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Properties

Label 16.0.20723941878...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 13^{8}\cdot 101^{4}$
Root discriminant $38.22$
Ramified primes $5, 13, 101$
Class number $32$ (GRH)
Class group $[4, 8]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T217)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -1920, 6320, -10664, 11552, -7810, 6741, -3351, 2674, -909, 706, -213, 104, -37, 14, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 14*x^14 - 37*x^13 + 104*x^12 - 213*x^11 + 706*x^10 - 909*x^9 + 2674*x^8 - 3351*x^7 + 6741*x^6 - 7810*x^5 + 11552*x^4 - 10664*x^3 + 6320*x^2 - 1920*x + 256)
 
gp: K = bnfinit(x^16 - 3*x^15 + 14*x^14 - 37*x^13 + 104*x^12 - 213*x^11 + 706*x^10 - 909*x^9 + 2674*x^8 - 3351*x^7 + 6741*x^6 - 7810*x^5 + 11552*x^4 - 10664*x^3 + 6320*x^2 - 1920*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 14 x^{14} - 37 x^{13} + 104 x^{12} - 213 x^{11} + 706 x^{10} - 909 x^{9} + 2674 x^{8} - 3351 x^{7} + 6741 x^{6} - 7810 x^{5} + 11552 x^{4} - 10664 x^{3} + 6320 x^{2} - 1920 x + 256 \)

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:  \(20723941878730254150390625=5^{12}\cdot 13^{8}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.22$
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:  $5, 13, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{20} a^{12} + \frac{3}{20} a^{11} - \frac{1}{5} a^{10} - \frac{3}{20} a^{9} + \frac{1}{10} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{9}{20} a^{5} - \frac{1}{10} a^{4} + \frac{1}{20} a^{3} - \frac{1}{4} a^{2} + \frac{3}{10} a - \frac{1}{5}$, $\frac{1}{40} a^{13} - \frac{1}{40} a^{12} + \frac{1}{10} a^{11} + \frac{3}{40} a^{10} - \frac{3}{20} a^{9} + \frac{7}{40} a^{8} - \frac{9}{40} a^{6} - \frac{2}{5} a^{5} - \frac{11}{40} a^{4} - \frac{9}{40} a^{3} - \frac{1}{10} a^{2} + \frac{3}{10} a + \frac{2}{5}$, $\frac{1}{160} a^{14} - \frac{1}{160} a^{13} - \frac{1}{40} a^{12} + \frac{19}{160} a^{11} - \frac{17}{80} a^{10} - \frac{9}{160} a^{9} - \frac{1}{10} a^{8} - \frac{29}{160} a^{7} + \frac{3}{20} a^{6} - \frac{39}{160} a^{5} + \frac{7}{160} a^{4} + \frac{7}{40} a^{3} + \frac{9}{20} a^{2} - \frac{9}{20} a + \frac{1}{5}$, $\frac{1}{1435325089126507245440} a^{15} + \frac{3426933671920475157}{1435325089126507245440} a^{14} - \frac{8570486367592257953}{717662544563253622720} a^{13} + \frac{21827836481091732083}{1435325089126507245440} a^{12} + \frac{17120533803017448173}{89707818070406702840} a^{11} + \frac{115512555533959175699}{1435325089126507245440} a^{10} - \frac{21627970604978243127}{143532508912650724544} a^{9} - \frac{1629493256568264163}{8594761012733576320} a^{8} - \frac{145506615886968827939}{717662544563253622720} a^{7} + \frac{467395791752548072129}{1435325089126507245440} a^{6} - \frac{111569667405040887059}{1435325089126507245440} a^{5} + \frac{245187479355251175927}{717662544563253622720} a^{4} + \frac{62885706746744774543}{179415636140813405680} a^{3} - \frac{12659358697635933501}{35883127228162681136} a^{2} + \frac{8360744043562608563}{17941563614081340568} a + \frac{8771567617254992061}{22426954517601675710}$

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

Class group and class number

$C_{4}\times C_{8}$, which has order $32$ (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:  \( 79444.6215571 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T217):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}, \sqrt{13})\), 8.0.1802913125.1, 8.8.45072828125.1, 8.0.4552355640625.9

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$101$101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$