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Properties

Label 16.0.24451371959...6617.1
Degree $16$
Signature $[0, 8]$
Discriminant $19^{12}\cdot 73^{7}$
Root discriminant $59.47$
Ramified primes $19, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2375353, -768343, 1018387, 341793, 349884, -15999, -18088, 292, 6731, -730, 1123, 294, -187, 8, -4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 4*x^14 + 8*x^13 - 187*x^12 + 294*x^11 + 1123*x^10 - 730*x^9 + 6731*x^8 + 292*x^7 - 18088*x^6 - 15999*x^5 + 349884*x^4 + 341793*x^3 + 1018387*x^2 - 768343*x + 2375353)
 
gp: K = bnfinit(x^16 - 3*x^15 - 4*x^14 + 8*x^13 - 187*x^12 + 294*x^11 + 1123*x^10 - 730*x^9 + 6731*x^8 + 292*x^7 - 18088*x^6 - 15999*x^5 + 349884*x^4 + 341793*x^3 + 1018387*x^2 - 768343*x + 2375353, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 4 x^{14} + 8 x^{13} - 187 x^{12} + 294 x^{11} + 1123 x^{10} - 730 x^{9} + 6731 x^{8} + 292 x^{7} - 18088 x^{6} - 15999 x^{5} + 349884 x^{4} + 341793 x^{3} + 1018387 x^{2} - 768343 x + 2375353 \)

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:  \(24451371959186803441564976617=19^{12}\cdot 73^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.47$
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:  $19, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{14} + \frac{1}{16} a^{13} + \frac{1}{16} a^{12} - \frac{3}{16} a^{11} + \frac{1}{8} a^{10} + \frac{3}{16} a^{9} + \frac{1}{16} a^{8} + \frac{5}{16} a^{7} + \frac{1}{16} a^{5} + \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{5}{16} a - \frac{1}{16}$, $\frac{1}{6157596473570583034392441113910910912406575648} a^{15} + \frac{5156757860569192487873785413691168327295853}{192424889799080719824763784809715966012705489} a^{14} + \frac{58925063356134197641061649858002303946316603}{769699559196322879299055139238863864050821956} a^{13} - \frac{76303343788497793975469703778128841454509717}{1539399118392645758598110278477727728101643912} a^{12} - \frac{448705934858752038153047717076365300589079803}{6157596473570583034392441113910910912406575648} a^{11} + \frac{1294505383561370610166368516673625345832939729}{6157596473570583034392441113910910912406575648} a^{10} - \frac{540369116397293362141145868192640418656853833}{3078798236785291517196220556955455456203287824} a^{9} + \frac{458787933182452271200875811479042667207039917}{1539399118392645758598110278477727728101643912} a^{8} - \frac{964255940655289830289579488394604082320562301}{6157596473570583034392441113910910912406575648} a^{7} + \frac{2665860775068320354997749797530125402521704873}{6157596473570583034392441113910910912406575648} a^{6} - \frac{452433141058416662019848842298647795724885645}{6157596473570583034392441113910910912406575648} a^{5} - \frac{17078398066971769350605012067389601602785301}{3078798236785291517196220556955455456203287824} a^{4} + \frac{177792202298648628083754693781157887536655279}{3078798236785291517196220556955455456203287824} a^{3} - \frac{1453461535998523387094731649125546780742283789}{6157596473570583034392441113910910912406575648} a^{2} - \frac{480714876922216490959629359533414990663657337}{1539399118392645758598110278477727728101643912} a + \frac{1538146849840783673959351253375648830483608225}{6157596473570583034392441113910910912406575648}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 20557469.1162 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-19}) \), 4.0.26353.1, 8.0.50697084457.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ $16$ $16$

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
19Data not computed
$73$73.8.7.7$x^{8} + 228125$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.0.1$x^{8} - x + 40$$1$$8$$0$$C_8$$[\ ]^{8}$