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Properties

Label 16.0.92600652331...0625.6
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 41^{14}$
Root discriminant $86.18$
Ramified primes $5, 41$
Class number $600$ (GRH)
Class group $[2, 300]$ (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![68181376, -58027856, 18758616, -22659160, 48482750, -19834849, 6850515, -3202300, 964626, -194525, 52755, -10599, 1475, -310, 46, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 46*x^14 - 310*x^13 + 1475*x^12 - 10599*x^11 + 52755*x^10 - 194525*x^9 + 964626*x^8 - 3202300*x^7 + 6850515*x^6 - 19834849*x^5 + 48482750*x^4 - 22659160*x^3 + 18758616*x^2 - 58027856*x + 68181376)
 
gp: K = bnfinit(x^16 - x^15 + 46*x^14 - 310*x^13 + 1475*x^12 - 10599*x^11 + 52755*x^10 - 194525*x^9 + 964626*x^8 - 3202300*x^7 + 6850515*x^6 - 19834849*x^5 + 48482750*x^4 - 22659160*x^3 + 18758616*x^2 - 58027856*x + 68181376, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 46 x^{14} - 310 x^{13} + 1475 x^{12} - 10599 x^{11} + 52755 x^{10} - 194525 x^{9} + 964626 x^{8} - 3202300 x^{7} + 6850515 x^{6} - 19834849 x^{5} + 48482750 x^{4} - 22659160 x^{3} + 18758616 x^{2} - 58027856 x + 68181376 \)

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:  \(9260065233133681348183837890625=5^{12}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.18$
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:  $5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{12} + \frac{1}{32} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{1}{8} a^{7} + \frac{5}{32} a^{6} - \frac{3}{16} a^{5} + \frac{3}{32} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{13} + \frac{1}{32} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} + \frac{5}{32} a^{7} - \frac{3}{16} a^{6} + \frac{3}{32} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{256} a^{14} + \frac{1}{256} a^{13} - \frac{1}{256} a^{12} - \frac{9}{256} a^{11} + \frac{1}{128} a^{10} - \frac{5}{128} a^{9} - \frac{3}{256} a^{8} - \frac{57}{256} a^{7} - \frac{13}{256} a^{6} + \frac{51}{256} a^{5} + \frac{11}{128} a^{4} + \frac{5}{32} a^{3} + \frac{7}{32} a^{2} + \frac{3}{16} a - \frac{1}{2}$, $\frac{1}{64413097790297105681853723198614292063932117326495744} a^{15} + \frac{209543252894598940356803523861240409122312026801}{519460466050783110337530025795276548902678365536256} a^{14} + \frac{460477416510219516215660684653952682168103066925053}{32206548895148552840926861599307146031966058663247872} a^{13} - \frac{4931834151422186941718129760460486917972278229225}{16103274447574276420463430799653573015983029331623936} a^{12} + \frac{1748217697832037964777320341771104141033045331124655}{64413097790297105681853723198614292063932117326495744} a^{11} + \frac{575024126651351134251325478313974322829621281947123}{16103274447574276420463430799653573015983029331623936} a^{10} - \frac{1675507062679322389221519305188173444641612986287185}{64413097790297105681853723198614292063932117326495744} a^{9} + \frac{3771190568902742600103102899848832945807170402930091}{32206548895148552840926861599307146031966058663247872} a^{8} + \frac{840086835729494913495529514247067200015534789415497}{4025818611893569105115857699913393253995757332905984} a^{7} + \frac{1846327548158270559136319389364413631783713083624533}{16103274447574276420463430799653573015983029331623936} a^{6} + \frac{5060289534838380228577030138977479999914610668959447}{64413097790297105681853723198614292063932117326495744} a^{5} - \frac{2143406166781812903124588642299430787161605410869267}{32206548895148552840926861599307146031966058663247872} a^{4} + \frac{229278539975745024314670032318790722317831729673211}{4025818611893569105115857699913393253995757332905984} a^{3} - \frac{78229651765603425183113815312391792755436344686957}{8051637223787138210231715399826786507991514665811968} a^{2} - \frac{1450211137044500083964939965447239242194945508316031}{4025818611893569105115857699913393253995757332905984} a - \frac{48253797974340172715826699535052531780009046645719}{503227326486696138139482212489174156749469666613248}$

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

Class group and class number

$C_{2}\times C_{300}$, which has order $600$ (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:  \( 697029494.906 \) (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

\(\Q(\sqrt{41}) \), 4.4.1723025.1, 4.0.8405.1, 4.0.344605.1, 8.8.3043035529390625.2, 8.0.3043035529390625.1, 8.0.2968815150625.3

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed