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Properties

Label 16.0.61575433521...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 5^{12}\cdot 83^{4}$
Root discriminant $23.01$
Ramified primes $3, 5, 83$
Class number $4$
Class group $[2, 2]$
Galois group $C_4\times S_4$ (as 16T181)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 43, -137, 298, -393, 649, -626, 463, -183, 66, -25, 48, -21, 7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 7*x^14 - 21*x^13 + 48*x^12 - 25*x^11 + 66*x^10 - 183*x^9 + 463*x^8 - 626*x^7 + 649*x^6 - 393*x^5 + 298*x^4 - 137*x^3 + 43*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 + 7*x^14 - 21*x^13 + 48*x^12 - 25*x^11 + 66*x^10 - 183*x^9 + 463*x^8 - 626*x^7 + 649*x^6 - 393*x^5 + 298*x^4 - 137*x^3 + 43*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 7 x^{14} - 21 x^{13} + 48 x^{12} - 25 x^{11} + 66 x^{10} - 183 x^{9} + 463 x^{8} - 626 x^{7} + 649 x^{6} - 393 x^{5} + 298 x^{4} - 137 x^{3} + 43 x^{2} - 8 x + 1 \)

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:  \(6157543352187744140625=3^{12}\cdot 5^{12}\cdot 83^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.01$
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:  $3, 5, 83$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{579933626724499081} a^{15} + \frac{190562059190212109}{579933626724499081} a^{14} - \frac{46460428671921069}{579933626724499081} a^{13} + \frac{278518794589618497}{579933626724499081} a^{12} + \frac{152769070401120832}{579933626724499081} a^{11} + \frac{195859667335606778}{579933626724499081} a^{10} + \frac{21283824189584195}{579933626724499081} a^{9} + \frac{272837263253445006}{579933626724499081} a^{8} - \frac{41839599545530287}{579933626724499081} a^{7} + \frac{3968198052041460}{579933626724499081} a^{6} + \frac{233417574244819308}{579933626724499081} a^{5} + \frac{168656895326527865}{579933626724499081} a^{4} + \frac{81203648647104722}{579933626724499081} a^{3} + \frac{271112442474976499}{579933626724499081} a^{2} + \frac{218789540078480301}{579933626724499081} a - \frac{270311263254507719}{579933626724499081}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( \frac{73029584819897249}{579933626724499081} a^{15} - \frac{162775257629643180}{579933626724499081} a^{14} + \frac{517094340249579661}{579933626724499081} a^{13} - \frac{1606345805009401631}{579933626724499081} a^{12} + \frac{3685058594269131720}{579933626724499081} a^{11} - \frac{2124085387859260981}{579933626724499081} a^{10} + \frac{4139394661666996272}{579933626724499081} a^{9} - \frac{14288086806047614416}{579933626724499081} a^{8} + \frac{35293955087986134994}{579933626724499081} a^{7} - \frac{49083804118573453558}{579933626724499081} a^{6} + \frac{47041820676990360675}{579933626724499081} a^{5} - \frac{27206203005309694641}{579933626724499081} a^{4} + \frac{16934801109233966399}{579933626724499081} a^{3} - \frac{10941782442828618849}{579933626724499081} a^{2} + \frac{643279930590324427}{579933626724499081} a + \frac{52169438270252402}{579933626724499081} \) (order $10$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 14789.2708979 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times S_4$ (as 16T181):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 20 conjugacy class representatives for $C_4\times S_4$
Character table for $C_4\times S_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.56025.1, 8.8.3138800625.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.12.12.23$x^{12} + 21 x^{11} + 21 x^{10} + 63 x^{9} + 36 x^{8} + 54 x^{7} + 90 x^{6} + 81 x^{3} - 81$$3$$4$$12$$S_3 \times C_4$$[3/2]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$83$83.4.0.1$x^{4} - x + 22$$1$$4$$0$$C_4$$[\ ]^{4}$
83.4.0.1$x^{4} - x + 22$$1$$4$$0$$C_4$$[\ ]^{4}$
83.8.4.1$x^{8} + 303116 x^{4} - 571787 x^{2} + 22969827364$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$