 4.0.153328.1, 8.0.479785216.2

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

Sibling fields

Degree 12 siblings: 12.4.8554542212938989568.1, 12.8.8554542212938989568.1, 12.8.57674573957496832.1, 12.4.8554542212938989568.2
Degree 16 sibling: Deg 16
Degree 24 siblings: data not computed
Degree 32 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 R ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$37$37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$