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Properties

Label 16.0.99435838830...7841.5
Degree $16$
Signature $[0, 8]$
Discriminant $29^{14}\cdot 109^{14}$
Root discriminant $1154.37$
Ramified primes $29, 109$
Class number $118546448$ (GRH)
Class group $[2, 2, 29636612]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5698054976725061, -3935185278274183, 1555861018641102, -337409531350162, 49584050475435, -3906739562893, 253622592936, 13050043542, -1447386238, 16350756, 20097237, -2405520, 198068, -5715, 289, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 289*x^14 - 5715*x^13 + 198068*x^12 - 2405520*x^11 + 20097237*x^10 + 16350756*x^9 - 1447386238*x^8 + 13050043542*x^7 + 253622592936*x^6 - 3906739562893*x^5 + 49584050475435*x^4 - 337409531350162*x^3 + 1555861018641102*x^2 - 3935185278274183*x + 5698054976725061)
 
gp: K = bnfinit(x^16 - 3*x^15 + 289*x^14 - 5715*x^13 + 198068*x^12 - 2405520*x^11 + 20097237*x^10 + 16350756*x^9 - 1447386238*x^8 + 13050043542*x^7 + 253622592936*x^6 - 3906739562893*x^5 + 49584050475435*x^4 - 337409531350162*x^3 + 1555861018641102*x^2 - 3935185278274183*x + 5698054976725061, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 289 x^{14} - 5715 x^{13} + 198068 x^{12} - 2405520 x^{11} + 20097237 x^{10} + 16350756 x^{9} - 1447386238 x^{8} + 13050043542 x^{7} + 253622592936 x^{6} - 3906739562893 x^{5} + 49584050475435 x^{4} - 337409531350162 x^{3} + 1555861018641102 x^{2} - 3935185278274183 x + 5698054976725061 \)

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:  \(9943583883093737930218051237711077235104795837841=29^{14}\cdot 109^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1154.37$
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:  $29, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{953} a^{14} - \frac{206}{953} a^{13} - \frac{31}{953} a^{12} + \frac{129}{953} a^{11} + \frac{56}{953} a^{10} + \frac{30}{953} a^{9} - \frac{99}{953} a^{8} - \frac{243}{953} a^{7} - \frac{180}{953} a^{6} - \frac{268}{953} a^{5} - \frac{382}{953} a^{4} + \frac{13}{953} a^{3} - \frac{3}{953} a^{2} + \frac{222}{953} a - \frac{202}{953}$, $\frac{1}{569136543253114502043782915206612684140657550457043699997799737354873167154731736394224033373958464128387114037} a^{15} + \frac{61206517617686890010519426857869431398295476870562436904415340721826688751861955230048791578647887999351877}{569136543253114502043782915206612684140657550457043699997799737354873167154731736394224033373958464128387114037} a^{14} - \frac{6687104432352230295377849545038718313903984134161537153160106319052891447324781152276833128270861085617054959}{569136543253114502043782915206612684140657550457043699997799737354873167154731736394224033373958464128387114037} a^{13} - \frac{182818020389089701565572801334440838814225674213476064115267213167854543409635834776574886764878595061669387841}{569136543253114502043782915206612684140657550457043699997799737354873167154731736394224033373958464128387114037} a^{12} - \frac{106831796140784039386757690073976942996416571715494490713435938225256617110659132812587898268378075281669635997}{569136543253114502043782915206612684140657550457043699997799737354873167154731736394224033373958464128387114037} a^{11} + \frac{213470940966127865431083002186243379317444724633408506287312368206645433417560072334888654804675137405849178388}{569136543253114502043782915206612684140657550457043699997799737354873167154731736394224033373958464128387114037} a^{10} + \frac{205513937626198055375060511695343961025026239682259844741910101994860557259779027531823015945354775686035173663}{569136543253114502043782915206612684140657550457043699997799737354873167154731736394224033373958464128387114037} a^{9} - \frac{20965295336668300868796210175316147495611937839928764849274134081325964034750691556183571685751585481323982370}{569136543253114502043782915206612684140657550457043699997799737354873167154731736394224033373958464128387114037} a^{8} + \frac{125635601645058710333432983414153341905386279820000522611895150542028775148818053767913062392696879428890901734}{569136543253114502043782915206612684140657550457043699997799737354873167154731736394224033373958464128387114037} a^{7} + \frac{65600829115034619813958210907627908656252209970632556218706070346322658395041748135239298094045227350453085368}{569136543253114502043782915206612684140657550457043699997799737354873167154731736394224033373958464128387114037} a^{6} - \frac{51447236950971273044820543103133687828765233199930010208012384112047756911399269060687096230228361702555247305}{569136543253114502043782915206612684140657550457043699997799737354873167154731736394224033373958464128387114037} a^{5} - \frac{158254110797558752888803362808266015579906649084377251588947298452249977823068876240146777576687282029348650316}{569136543253114502043782915206612684140657550457043699997799737354873167154731736394224033373958464128387114037} a^{4} - \frac{73440041648202426552958006291966388686963383506327100041612292538019731523551838902439061566612561484785511108}{569136543253114502043782915206612684140657550457043699997799737354873167154731736394224033373958464128387114037} a^{3} - \frac{208752578960125142479410897075860456967838062599287347401882703467119947635572562593963509402916938641564571171}{569136543253114502043782915206612684140657550457043699997799737354873167154731736394224033373958464128387114037} a^{2} + \frac{8650318086227178928494928762698398067105069454965406324651866236645649114827881563986495730542977579875088445}{569136543253114502043782915206612684140657550457043699997799737354873167154731736394224033373958464128387114037} a - \frac{280846183937792520142310766020748062113424596194193851274317491308922811169783632335736424450197120413646387518}{569136543253114502043782915206612684140657550457043699997799737354873167154731736394224033373958464128387114037}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{29636612}$, which has order $118546448$ (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:  \( 2856304493.41 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{109}) \), \(\Q(\sqrt{3161}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{29}, \sqrt{109})\), 8.8.997578257579911722961.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$29$29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$109$109.8.7.2$x^{8} - 3924$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
109.8.7.2$x^{8} - 3924$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$