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Properties

Label 16.0.12227568666...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{4}\cdot 89^{14}$
Root discriminant $75.94$
Ramified primes $5, 89$
Class number $7345$ (GRH)
Class group $[7345]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![229888, 304734, 365021, 305600, 367768, 190602, 126202, 54742, 28477, 4546, 4361, -36, 554, -26, 32, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 32*x^14 - 26*x^13 + 554*x^12 - 36*x^11 + 4361*x^10 + 4546*x^9 + 28477*x^8 + 54742*x^7 + 126202*x^6 + 190602*x^5 + 367768*x^4 + 305600*x^3 + 365021*x^2 + 304734*x + 229888)
 
gp: K = bnfinit(x^16 - 2*x^15 + 32*x^14 - 26*x^13 + 554*x^12 - 36*x^11 + 4361*x^10 + 4546*x^9 + 28477*x^8 + 54742*x^7 + 126202*x^6 + 190602*x^5 + 367768*x^4 + 305600*x^3 + 365021*x^2 + 304734*x + 229888, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 32 x^{14} - 26 x^{13} + 554 x^{12} - 36 x^{11} + 4361 x^{10} + 4546 x^{9} + 28477 x^{8} + 54742 x^{7} + 126202 x^{6} + 190602 x^{5} + 367768 x^{4} + 305600 x^{3} + 365021 x^{2} + 304734 x + 229888 \)

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:  \(1222756866650275883340118650625=5^{4}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.94$
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:  $5, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{10} - \frac{3}{32} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{3}{16} a^{5} - \frac{7}{32} a^{4} + \frac{1}{4} a^{3} + \frac{13}{32} a^{2} - \frac{5}{16} a$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{3}{64} a^{9} - \frac{1}{64} a^{8} - \frac{1}{8} a^{7} - \frac{3}{32} a^{6} + \frac{11}{64} a^{5} + \frac{7}{64} a^{4} - \frac{27}{64} a^{3} + \frac{1}{64} a^{2} + \frac{13}{32} a$, $\frac{1}{256} a^{12} - \frac{1}{64} a^{10} + \frac{3}{64} a^{9} - \frac{25}{256} a^{8} + \frac{1}{128} a^{7} - \frac{27}{256} a^{6} + \frac{1}{128} a^{5} + \frac{7}{64} a^{4} + \frac{43}{128} a^{3} - \frac{101}{256} a^{2} + \frac{13}{128} a$, $\frac{1}{1024} a^{13} + \frac{1}{1024} a^{12} - \frac{1}{256} a^{11} + \frac{1}{128} a^{10} + \frac{51}{1024} a^{9} - \frac{23}{1024} a^{8} - \frac{89}{1024} a^{7} + \frac{39}{1024} a^{6} + \frac{79}{512} a^{5} + \frac{25}{512} a^{4} + \frac{113}{1024} a^{3} + \frac{437}{1024} a^{2} - \frac{115}{512} a - \frac{1}{2}$, $\frac{1}{8192} a^{14} + \frac{1}{4096} a^{13} - \frac{3}{8192} a^{12} - \frac{15}{2048} a^{11} - \frac{5}{8192} a^{10} + \frac{55}{2048} a^{9} - \frac{11}{512} a^{8} - \frac{153}{4096} a^{7} - \frac{955}{8192} a^{6} - \frac{127}{512} a^{5} - \frac{669}{8192} a^{4} + \frac{883}{4096} a^{3} + \frac{1295}{8192} a^{2} + \frac{717}{4096} a + \frac{3}{16}$, $\frac{1}{12173129256481906325504032768} a^{15} + \frac{713778843267945075138627}{12173129256481906325504032768} a^{14} + \frac{2111877371106011828227071}{12173129256481906325504032768} a^{13} - \frac{5297429032168921694528895}{12173129256481906325504032768} a^{12} - \frac{95093605566293562702085185}{12173129256481906325504032768} a^{11} + \frac{54171177924286605148277911}{12173129256481906325504032768} a^{10} - \frac{8901584819155392370739487}{276662028556406961943273472} a^{9} + \frac{532080912918681513058278735}{6086564628240953162752016384} a^{8} - \frac{982537323297610565635469549}{12173129256481906325504032768} a^{7} + \frac{40854254772274771885817877}{12173129256481906325504032768} a^{6} + \frac{2760994315821833406782447987}{12173129256481906325504032768} a^{5} - \frac{54460448267285639948879941}{1106648114225627847773093888} a^{4} - \frac{2644719708125847047526623755}{12173129256481906325504032768} a^{3} + \frac{4183592037899982594703923945}{12173129256481906325504032768} a^{2} + \frac{171060768226076947781970887}{553324057112813923886546944} a - \frac{4725471459672012051405181}{23775643079066223292000064}$

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

Class group and class number

$C_{7345}$, which has order $7345$ (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:  \( 15350212.3839 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.39605.1, 4.4.704969.1, 4.4.3524845.1, 8.0.1105783372388225.1, 8.0.44231334895529.1, 8.8.12424532274025.1

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 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
89Data not computed