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Properties

Label 16.0.12825701184...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{2}\cdot 89^{2}\cdot 149^{4}$
Root discriminant $20.86$
Ramified primes $5, 29, 89, 149$
Class number $1$
Class group Trivial
Galois group 16T1340

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, -9, 67, -204, 387, -429, 302, -126, 46, -33, 30, -20, 10, -6, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 - 6*x^13 + 10*x^12 - 20*x^11 + 30*x^10 - 33*x^9 + 46*x^8 - 126*x^7 + 302*x^6 - 429*x^5 + 387*x^4 - 204*x^3 + 67*x^2 - 9*x + 11)
 
gp: K = bnfinit(x^16 - 4*x^15 + 6*x^14 - 6*x^13 + 10*x^12 - 20*x^11 + 30*x^10 - 33*x^9 + 46*x^8 - 126*x^7 + 302*x^6 - 429*x^5 + 387*x^4 - 204*x^3 + 67*x^2 - 9*x + 11, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 6 x^{14} - 6 x^{13} + 10 x^{12} - 20 x^{11} + 30 x^{10} - 33 x^{9} + 46 x^{8} - 126 x^{7} + 302 x^{6} - 429 x^{5} + 387 x^{4} - 204 x^{3} + 67 x^{2} - 9 x + 11 \)

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:  \(1282570118441391015625=5^{8}\cdot 29^{2}\cdot 89^{2}\cdot 149^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.86$
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:  $5, 29, 89, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{136618016075617} a^{15} + \frac{3988938157921}{8036353886801} a^{14} - \frac{20073956265373}{136618016075617} a^{13} + \frac{20022874266866}{136618016075617} a^{12} - \frac{30908773879752}{136618016075617} a^{11} + \frac{11387746692298}{136618016075617} a^{10} - \frac{53035677840640}{136618016075617} a^{9} - \frac{74438207619}{277115651269} a^{8} - \frac{474141658945}{136618016075617} a^{7} + \frac{4636114013211}{136618016075617} a^{6} + \frac{44085045505885}{136618016075617} a^{5} - \frac{38750061738011}{136618016075617} a^{4} + \frac{67788286481564}{136618016075617} a^{3} - \frac{49082223183822}{136618016075617} a^{2} - \frac{23023431489077}{136618016075617} a + \frac{41805035336893}{136618016075617}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( 10248.8584011 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1340:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 119 conjugacy class representatives for t16n1340 are not computed
Character table for t16n1340 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.3725.1, 8.0.35812988125.2, 8.0.1234930625.1, 8.0.402393125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
$149$149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$