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Properties

Label 16.0.68912132671...8833.1
Degree $16$
Signature $[0, 8]$
Discriminant $31^{8}\cdot 97^{7}$
Root discriminant $41.20$
Ramified primes $31, 97$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1327, 926, -1839, -1363, 2338, 3020, 34, -1692, -438, 621, 329, -108, -65, 10, 9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 9*x^14 + 10*x^13 - 65*x^12 - 108*x^11 + 329*x^10 + 621*x^9 - 438*x^8 - 1692*x^7 + 34*x^6 + 3020*x^5 + 2338*x^4 - 1363*x^3 - 1839*x^2 + 926*x + 1327)
 
gp: K = bnfinit(x^16 - x^15 + 9*x^14 + 10*x^13 - 65*x^12 - 108*x^11 + 329*x^10 + 621*x^9 - 438*x^8 - 1692*x^7 + 34*x^6 + 3020*x^5 + 2338*x^4 - 1363*x^3 - 1839*x^2 + 926*x + 1327, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 9 x^{14} + 10 x^{13} - 65 x^{12} - 108 x^{11} + 329 x^{10} + 621 x^{9} - 438 x^{8} - 1692 x^{7} + 34 x^{6} + 3020 x^{5} + 2338 x^{4} - 1363 x^{3} - 1839 x^{2} + 926 x + 1327 \)

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:  \(68912132671990843828028833=31^{8}\cdot 97^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.20$
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:  $31, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{144535554659799487495266581} a^{15} + \frac{38501351733415472852366633}{144535554659799487495266581} a^{14} - \frac{70949026191625236273891035}{144535554659799487495266581} a^{13} + \frac{64002021977633733662844129}{144535554659799487495266581} a^{12} + \frac{10102623703653124687909008}{144535554659799487495266581} a^{11} + \frac{32380096256411669861315890}{144535554659799487495266581} a^{10} + \frac{67421675945443878738524275}{144535554659799487495266581} a^{9} + \frac{21676162692505181914448452}{144535554659799487495266581} a^{8} + \frac{525748980676908312239606}{144535554659799487495266581} a^{7} - \frac{50841371103143231736699847}{144535554659799487495266581} a^{6} - \frac{46597362716525340871620933}{144535554659799487495266581} a^{5} + \frac{32897154253886157913448388}{144535554659799487495266581} a^{4} - \frac{29106803981258014693862142}{144535554659799487495266581} a^{3} - \frac{55700459397150448606074218}{144535554659799487495266581} a^{2} - \frac{35350017235826086692306172}{144535554659799487495266581} a - \frac{39489883875382544293948623}{144535554659799487495266581}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 344989.006574 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-31}) \), 4.0.93217.2, 8.0.842872681633.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ $16$ $16$ R $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
31Data not computed
$97$97.8.7.8$x^{8} + 7578125$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$