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Properties

Label 16.0.89919474020...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{20}\cdot 5^{12}\cdot 37^{8}$
Root discriminant $48.37$
Ramified primes $2, 5, 37$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^3:S_4.C_2$ (as 16T764)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![83022676, -29920920, -32073644, 17845836, 1626709, -2844928, 904173, 6084, -105028, 32208, -1171, -658, 308, -90, 17, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 17*x^14 - 90*x^13 + 308*x^12 - 658*x^11 - 1171*x^10 + 32208*x^9 - 105028*x^8 + 6084*x^7 + 904173*x^6 - 2844928*x^5 + 1626709*x^4 + 17845836*x^3 - 32073644*x^2 - 29920920*x + 83022676)
 
gp: K = bnfinit(x^16 - 2*x^15 + 17*x^14 - 90*x^13 + 308*x^12 - 658*x^11 - 1171*x^10 + 32208*x^9 - 105028*x^8 + 6084*x^7 + 904173*x^6 - 2844928*x^5 + 1626709*x^4 + 17845836*x^3 - 32073644*x^2 - 29920920*x + 83022676, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 17 x^{14} - 90 x^{13} + 308 x^{12} - 658 x^{11} - 1171 x^{10} + 32208 x^{9} - 105028 x^{8} + 6084 x^{7} + 904173 x^{6} - 2844928 x^{5} + 1626709 x^{4} + 17845836 x^{3} - 32073644 x^{2} - 29920920 x + 83022676 \)

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:  \(899194740203776000000000000=2^{20}\cdot 5^{12}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.37$
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:  $2, 5, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{8} a^{14} + \frac{3}{8} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{3}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{21328259401004254129853937873925945801918213483586264048} a^{15} + \frac{13305204267038852970548587365453318385419977795495087}{1333016212562765883115871117120371612619888342724141503} a^{14} + \frac{1213734193171482167500241650660740461154624860918319339}{21328259401004254129853937873925945801918213483586264048} a^{13} - \frac{1558167139165068380384890824573219817776771692553174969}{5332064850251063532463484468481486450479553370896566012} a^{12} + \frac{535935372455152376913305103698519048883698982181125357}{10664129700502127064926968936962972900959106741793132024} a^{11} + \frac{3925633905856427531335221565568582447123679208699676661}{10664129700502127064926968936962972900959106741793132024} a^{10} + \frac{4660614338277437081577492454104785606493780583287219093}{21328259401004254129853937873925945801918213483586264048} a^{9} + \frac{4023840447361420345022153797727770830930651203782607687}{10664129700502127064926968936962972900959106741793132024} a^{8} - \frac{4315463011118341751485913483663737116950564169005676683}{10664129700502127064926968936962972900959106741793132024} a^{7} + \frac{1279436462660363911068648916175082284735407166449710085}{5332064850251063532463484468481486450479553370896566012} a^{6} - \frac{3047837165862568682159151594553683977171296425010596695}{21328259401004254129853937873925945801918213483586264048} a^{5} + \frac{1687026848781207770709046796777025567282486505526547261}{10664129700502127064926968936962972900959106741793132024} a^{4} + \frac{3844384748310589506976689571999660107225623314772146947}{21328259401004254129853937873925945801918213483586264048} a^{3} - \frac{5076934527342731789652885357461076974431479967180490093}{10664129700502127064926968936962972900959106741793132024} a^{2} - \frac{3957335233989692190087670093274510702116595329915571793}{10664129700502127064926968936962972900959106741793132024} a - \frac{248266011566361641963637580212480079969173921020121992}{1333016212562765883115871117120371612619888342724141503}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 2581734.76273 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:S_4.C_2$ (as 16T764):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 23 conjugacy class representatives for $C_2^3:S_4.C_2$
Character table for $C_2^3:S_4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.3700.1, 8.0.13690000.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.12.16.19$x^{12} + x^{10} - 2 x^{8} - 3 x^{6} + 2 x^{4} - 3 x^{2} + 1$$6$$2$$16$$C_2\times S_4$$[4/3, 4/3, 2]_{3}^{2}$
5Data not computed
$37$37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.8.6.2$x^{8} + 333 x^{4} + 34225$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$