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Properties

Label 16.0.40057678793...8433.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{11}\cdot 43^{8}$
Root discriminant $45.99$
Ramified primes $17, 43$
Class number $2$ (GRH)
Class group $[2]$ (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![78949, -258800, 493567, -304085, 57683, 15701, 35757, -26244, -2631, 8314, -975, -1167, 257, 75, -23, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 23*x^14 + 75*x^13 + 257*x^12 - 1167*x^11 - 975*x^10 + 8314*x^9 - 2631*x^8 - 26244*x^7 + 35757*x^6 + 15701*x^5 + 57683*x^4 - 304085*x^3 + 493567*x^2 - 258800*x + 78949)
 
gp: K = bnfinit(x^16 - 2*x^15 - 23*x^14 + 75*x^13 + 257*x^12 - 1167*x^11 - 975*x^10 + 8314*x^9 - 2631*x^8 - 26244*x^7 + 35757*x^6 + 15701*x^5 + 57683*x^4 - 304085*x^3 + 493567*x^2 - 258800*x + 78949, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 23 x^{14} + 75 x^{13} + 257 x^{12} - 1167 x^{11} - 975 x^{10} + 8314 x^{9} - 2631 x^{8} - 26244 x^{7} + 35757 x^{6} + 15701 x^{5} + 57683 x^{4} - 304085 x^{3} + 493567 x^{2} - 258800 x + 78949 \)

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:  \(400576787936788717495228433=17^{11}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.99$
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:  $17, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{984864623754887442096425145758959641557263} a^{15} - \frac{386656367417620206557610477385090298415370}{984864623754887442096425145758959641557263} a^{14} - \frac{282890326118648939019511717660855009753865}{984864623754887442096425145758959641557263} a^{13} + \frac{104431630223725164720184127368743390850385}{984864623754887442096425145758959641557263} a^{12} - \frac{167614115202013954774792383496456394062005}{984864623754887442096425145758959641557263} a^{11} - \frac{10055552789133250657616011621286319451469}{75758817211914418622801934289150741658251} a^{10} + \frac{532638015058069879891537714230630053704}{1284047749354481671572914140494080367089} a^{9} + \frac{421841103933931043812837087681958505923714}{984864623754887442096425145758959641557263} a^{8} + \frac{471249491078702451896471609937341035743739}{984864623754887442096425145758959641557263} a^{7} - \frac{489655926965018811188494725466282636240048}{984864623754887442096425145758959641557263} a^{6} - \frac{128584279791791002974605238581822653484627}{984864623754887442096425145758959641557263} a^{5} + \frac{445094583992662258888358320994858627691478}{984864623754887442096425145758959641557263} a^{4} + \frac{348910370728190706526425370470284073066019}{984864623754887442096425145758959641557263} a^{3} + \frac{482125803907523936451461421545860437360213}{984864623754887442096425145758959641557263} a^{2} - \frac{88158405220898102169855465126333716994380}{984864623754887442096425145758959641557263} a - \frac{30623735171470127959275028248883590879536}{75758817211914418622801934289150741658251}$

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:  \( 2382782.87662 \) (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

\(\Q(\sqrt{-43}) \), 4.0.31433.1, 8.0.16796569313.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$17$17.8.4.2$x^{8} - 4913 x^{2} + 918731$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
17.8.7.6$x^{8} + 37179$$8$$1$$7$$C_8$$[\ ]_{8}$
43Data not computed