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Properties

Label 16.16.1269339715...5625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{12}\cdot 11^{6}\cdot 4139^{4}$
Root discriminant $65.91$
Ramified primes $5, 11, 4139$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1496

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10021, -61303, 103419, 53446, -297890, 174664, 167394, -184970, 532, 45684, -7319, -4752, 1027, 225, -54, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 54*x^14 + 225*x^13 + 1027*x^12 - 4752*x^11 - 7319*x^10 + 45684*x^9 + 532*x^8 - 184970*x^7 + 167394*x^6 + 174664*x^5 - 297890*x^4 + 53446*x^3 + 103419*x^2 - 61303*x + 10021)
 
gp: K = bnfinit(x^16 - 4*x^15 - 54*x^14 + 225*x^13 + 1027*x^12 - 4752*x^11 - 7319*x^10 + 45684*x^9 + 532*x^8 - 184970*x^7 + 167394*x^6 + 174664*x^5 - 297890*x^4 + 53446*x^3 + 103419*x^2 - 61303*x + 10021, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 54 x^{14} + 225 x^{13} + 1027 x^{12} - 4752 x^{11} - 7319 x^{10} + 45684 x^{9} + 532 x^{8} - 184970 x^{7} + 167394 x^{6} + 174664 x^{5} - 297890 x^{4} + 53446 x^{3} + 103419 x^{2} - 61303 x + 10021 \)

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:  \(126933971543808993896728515625=5^{12}\cdot 11^{6}\cdot 4139^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.91$
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:  $5, 11, 4139$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} - \frac{4}{11} a^{13} + \frac{2}{11} a^{12} + \frac{1}{11} a^{11} - \frac{5}{11} a^{10} + \frac{1}{11} a^{9} + \frac{2}{11} a^{8} + \frac{2}{11} a^{7} - \frac{5}{11} a^{6} - \frac{3}{11} a^{5} + \frac{2}{11} a^{4} + \frac{3}{11} a^{3} + \frac{3}{11} a^{2}$, $\frac{1}{186578975740879147119148513619} a^{15} + \frac{32711833756881850021372625}{997748533373685278712024137} a^{14} - \frac{29781067726513485226780332312}{186578975740879147119148513619} a^{13} - \frac{15105880403355525327047790640}{186578975740879147119148513619} a^{12} - \frac{15461226428444276481703438342}{186578975740879147119148513619} a^{11} + \frac{45708227623117001446207186046}{186578975740879147119148513619} a^{10} + \frac{81097984627344508520891909578}{186578975740879147119148513619} a^{9} + \frac{25706129222597641800505106198}{186578975740879147119148513619} a^{8} + \frac{15027535327424526508839566871}{186578975740879147119148513619} a^{7} + \frac{949307823571588613007113368}{10975233867110538065832265507} a^{6} - \frac{39252893779093628524527171072}{186578975740879147119148513619} a^{5} - \frac{4518574952777472059210525127}{16961725067352649738104410329} a^{4} - \frac{82707357418886761179476430448}{186578975740879147119148513619} a^{3} + \frac{44351708781198964121682790041}{186578975740879147119148513619} a^{2} + \frac{2037017922403646559220578110}{16961725067352649738104410329} a + \frac{6688328163971893328460197}{18618798098081942632386839}$

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:  $15$
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:  \( 3104183176.85 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1496:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 40 conjugacy class representatives for t16n1496
Character table for t16n1496 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.5691125.1, 8.8.86078265625.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ R ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
4139Data not computed