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Properties

Label 16.16.1171598758...5009.2
Degree $16$
Signature $[16, 0]$
Discriminant $13^{14}\cdot 29^{14}$
Root discriminant $179.60$
Ramified primes $13, 29$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-297452, 2153042, -1117197, -12357337, 10383990, 14524949, -6310742, -5189481, 1461579, 700390, -159987, -35545, 7442, 723, -146, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 146*x^14 + 723*x^13 + 7442*x^12 - 35545*x^11 - 159987*x^10 + 700390*x^9 + 1461579*x^8 - 5189481*x^7 - 6310742*x^6 + 14524949*x^5 + 10383990*x^4 - 12357337*x^3 - 1117197*x^2 + 2153042*x - 297452)
 
gp: K = bnfinit(x^16 - 5*x^15 - 146*x^14 + 723*x^13 + 7442*x^12 - 35545*x^11 - 159987*x^10 + 700390*x^9 + 1461579*x^8 - 5189481*x^7 - 6310742*x^6 + 14524949*x^5 + 10383990*x^4 - 12357337*x^3 - 1117197*x^2 + 2153042*x - 297452, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 146 x^{14} + 723 x^{13} + 7442 x^{12} - 35545 x^{11} - 159987 x^{10} + 700390 x^{9} + 1461579 x^{8} - 5189481 x^{7} - 6310742 x^{6} + 14524949 x^{5} + 10383990 x^{4} - 12357337 x^{3} - 1117197 x^{2} + 2153042 x - 297452 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1171598758708107367475386427203165009=13^{14}\cdot 29^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $179.60$
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:  $13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{32} a^{12} + \frac{1}{32} a^{11} + \frac{1}{32} a^{9} - \frac{1}{32} a^{8} + \frac{1}{16} a^{7} + \frac{7}{32} a^{6} - \frac{5}{32} a^{5} - \frac{1}{4} a^{4} + \frac{3}{32} a^{3} - \frac{11}{32} a^{2} + \frac{7}{16} a - \frac{1}{8}$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{11} + \frac{1}{64} a^{10} - \frac{1}{32} a^{9} + \frac{3}{64} a^{8} - \frac{11}{64} a^{7} - \frac{3}{16} a^{6} + \frac{13}{64} a^{5} - \frac{5}{64} a^{4} + \frac{9}{32} a^{3} + \frac{9}{64} a^{2} + \frac{7}{32} a - \frac{7}{16}$, $\frac{1}{7936} a^{14} + \frac{33}{7936} a^{13} - \frac{49}{7936} a^{12} + \frac{45}{992} a^{11} + \frac{439}{7936} a^{10} + \frac{105}{7936} a^{9} - \frac{17}{496} a^{8} + \frac{153}{7936} a^{7} + \frac{47}{256} a^{6} + \frac{1}{248} a^{5} - \frac{1739}{7936} a^{4} + \frac{3123}{7936} a^{3} + \frac{2591}{7936} a^{2} + \frac{401}{3968} a - \frac{607}{1984}$, $\frac{1}{52314133227392485990621257996111645661184} a^{15} - \frac{96908862290510257378882479087577743}{1634816663356015187206914312378488926912} a^{14} + \frac{114188030234224429631968855344349076135}{26157066613696242995310628998055822830592} a^{13} - \frac{39469091154410788307495603730596463975}{52314133227392485990621257996111645661184} a^{12} + \frac{1662271128416508612281792439007014200815}{52314133227392485990621257996111645661184} a^{11} + \frac{463929258761057569049713288166843888697}{26157066613696242995310628998055822830592} a^{10} - \frac{1367655943183779635762670418431801974137}{52314133227392485990621257996111645661184} a^{9} - \frac{4059960110886264016484434940999031524311}{52314133227392485990621257996111645661184} a^{8} - \frac{280446487920780104451018620453892572417}{6539266653424060748827657249513955707648} a^{7} - \frac{11311449777239020184503871633012001124881}{52314133227392485990621257996111645661184} a^{6} - \frac{7989762571789673927869834420654636768139}{52314133227392485990621257996111645661184} a^{5} - \frac{4945049272927857655770145522402468217825}{26157066613696242995310628998055822830592} a^{4} + \frac{4283057365092078436036675606830392431259}{13078533306848121497655314499027911415296} a^{3} - \frac{25924793830155920944959578332828794739453}{52314133227392485990621257996111645661184} a^{2} - \frac{233918222967566758799571432629874258403}{706947746316114675548935918866373590016} a + \frac{1428146559630479095765808899316262212767}{13078533306848121497655314499027911415296}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $15$
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:  \( 15637438683500 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{377}) \), \(\Q(\sqrt{29}) \), 4.4.53582633.2, 4.4.53582633.1, \(\Q(\sqrt{13}, \sqrt{29})\), 8.8.1082404156823183753.2 x2, 8.8.1082404156823183753.1 x2, 8.8.2871098559212689.3

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.8.7.1$x^{8} - 13$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.1$x^{8} - 13$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$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}$