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Properties

Label 16.8.32454990969...0912.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{67}\cdot 17^{6}\cdot 977^{4}$
Root discriminant $294.75$
Ramified primes $2, 17, 977$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group 16T937

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2106522414576392, 0, -142557119271712, 0, -1502825730064, 0, 105806192448, 0, 1098627660, 0, -9455408, 0, -81140, 0, 272, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 272*x^14 - 81140*x^12 - 9455408*x^10 + 1098627660*x^8 + 105806192448*x^6 - 1502825730064*x^4 - 142557119271712*x^2 + 2106522414576392)
 
gp: K = bnfinit(x^16 + 272*x^14 - 81140*x^12 - 9455408*x^10 + 1098627660*x^8 + 105806192448*x^6 - 1502825730064*x^4 - 142557119271712*x^2 + 2106522414576392, 1)
 

Normalized defining polynomial

\( x^{16} + 272 x^{14} - 81140 x^{12} - 9455408 x^{10} + 1098627660 x^{8} + 105806192448 x^{6} - 1502825730064 x^{4} - 142557119271712 x^{2} + 2106522414576392 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3245499096990365116593765338925510950912=2^{67}\cdot 17^{6}\cdot 977^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $294.75$
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:  $2, 17, 977$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{1954} a^{10} + \frac{136}{977} a^{8} - \frac{49}{1954} a^{6} - \frac{1}{977} a^{4} + \frac{465}{977} a^{2}$, $\frac{1}{3908} a^{11} + \frac{68}{977} a^{9} + \frac{232}{977} a^{7} + \frac{488}{977} a^{5} - \frac{256}{977} a^{3}$, $\frac{1}{3818116} a^{12} + \frac{68}{954529} a^{10} - \frac{20285}{954529} a^{8} + \frac{44941}{1909058} a^{6} - \frac{247437}{954529} a^{4} - \frac{368}{977} a^{2}$, $\frac{1}{3818116} a^{13} + \frac{68}{954529} a^{11} - \frac{20285}{954529} a^{9} + \frac{44941}{1909058} a^{7} - \frac{247437}{954529} a^{5} - \frac{368}{977} a^{3}$, $\frac{1}{915869672053792941058887966600668092542953692} a^{14} - \frac{617177682332267548583222592130946769}{53874686591399584768169880388274593678997276} a^{12} - \frac{2590007536086324033685278528939141237281}{17612878308726787328055537819243617164287571} a^{10} - \frac{103910587804346137533452396067876075608927229}{457934836026896470529443983300334046271476846} a^{8} - \frac{8327728887004611821421384822602961642252291}{65419262289556638647063426185762006610210978} a^{6} + \frac{6776071841644115959261762023847735992535}{234357643821339033024280441811839327672199} a^{4} - \frac{34104856102847837962991177672629223179}{239874763379057352123112018231155913687} a^{2} + \frac{6856560896908881937756278893774717}{14442456702935598297496057452655543}$, $\frac{1}{915869672053792941058887966600668092542953692} a^{15} - \frac{617177682332267548583222592130946769}{53874686591399584768169880388274593678997276} a^{13} + \frac{7667480918834629482511227562077229487199}{70451513234907149312222151276974468657150284} a^{11} - \frac{72037948244644029042150255981465927045508165}{457934836026896470529443983300334046271476846} a^{9} + \frac{7206834932009861224759490177496102363447757}{65419262289556638647063426185762006610210978} a^{7} - \frac{110522687450714929228940014891187505800408}{234357643821339033024280441811839327672199} a^{5} - \frac{96958427674023561753694019706586146315}{239874763379057352123112018231155913687} a^{3} + \frac{6856560896908881937756278893774717}{14442456702935598297496057452655543} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  $11$
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:  \( 10670340578800 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T937:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n937
Character table for t16n937 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.4352.1, 8.8.9697230848.1

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 R $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$17$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
977Data not computed