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Properties

Label 16.8.74787115857...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 5^{8}\cdot 13^{6}\cdot 31^{4}$
Root discriminant $55.22$
Ramified primes $2, 5, 13, 31$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times C_4).C_2^4$ (as 16T205)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![185071, 4371640, -8339980, 3390956, -62674, -214428, 153132, -24488, 21998, -8760, -678, 244, -511, -16, -14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 14*x^14 - 16*x^13 - 511*x^12 + 244*x^11 - 678*x^10 - 8760*x^9 + 21998*x^8 - 24488*x^7 + 153132*x^6 - 214428*x^5 - 62674*x^4 + 3390956*x^3 - 8339980*x^2 + 4371640*x + 185071)
 
gp: K = bnfinit(x^16 - 4*x^15 - 14*x^14 - 16*x^13 - 511*x^12 + 244*x^11 - 678*x^10 - 8760*x^9 + 21998*x^8 - 24488*x^7 + 153132*x^6 - 214428*x^5 - 62674*x^4 + 3390956*x^3 - 8339980*x^2 + 4371640*x + 185071, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 14 x^{14} - 16 x^{13} - 511 x^{12} + 244 x^{11} - 678 x^{10} - 8760 x^{9} + 21998 x^{8} - 24488 x^{7} + 153132 x^{6} - 214428 x^{5} - 62674 x^{4} + 3390956 x^{3} - 8339980 x^{2} + 4371640 x + 185071 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7478711585794844262400000000=2^{32}\cdot 5^{8}\cdot 13^{6}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.22$
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:  $2, 5, 13, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{16334633703436526499907859485172457011227413569848451} a^{15} + \frac{7065946919775397171441194759397315449373560087802232}{16334633703436526499907859485172457011227413569848451} a^{14} + \frac{5632834559332528683229412785025411177198670089221144}{16334633703436526499907859485172457011227413569848451} a^{13} - \frac{5226554540888825350400499403382473514589242173817126}{16334633703436526499907859485172457011227413569848451} a^{12} - \frac{5600968704838558312255140978237856904219378066238289}{16334633703436526499907859485172457011227413569848451} a^{11} - \frac{16221743981602806624916746528670439966780693293319}{960860806084501558818109381480732765366318445285203} a^{10} - \frac{4599866985795436111345455475667035161217583989948569}{16334633703436526499907859485172457011227413569848451} a^{9} + \frac{7942461916385100487881141913736230408006094448455830}{16334633703436526499907859485172457011227413569848451} a^{8} + \frac{3771952966795828145949834366916376480254963965853276}{16334633703436526499907859485172457011227413569848451} a^{7} + \frac{8047172537303971055203370089311398121160955313653128}{16334633703436526499907859485172457011227413569848451} a^{6} - \frac{6077764232696337850449597630399080921891441922437688}{16334633703436526499907859485172457011227413569848451} a^{5} - \frac{573436368622050593350422767642558632774457464181088}{16334633703436526499907859485172457011227413569848451} a^{4} - \frac{2713143021937495213285343448746617862916793999938346}{16334633703436526499907859485172457011227413569848451} a^{3} + \frac{5767966923385247235088347815071889914273343611121694}{16334633703436526499907859485172457011227413569848451} a^{2} + \frac{7172048697184751735431818435086804886319207668512546}{16334633703436526499907859485172457011227413569848451} a + \frac{688717916720972482355948387420788319178619010626883}{16334633703436526499907859485172457011227413569848451}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 43633048.4531 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).C_2^4$ (as 16T205):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $(C_2\times C_4).C_2^4$
Character table for $(C_2\times C_4).C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.432640000.1

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

Sibling fields

Degree 16 sibling: 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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
5Data not computed
$13$13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
13.8.6.3$x^{8} + 65 x^{4} + 1352$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
31Data not computed