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Properties

Label 16.0.56148868668...6129.3
Degree $16$
Signature $[0, 8]$
Discriminant $23^{6}\cdot 41^{14}$
Root discriminant $83.53$
Ramified primes $23, 41$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T258)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![364951, 752231, -504530, -1675190, 2060975, -811880, 259075, -78822, -7306, 3679, -3371, 1172, 357, -37, 49, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 49*x^14 - 37*x^13 + 357*x^12 + 1172*x^11 - 3371*x^10 + 3679*x^9 - 7306*x^8 - 78822*x^7 + 259075*x^6 - 811880*x^5 + 2060975*x^4 - 1675190*x^3 - 504530*x^2 + 752231*x + 364951)
 
gp: K = bnfinit(x^16 - 3*x^15 + 49*x^14 - 37*x^13 + 357*x^12 + 1172*x^11 - 3371*x^10 + 3679*x^9 - 7306*x^8 - 78822*x^7 + 259075*x^6 - 811880*x^5 + 2060975*x^4 - 1675190*x^3 - 504530*x^2 + 752231*x + 364951, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 49 x^{14} - 37 x^{13} + 357 x^{12} + 1172 x^{11} - 3371 x^{10} + 3679 x^{9} - 7306 x^{8} - 78822 x^{7} + 259075 x^{6} - 811880 x^{5} + 2060975 x^{4} - 1675190 x^{3} - 504530 x^{2} + 752231 x + 364951 \)

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:  \(5614886866882301027209678756129=23^{6}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.53$
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:  $23, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{12} - \frac{1}{5} a^{11} - \frac{3}{10} a^{9} - \frac{3}{10} a^{8} + \frac{2}{5} a^{7} - \frac{1}{10} a^{6} - \frac{2}{5} a^{5} + \frac{3}{10} a^{4} + \frac{1}{10} a^{3} - \frac{3}{10} a^{2} + \frac{3}{10} a - \frac{1}{5}$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{10} a^{6} - \frac{1}{2} a^{5} - \frac{3}{10} a^{4} + \frac{2}{5} a^{3} - \frac{3}{10} a^{2} - \frac{1}{10} a - \frac{2}{5}$, $\frac{1}{17020} a^{14} + \frac{39}{1702} a^{13} - \frac{243}{8510} a^{12} - \frac{4179}{17020} a^{11} + \frac{734}{4255} a^{10} + \frac{369}{17020} a^{9} + \frac{919}{8510} a^{8} - \frac{2101}{4255} a^{7} + \frac{103}{4255} a^{6} + \frac{493}{1702} a^{5} - \frac{2917}{17020} a^{4} + \frac{19}{92} a^{3} - \frac{439}{3404} a^{2} - \frac{157}{851} a + \frac{2659}{17020}$, $\frac{1}{54693406499448325975775734972570742453901740} a^{15} + \frac{466144436155152438308269695279815206327}{27346703249724162987887867486285371226950870} a^{14} + \frac{543523557682223382219417251591152221327209}{27346703249724162987887867486285371226950870} a^{13} - \frac{2475731535088665421447253087760773641810927}{54693406499448325975775734972570742453901740} a^{12} + \frac{1442364533875750260539174583172928368069811}{27346703249724162987887867486285371226950870} a^{11} + \frac{1683831825079377394591108420082640883811291}{54693406499448325975775734972570742453901740} a^{10} + \frac{635507938843365906415944381664579649877293}{2734670324972416298788786748628537122695087} a^{9} + \frac{6391702232592382283653959806555619365529633}{13673351624862081493943933743142685613475435} a^{8} + \frac{7410921561187898664262949883416590352789959}{27346703249724162987887867486285371226950870} a^{7} + \frac{974796418332582066566955733319856997750722}{13673351624862081493943933743142685613475435} a^{6} - \frac{15724183290091360336202193285938423649619541}{54693406499448325975775734972570742453901740} a^{5} - \frac{16641427045907906055507268292836559082168957}{54693406499448325975775734972570742453901740} a^{4} - \frac{2011537945952357811425315913544880754416993}{54693406499448325975775734972570742453901740} a^{3} + \frac{582684086205303911638241220321991208605701}{2734670324972416298788786748628537122695087} a^{2} + \frac{21280506377221658480112977884927684231435153}{54693406499448325975775734972570742453901740} a + \frac{1074949189702163854934280973355306079818937}{13673351624862081493943933743142685613475435}$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( 32641463.1312 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T258):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.6.57794518300247.1, 8.0.194754273881.1, 8.2.2369575250310127.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$