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Properties

Label 16.0.14906424550...3344.8
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 3^{14}\cdot 11^{6}$
Root discriminant $43.24$
Ramified primes $2, 3, 11$
Class number $64$ (GRH)
Class group $[2, 2, 2, 8]$ (GRH)
Galois group $C_2^3.(C_2\times D_4)$ (as 16T408)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2797, -748, 9024, -1892, 9350, -2148, 2896, -556, 390, 284, 184, -156, 50, 4, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^13 + 50*x^12 - 156*x^11 + 184*x^10 + 284*x^9 + 390*x^8 - 556*x^7 + 2896*x^6 - 2148*x^5 + 9350*x^4 - 1892*x^3 + 9024*x^2 - 748*x + 2797)
 
gp: K = bnfinit(x^16 - 4*x^15 + 4*x^13 + 50*x^12 - 156*x^11 + 184*x^10 + 284*x^9 + 390*x^8 - 556*x^7 + 2896*x^6 - 2148*x^5 + 9350*x^4 - 1892*x^3 + 9024*x^2 - 748*x + 2797, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 4 x^{13} + 50 x^{12} - 156 x^{11} + 184 x^{10} + 284 x^{9} + 390 x^{8} - 556 x^{7} + 2896 x^{6} - 2148 x^{5} + 9350 x^{4} - 1892 x^{3} + 9024 x^{2} - 748 x + 2797 \)

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:  \(149064245508482666116153344=2^{44}\cdot 3^{14}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.24$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{9} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{6} a^{13} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{66} a^{14} - \frac{2}{33} a^{13} - \frac{1}{66} a^{12} - \frac{1}{22} a^{11} - \frac{2}{33} a^{10} + \frac{1}{66} a^{9} + \frac{1}{66} a^{8} - \frac{7}{33} a^{7} - \frac{29}{66} a^{6} + \frac{5}{11} a^{5} - \frac{1}{66} a^{4} + \frac{1}{6} a^{3} - \frac{16}{33} a^{2} + \frac{1}{6} a + \frac{1}{22}$, $\frac{1}{510682627348852345720146} a^{15} - \frac{244441378505408783009}{85113771224808724286691} a^{14} - \frac{269422519339456792306}{7737615565891702207881} a^{13} - \frac{232212223054312886101}{46425693395350213247286} a^{12} - \frac{6649206087495172768447}{85113771224808724286691} a^{11} + \frac{4041540354193534582345}{56742514149872482857794} a^{10} + \frac{14318290982615828591228}{255341313674426172860073} a^{9} + \frac{191358854089246927720}{85113771224808724286691} a^{8} - \frac{5317053118696081240679}{170227542449617448573382} a^{7} - \frac{63188351438049117954797}{255341313674426172860073} a^{6} + \frac{2572800868800282680690}{28371257074936241428897} a^{5} - \frac{27704115058486418237971}{170227542449617448573382} a^{4} + \frac{110071725594228662493037}{255341313674426172860073} a^{3} + \frac{44854208352012916635481}{170227542449617448573382} a^{2} + \frac{11903505168867290188964}{85113771224808724286691} a - \frac{38123376474371131976306}{255341313674426172860073}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{8}$, which has order $64$ (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:  \( 76940.9577425 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.(C_2\times D_4)$ (as 16T408):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^3.(C_2\times D_4)$
Character table for $C_2^3.(C_2\times D_4)$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.13824.1, 4.4.4752.1, 4.4.50688.2, 8.0.190768545792.5, 8.0.763074183168.14, 8.8.23123460096.1

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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$