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Properties

Label 16.0.46930925056...6433.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 97^{11}$
Root discriminant $40.22$
Ramified primes $3, 97$
Class number $2$ (GRH)
Class group $[2]$ (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![1181299, -2986939, 2505865, -69749, -1015471, 372630, 142619, -79804, -22030, 13003, 3340, -2068, -123, 171, -12, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 12*x^14 + 171*x^13 - 123*x^12 - 2068*x^11 + 3340*x^10 + 13003*x^9 - 22030*x^8 - 79804*x^7 + 142619*x^6 + 372630*x^5 - 1015471*x^4 - 69749*x^3 + 2505865*x^2 - 2986939*x + 1181299)
 
gp: K = bnfinit(x^16 - 5*x^15 - 12*x^14 + 171*x^13 - 123*x^12 - 2068*x^11 + 3340*x^10 + 13003*x^9 - 22030*x^8 - 79804*x^7 + 142619*x^6 + 372630*x^5 - 1015471*x^4 - 69749*x^3 + 2505865*x^2 - 2986939*x + 1181299, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 12 x^{14} + 171 x^{13} - 123 x^{12} - 2068 x^{11} + 3340 x^{10} + 13003 x^{9} - 22030 x^{8} - 79804 x^{7} + 142619 x^{6} + 372630 x^{5} - 1015471 x^{4} - 69749 x^{3} + 2505865 x^{2} - 2986939 x + 1181299 \)

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:  \(46930925056608955875626433=3^{8}\cdot 97^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.22$
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:  $3, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} + \frac{2}{7} a^{12} + \frac{1}{7} a^{10} + \frac{2}{7} a^{8} + \frac{3}{7} a^{7} - \frac{3}{7} a^{5} + \frac{2}{7} a^{4} + \frac{3}{7} a^{2} + \frac{2}{7} a$, $\frac{1}{7} a^{14} + \frac{3}{7} a^{12} + \frac{1}{7} a^{11} - \frac{2}{7} a^{10} + \frac{2}{7} a^{9} - \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6} + \frac{1}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{3} + \frac{3}{7} a^{2} + \frac{3}{7} a$, $\frac{1}{62253137325057693917813022659804226124421} a^{15} - \frac{88944184117340230519694269264045776553}{1270472190307299875873735156322535227029} a^{14} - \frac{364095984198930960255994093130251741239}{62253137325057693917813022659804226124421} a^{13} - \frac{1787350141228446039186655349267013145513}{62253137325057693917813022659804226124421} a^{12} + \frac{27099379789093571312001496475119871721263}{62253137325057693917813022659804226124421} a^{11} + \frac{23852726136291167206256212888468262240936}{62253137325057693917813022659804226124421} a^{10} - \frac{4961900436572170881074432873073769735469}{62253137325057693917813022659804226124421} a^{9} + \frac{22636336756788348647150050604306277984578}{62253137325057693917813022659804226124421} a^{8} - \frac{20602407184093246145284163202128224113964}{62253137325057693917813022659804226124421} a^{7} - \frac{19106678807858405107414205413094294081141}{62253137325057693917813022659804226124421} a^{6} - \frac{4395790984117199448938194972275291917220}{62253137325057693917813022659804226124421} a^{5} - \frac{1570792767312574448709235579560120746523}{62253137325057693917813022659804226124421} a^{4} + \frac{13159542126540920138042940452579002058}{62253137325057693917813022659804226124421} a^{3} + \frac{150422101538465168823629315951541299696}{1270472190307299875873735156322535227029} a^{2} + \frac{5123399141325311340822501155922318596649}{62253137325057693917813022659804226124421} a + \frac{52990740992452801074329238587461294942}{240359603571651327868003948493452610519}$

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:  \( -\frac{7590649459071325128672}{321860129163246490862993027} a^{15} + \frac{29842295251021789444615}{321860129163246490862993027} a^{14} + \frac{103828837829282713560499}{321860129163246490862993027} a^{13} - \frac{1069006388665281795102281}{321860129163246490862993027} a^{12} - \frac{171195161344480173746624}{321860129163246490862993027} a^{11} + \frac{12530662820494138171575890}{321860129163246490862993027} a^{10} - \frac{6026580505030282324056920}{321860129163246490862993027} a^{9} - \frac{80905853894158716587059182}{321860129163246490862993027} a^{8} + \frac{5348197197574172274243950}{321860129163246490862993027} a^{7} + \frac{517551518856638408998154006}{321860129163246490862993027} a^{6} - \frac{211420026560190033894185174}{321860129163246490862993027} a^{5} - \frac{2144061830656040599284832760}{321860129163246490862993027} a^{4} + \frac{2513962889959976928596781359}{321860129163246490862993027} a^{3} + \frac{1888793536028869172257108209}{321860129163246490862993027} a^{2} - \frac{4260763862398426437908028467}{321860129163246490862993027} a + \frac{1893210192121737394159310470}{321860129163246490862993027} \) (order $6$)
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:  \( 5787832.84764 \) (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

\(\Q(\sqrt{-3}) \), 4.0.873.1, 8.0.73926513.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
3Data not computed
$97$97.8.4.2$x^{8} - 912673 x^{2} + 2036173463$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
97.8.7.6$x^{8} + 12125$$8$$1$$7$$C_8$$[\ ]_{8}$