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Properties

Label 16.0.42917363397...6224.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{52}\cdot 3^{4}\cdot 7^{6}$
Root discriminant $25.97$
Ramified primes $2, 3, 7$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![367, -1920, 3976, -7272, 16530, -21536, 18992, -11144, 4135, -544, -256, 96, 62, -72, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 72*x^13 + 62*x^12 + 96*x^11 - 256*x^10 - 544*x^9 + 4135*x^8 - 11144*x^7 + 18992*x^6 - 21536*x^5 + 16530*x^4 - 7272*x^3 + 3976*x^2 - 1920*x + 367)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 72*x^13 + 62*x^12 + 96*x^11 - 256*x^10 - 544*x^9 + 4135*x^8 - 11144*x^7 + 18992*x^6 - 21536*x^5 + 16530*x^4 - 7272*x^3 + 3976*x^2 - 1920*x + 367, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 72 x^{13} + 62 x^{12} + 96 x^{11} - 256 x^{10} - 544 x^{9} + 4135 x^{8} - 11144 x^{7} + 18992 x^{6} - 21536 x^{5} + 16530 x^{4} - 7272 x^{3} + 3976 x^{2} - 1920 x + 367 \)

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:  \(42917363397401430196224=2^{52}\cdot 3^{4}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.97$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{697} a^{14} + \frac{52}{697} a^{13} + \frac{270}{697} a^{12} + \frac{88}{697} a^{11} + \frac{175}{697} a^{10} + \frac{233}{697} a^{9} + \frac{62}{697} a^{8} + \frac{93}{697} a^{7} - \frac{295}{697} a^{6} + \frac{52}{697} a^{5} - \frac{342}{697} a^{4} - \frac{245}{697} a^{3} - \frac{175}{697} a^{2} - \frac{318}{697} a - \frac{48}{697}$, $\frac{1}{15474622355902044612636791} a^{15} - \frac{281000022780057279747}{910271903288355565449223} a^{14} + \frac{5157011984414561653342399}{15474622355902044612636791} a^{13} - \frac{2250753464307317600200765}{15474622355902044612636791} a^{12} + \frac{417344757355882695716456}{15474622355902044612636791} a^{11} + \frac{3396732782873252084947038}{15474622355902044612636791} a^{10} - \frac{4155478962331117513879493}{15474622355902044612636791} a^{9} + \frac{4028132457796542442356035}{15474622355902044612636791} a^{8} + \frac{3947251642832374244647973}{15474622355902044612636791} a^{7} + \frac{4175700165895007176157629}{15474622355902044612636791} a^{6} - \frac{7218907354960222697067710}{15474622355902044612636791} a^{5} - \frac{2079599738879470385934193}{15474622355902044612636791} a^{4} - \frac{6108366908002122114348162}{15474622355902044612636791} a^{3} - \frac{2941204699349040007191419}{15474622355902044612636791} a^{2} - \frac{6722790709554249635586598}{15474622355902044612636791} a + \frac{161038503947329713708893}{377429813558586453966751}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 22602.3171027 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n781 are not computed
Character table for t16n781 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.2.448.1, \(\Q(\zeta_{16})^+\), 4.2.14336.1, 8.4.205520896.3

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$3$3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$