Properties

Label 16.16.140...313.2
Degree $16$
Signature $[16, 0]$
Discriminant $1.406\times 10^{39}$
Root discriminant $279.74$
Ramified primes $17, 53$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 332*x^14 + 1473*x^13 + 38365*x^12 - 128109*x^11 - 2026194*x^10 + 3852972*x^9 + 53239647*x^8 - 18701219*x^7 - 668393921*x^6 - 732038184*x^5 + 2730181967*x^4 + 7553452254*x^3 + 7277621373*x^2 + 2991300154*x + 410069071)
 
gp: K = bnfinit(x^16 - 5*x^15 - 332*x^14 + 1473*x^13 + 38365*x^12 - 128109*x^11 - 2026194*x^10 + 3852972*x^9 + 53239647*x^8 - 18701219*x^7 - 668393921*x^6 - 732038184*x^5 + 2730181967*x^4 + 7553452254*x^3 + 7277621373*x^2 + 2991300154*x + 410069071, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![410069071, 2991300154, 7277621373, 7553452254, 2730181967, -732038184, -668393921, -18701219, 53239647, 3852972, -2026194, -128109, 38365, 1473, -332, -5, 1]);
 

\( x^{16} - 5 x^{15} - 332 x^{14} + 1473 x^{13} + 38365 x^{12} - 128109 x^{11} - 2026194 x^{10} + 3852972 x^{9} + 53239647 x^{8} - 18701219 x^{7} - 668393921 x^{6} - 732038184 x^{5} + 2730181967 x^{4} + 7553452254 x^{3} + 7277621373 x^{2} + 2991300154 x + 410069071 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[16, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(1406190811803906514680942949803142045313\)\(\medspace = 17^{15}\cdot 53^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $279.74$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $17, 53$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
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}{31874916053439751006733544828352725415458970530048718587411115980637} a^{15} + \frac{2274107284253035661951343747272000912639045234268812900875699789677}{31874916053439751006733544828352725415458970530048718587411115980637} a^{14} - \frac{12744646952280441714661403988232492020383106810059712927344427155368}{31874916053439751006733544828352725415458970530048718587411115980637} a^{13} + \frac{13294137692464066074163405338435255546488193924857434907895378343862}{31874916053439751006733544828352725415458970530048718587411115980637} a^{12} - \frac{7898799415663777077472421312398025482172076624248810985479813312151}{31874916053439751006733544828352725415458970530048718587411115980637} a^{11} + \frac{2372621209877480646165577008537363791641898512156486282429032559906}{31874916053439751006733544828352725415458970530048718587411115980637} a^{10} + \frac{13228230858193600907000248640426899071739577427249373795217394109446}{31874916053439751006733544828352725415458970530048718587411115980637} a^{9} + \frac{11447181360950390624824532760633634148989101279581729319665735320954}{31874916053439751006733544828352725415458970530048718587411115980637} a^{8} + \frac{15730063303727072005509929381493652801438520787631546383345397864365}{31874916053439751006733544828352725415458970530048718587411115980637} a^{7} + \frac{1423912729479272323529873250659508673225182673400403008768811221957}{31874916053439751006733544828352725415458970530048718587411115980637} a^{6} - \frac{14751546628791365323492307189910595616841958553277528871889608198118}{31874916053439751006733544828352725415458970530048718587411115980637} a^{5} - \frac{5143178893942906706264531131320341334280035947943798739855469301527}{31874916053439751006733544828352725415458970530048718587411115980637} a^{4} + \frac{2151299322334183749046002075861262483940784552169382699719697534888}{31874916053439751006733544828352725415458970530048718587411115980637} a^{3} - \frac{5942223379334535407582636554292241247598040260560900389353329411261}{31874916053439751006733544828352725415458970530048718587411115980637} a^{2} + \frac{12922051917023180871890920393812944820521175204507759148805833111916}{31874916053439751006733544828352725415458970530048718587411115980637} a + \frac{3404639881038766022573372691003196742598485432979396680771564651296}{31874916053439751006733544828352725415458970530048718587411115980637}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 34530035498800 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{16}\cdot(2\pi)^{0}\cdot 34530035498800 \cdot 4}{2\sqrt{1406190811803906514680942949803142045313}}\approx 0.120693761667060$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.8.3237769502871713.1

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
17Data not computed
$53$53.8.6.4$x^{8} + 742 x^{4} + 351125$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
53.8.6.3$x^{8} - 53 x^{4} + 14045$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$