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Properties

Label 16.12.2429203788...1449.1
Degree $16$
Signature $[12, 2]$
Discriminant $61^{2}\cdot 97^{14}$
Root discriminant $91.54$
Ramified primes $61, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1194

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![673, 13177, 50232, -24747, -267330, -296732, -7339, 149437, 55928, -22454, -14262, 786, 1350, 37, -58, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 58*x^14 + 37*x^13 + 1350*x^12 + 786*x^11 - 14262*x^10 - 22454*x^9 + 55928*x^8 + 149437*x^7 - 7339*x^6 - 296732*x^5 - 267330*x^4 - 24747*x^3 + 50232*x^2 + 13177*x + 673)
 
gp: K = bnfinit(x^16 - 2*x^15 - 58*x^14 + 37*x^13 + 1350*x^12 + 786*x^11 - 14262*x^10 - 22454*x^9 + 55928*x^8 + 149437*x^7 - 7339*x^6 - 296732*x^5 - 267330*x^4 - 24747*x^3 + 50232*x^2 + 13177*x + 673, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 58 x^{14} + 37 x^{13} + 1350 x^{12} + 786 x^{11} - 14262 x^{10} - 22454 x^{9} + 55928 x^{8} + 149437 x^{7} - 7339 x^{6} - 296732 x^{5} - 267330 x^{4} - 24747 x^{3} + 50232 x^{2} + 13177 x + 673 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(24292037884309209334711647701449=61^{2}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.54$
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:  $61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{183} a^{14} - \frac{16}{183} a^{13} - \frac{29}{183} a^{12} + \frac{86}{183} a^{11} + \frac{2}{61} a^{10} - \frac{25}{183} a^{9} + \frac{46}{183} a^{8} - \frac{15}{61} a^{7} - \frac{53}{183} a^{6} - \frac{73}{183} a^{5} - \frac{8}{183} a^{4} - \frac{16}{183} a^{3} + \frac{16}{61} a^{2} - \frac{34}{183} a - \frac{71}{183}$, $\frac{1}{140357228701230661618264233} a^{15} + \frac{13032633002773203553735}{140357228701230661618264233} a^{14} - \frac{17424851102060579459271743}{140357228701230661618264233} a^{13} - \frac{7261284268959031793414236}{140357228701230661618264233} a^{12} + \frac{18011132145665111725391756}{140357228701230661618264233} a^{11} - \frac{3805477992677189469551816}{140357228701230661618264233} a^{10} - \frac{15908916342678165828740595}{46785742900410220539421411} a^{9} - \frac{130196192901221154003719}{312599618488264279773417} a^{8} - \frac{25830664026736364751271256}{140357228701230661618264233} a^{7} + \frac{42792329246253971037966430}{140357228701230661618264233} a^{6} - \frac{62601269049703245225756065}{140357228701230661618264233} a^{5} + \frac{66474436897267407604604987}{140357228701230661618264233} a^{4} - \frac{23194789589298692004828076}{140357228701230661618264233} a^{3} - \frac{33362742378173152883798846}{140357228701230661618264233} a^{2} - \frac{18686329650462435168268604}{46785742900410220539421411} a - \frac{13061385338828930609541286}{140357228701230661618264233}$

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:  $13$
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:  \( 10188441972.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1194:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1194
Character table for t16n1194 is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
61Data not computed
$97$97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$