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Properties

Label 16.0.25554788874...0409.4
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 37^{10}$
Root discriminant $21.78$
Ramified primes $3, 37$
Class number $4$
Class group $[4]$
Galois group $C_4\wr C_2$ (as 16T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729, -729, -972, -324, 3186, -4131, 4419, -3348, 2397, -1296, 693, -279, 124, -31, 15, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 15*x^14 - 31*x^13 + 124*x^12 - 279*x^11 + 693*x^10 - 1296*x^9 + 2397*x^8 - 3348*x^7 + 4419*x^6 - 4131*x^5 + 3186*x^4 - 324*x^3 - 972*x^2 - 729*x + 729)
 
gp: K = bnfinit(x^16 - x^15 + 15*x^14 - 31*x^13 + 124*x^12 - 279*x^11 + 693*x^10 - 1296*x^9 + 2397*x^8 - 3348*x^7 + 4419*x^6 - 4131*x^5 + 3186*x^4 - 324*x^3 - 972*x^2 - 729*x + 729, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 15 x^{14} - 31 x^{13} + 124 x^{12} - 279 x^{11} + 693 x^{10} - 1296 x^{9} + 2397 x^{8} - 3348 x^{7} + 4419 x^{6} - 4131 x^{5} + 3186 x^{4} - 324 x^{3} - 972 x^{2} - 729 x + 729 \)

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:  \(2555478887462114090409=3^{12}\cdot 37^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.78$
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:  $3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{2}{9} a^{7} - \frac{2}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{1}{3} a^{7} + \frac{4}{9} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{27} a^{12} - \frac{1}{27} a^{11} + \frac{2}{27} a^{9} - \frac{2}{27} a^{8} - \frac{4}{9} a^{7} + \frac{1}{9} a^{6} + \frac{1}{3} a^{5} + \frac{1}{9} a^{4}$, $\frac{1}{27} a^{13} - \frac{1}{27} a^{11} - \frac{1}{27} a^{10} + \frac{1}{9} a^{9} + \frac{4}{27} a^{8} - \frac{2}{9} a^{7} - \frac{1}{3} a^{6} + \frac{1}{9} a^{5} + \frac{4}{9} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{81} a^{14} - \frac{1}{81} a^{13} + \frac{2}{81} a^{11} - \frac{2}{81} a^{10} - \frac{4}{27} a^{9} + \frac{1}{27} a^{8} + \frac{1}{9} a^{7} - \frac{8}{27} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{470280248330575419} a^{15} + \frac{2327072792398796}{470280248330575419} a^{14} + \frac{3803212873753}{156760082776858473} a^{13} - \frac{4710140011363414}{470280248330575419} a^{12} - \frac{871495038458773}{42752749848234129} a^{11} - \frac{290068938621071}{156760082776858473} a^{10} + \frac{494254140000656}{17417786975206497} a^{9} + \frac{4792108088244274}{52253360925619491} a^{8} - \frac{12802717236271682}{156760082776858473} a^{7} - \frac{11631039638577164}{52253360925619491} a^{6} - \frac{25850176795223926}{52253360925619491} a^{5} - \frac{661647551199034}{5805928991735499} a^{4} - \frac{3218302807508201}{17417786975206497} a^{3} + \frac{331066601376889}{1935309663911833} a^{2} - \frac{749567845308923}{1935309663911833} a + \frac{542247619313597}{1935309663911833}$

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

Class group and class number

$C_{4}$, 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:  \( \frac{40115937572}{17807575005891} a^{15} - \frac{13167989095}{17807575005891} a^{14} + \frac{594064940792}{17807575005891} a^{13} - \frac{850985301569}{17807575005891} a^{12} + \frac{402805705061}{1618870455081} a^{11} - \frac{8320256871350}{17807575005891} a^{10} + \frac{7534119932597}{5935858335297} a^{9} - \frac{1398969721465}{659539815033} a^{8} + \frac{24495414544313}{5935858335297} a^{7} - \frac{30184263512378}{5935858335297} a^{6} + \frac{1575204576139}{219846605011} a^{5} - \frac{10868286226390}{1978619445099} a^{4} + \frac{3297898235777}{659539815033} a^{3} + \frac{781540102837}{659539815033} a^{2} - \frac{26162374961}{219846605011} a - \frac{343863337069}{219846605011} \) (order $6$)
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:  \( 21312.5774141 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{37}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-111}) \), 4.2.4107.1 x2, 4.0.333.1 x2, \(\Q(\sqrt{-3}, \sqrt{37})\), 8.0.36926037.1, 8.0.50551744653.2, 8.0.151807041.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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

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$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
37Data not computed