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Properties

Label 16.4.23302078379...3161.1
Degree $16$
Signature $[4, 6]$
Discriminant $23^{8}\cdot 29^{14}$
Root discriminant $91.30$
Ramified primes $23, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![919829, 136350, -236831, 187219, -55815, -53943, 28943, -1894, -10696, 10244, -4856, 1526, -224, -42, 26, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 26*x^14 - 42*x^13 - 224*x^12 + 1526*x^11 - 4856*x^10 + 10244*x^9 - 10696*x^8 - 1894*x^7 + 28943*x^6 - 53943*x^5 - 55815*x^4 + 187219*x^3 - 236831*x^2 + 136350*x + 919829)
 
gp: K = bnfinit(x^16 - 8*x^15 + 26*x^14 - 42*x^13 - 224*x^12 + 1526*x^11 - 4856*x^10 + 10244*x^9 - 10696*x^8 - 1894*x^7 + 28943*x^6 - 53943*x^5 - 55815*x^4 + 187219*x^3 - 236831*x^2 + 136350*x + 919829, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 26 x^{14} - 42 x^{13} - 224 x^{12} + 1526 x^{11} - 4856 x^{10} + 10244 x^{9} - 10696 x^{8} - 1894 x^{7} + 28943 x^{6} - 53943 x^{5} - 55815 x^{4} + 187219 x^{3} - 236831 x^{2} + 136350 x + 919829 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(23302078379314905029568288023161=23^{8}\cdot 29^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.30$
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:  $23, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{5} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{495925} a^{12} - \frac{6}{495925} a^{11} - \frac{45856}{495925} a^{10} + \frac{6193}{99185} a^{9} + \frac{26098}{495925} a^{8} + \frac{205677}{495925} a^{7} - \frac{109048}{495925} a^{6} - \frac{14831}{99185} a^{5} - \frac{28972}{99185} a^{4} + \frac{111911}{495925} a^{3} + \frac{3052}{495925} a^{2} - \frac{202149}{495925} a - \frac{142744}{495925}$, $\frac{1}{495925} a^{13} - \frac{45892}{495925} a^{11} - \frac{45801}{495925} a^{10} + \frac{13518}{495925} a^{9} - \frac{1379}{19837} a^{8} - \frac{65206}{495925} a^{7} + \frac{164222}{495925} a^{6} - \frac{7722}{19837} a^{5} + \frac{135416}{495925} a^{4} - \frac{19777}{495925} a^{3} + \frac{113718}{495925} a^{2} + \frac{231322}{495925} a + \frac{135386}{495925}$, $\frac{1}{271382558125} a^{14} - \frac{1}{38768936875} a^{13} - \frac{50293}{271382558125} a^{12} + \frac{301849}{271382558125} a^{11} + \frac{16030683791}{271382558125} a^{10} - \frac{25879674446}{271382558125} a^{9} + \frac{7343138156}{271382558125} a^{8} + \frac{125908814677}{271382558125} a^{7} - \frac{63722980191}{271382558125} a^{6} + \frac{65146926566}{271382558125} a^{5} + \frac{134395270101}{271382558125} a^{4} - \frac{70161101559}{271382558125} a^{3} + \frac{132468102239}{271382558125} a^{2} + \frac{4129638866}{271382558125} a - \frac{3362943908}{271382558125}$, $\frac{1}{67623377216145625} a^{15} + \frac{1501}{814739484531875} a^{14} + \frac{54439756702}{67623377216145625} a^{13} - \frac{19234935521}{67623377216145625} a^{12} + \frac{6708935856548076}{67623377216145625} a^{11} - \frac{3204751902735381}{67623377216145625} a^{10} + \frac{4750934273504516}{67623377216145625} a^{9} - \frac{6089882442785158}{67623377216145625} a^{8} - \frac{6834917520915136}{67623377216145625} a^{7} + \frac{23979611739491376}{67623377216145625} a^{6} + \frac{9988908288565666}{67623377216145625} a^{5} + \frac{12802161160945906}{67623377216145625} a^{4} - \frac{9600092425980946}{67623377216145625} a^{3} + \frac{245872641008997}{814739484531875} a^{2} - \frac{2993347290523468}{67623377216145625} a + \frac{5479564211240256}{13524675443229125}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $9$
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:  \( 5321142774.02 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{29}) \), 4.4.12901781.1, 4.2.560947.1, 4.2.19343.1, 8.4.4827222636186869.1 x2, 8.4.166455952971961.1

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 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$23$23.4.2.1$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$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29Data not computed