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Properties

Label 16.0.98586762102...169.12
Degree $16$
Signature $[0, 8]$
Discriminant $37^{12}\cdot 157^{6}$
Root discriminant $99.91$
Ramified primes $37, 157$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![104368793, -279329589, 244561025, -84988496, 27751164, -12157062, 5475094, -1860278, 579189, -179757, 42902, -11500, 2451, -371, 80, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 80*x^14 - 371*x^13 + 2451*x^12 - 11500*x^11 + 42902*x^10 - 179757*x^9 + 579189*x^8 - 1860278*x^7 + 5475094*x^6 - 12157062*x^5 + 27751164*x^4 - 84988496*x^3 + 244561025*x^2 - 279329589*x + 104368793)
 
gp: K = bnfinit(x^16 - 5*x^15 + 80*x^14 - 371*x^13 + 2451*x^12 - 11500*x^11 + 42902*x^10 - 179757*x^9 + 579189*x^8 - 1860278*x^7 + 5475094*x^6 - 12157062*x^5 + 27751164*x^4 - 84988496*x^3 + 244561025*x^2 - 279329589*x + 104368793, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 80 x^{14} - 371 x^{13} + 2451 x^{12} - 11500 x^{11} + 42902 x^{10} - 179757 x^{9} + 579189 x^{8} - 1860278 x^{7} + 5475094 x^{6} - 12157062 x^{5} + 27751164 x^{4} - 84988496 x^{3} + 244561025 x^{2} - 279329589 x + 104368793 \)

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:  \(98586762102441645314452445352169=37^{12}\cdot 157^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.91$
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:  $37, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{11} - \frac{3}{7} a^{10} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} - \frac{3}{7} a^{6} - \frac{1}{7} a^{5} + \frac{3}{7} a^{4} - \frac{2}{7} a^{2} + \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{13} + \frac{3}{7} a^{11} + \frac{3}{7} a^{10} + \frac{2}{7} a^{9} + \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{2}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{7} a^{4} - \frac{2}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7}$, $\frac{1}{21} a^{14} + \frac{1}{3} a^{11} - \frac{10}{21} a^{10} + \frac{10}{21} a^{9} + \frac{1}{21} a^{7} - \frac{1}{21} a^{6} + \frac{1}{7} a^{4} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{8}{21} a + \frac{1}{21}$, $\frac{1}{62801052331149539613095291492651053802552583448013782270527} a^{15} - \frac{1321359557145168736750319399472012871886692342775898033047}{62801052331149539613095291492651053802552583448013782270527} a^{14} - \frac{1480868587820854451410115347853328339141687465649621820232}{20933684110383179871031763830883684600850861149337927423509} a^{13} + \frac{411523160059619135422724577254092080025049408790204919921}{8971578904449934230442184498950150543221797635430540324361} a^{12} - \frac{239904447759585162881285729151051507030190339389311009559}{20933684110383179871031763830883684600850861149337927423509} a^{11} - \frac{6386262916151048995849824708105382460429766639395659201755}{20933684110383179871031763830883684600850861149337927423509} a^{10} + \frac{9579455199606002263426597168077926475153862671223696956292}{62801052331149539613095291492651053802552583448013782270527} a^{9} + \frac{8132928265343787619961033475710050223496594940847922048202}{62801052331149539613095291492651053802552583448013782270527} a^{8} + \frac{14508368683378648651982545677222038128489166140831947123}{20933684110383179871031763830883684600850861149337927423509} a^{7} - \frac{25454825828011327411984513244256252032055197173690624354114}{62801052331149539613095291492651053802552583448013782270527} a^{6} - \frac{2584735563101885828000485105183433611058022631438096134349}{20933684110383179871031763830883684600850861149337927423509} a^{5} + \frac{7740461332857178496901161592856865807228792603061768040951}{20933684110383179871031763830883684600850861149337927423509} a^{4} - \frac{4832437273099144339761680738461021537065875750006964107454}{20933684110383179871031763830883684600850861149337927423509} a^{3} - \frac{9555961877846709870515037145720492403649059408738718596798}{62801052331149539613095291492651053802552583448013782270527} a^{2} - \frac{3292880031666316054127621127807905417617420473432899564518}{62801052331149539613095291492651053802552583448013782270527} a - \frac{5652005583900306720066786865304013391202823415793348115623}{62801052331149539613095291492651053802552583448013782270527}$

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:  \( -1 \) (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:  \( 1996725748.59 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 4.0.50653.1, 8.0.402819046213.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$37$37.8.6.1$x^{8} - 1147 x^{4} + 855625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
37.8.6.1$x^{8} - 1147 x^{4} + 855625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
157Data not computed