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Properties

Label 16.0.81950192277...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 5^{12}\cdot 29^{2}\cdot 241^{2}$
Root discriminant $48.09$
Ramified primes $2, 5, 29, 241$
Class number $760$ (GRH)
Class group $[2, 2, 190]$ (GRH)
Galois group 16T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![251761, 99788, 141578, 167908, 57936, 19260, 78102, -46980, 46707, -22360, 11742, -4000, 1366, -308, 68, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 68*x^14 - 308*x^13 + 1366*x^12 - 4000*x^11 + 11742*x^10 - 22360*x^9 + 46707*x^8 - 46980*x^7 + 78102*x^6 + 19260*x^5 + 57936*x^4 + 167908*x^3 + 141578*x^2 + 99788*x + 251761)
 
gp: K = bnfinit(x^16 - 8*x^15 + 68*x^14 - 308*x^13 + 1366*x^12 - 4000*x^11 + 11742*x^10 - 22360*x^9 + 46707*x^8 - 46980*x^7 + 78102*x^6 + 19260*x^5 + 57936*x^4 + 167908*x^3 + 141578*x^2 + 99788*x + 251761, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 68 x^{14} - 308 x^{13} + 1366 x^{12} - 4000 x^{11} + 11742 x^{10} - 22360 x^{9} + 46707 x^{8} - 46980 x^{7} + 78102 x^{6} + 19260 x^{5} + 57936 x^{4} + 167908 x^{3} + 141578 x^{2} + 99788 x + 251761 \)

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:  \(819501922779136000000000000=2^{36}\cdot 5^{12}\cdot 29^{2}\cdot 241^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.09$
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:  $2, 5, 29, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{41} a^{14} + \frac{6}{41} a^{13} + \frac{17}{41} a^{12} - \frac{19}{41} a^{11} - \frac{6}{41} a^{10} - \frac{2}{41} a^{9} + \frac{19}{41} a^{8} - \frac{12}{41} a^{7} - \frac{19}{41} a^{6} + \frac{7}{41} a^{5} - \frac{5}{41} a^{4} - \frac{19}{41} a^{2} - \frac{7}{41} a + \frac{12}{41}$, $\frac{1}{4115425511730528812042106297762400921} a^{15} + \frac{7496676514027455975867372359735335}{4115425511730528812042106297762400921} a^{14} - \frac{585372743501812151234288221554631701}{4115425511730528812042106297762400921} a^{13} + \frac{1525175891053849720016423139798065658}{4115425511730528812042106297762400921} a^{12} - \frac{267742353433259385293765367282466765}{4115425511730528812042106297762400921} a^{11} + \frac{202661105807560293839132260991279459}{4115425511730528812042106297762400921} a^{10} - \frac{1795530935736641218076994720898581720}{4115425511730528812042106297762400921} a^{9} + \frac{1849037822434718183081207657182383793}{4115425511730528812042106297762400921} a^{8} - \frac{1715555879965539420855740207529361978}{4115425511730528812042106297762400921} a^{7} + \frac{2055640952585910949810471098026917963}{4115425511730528812042106297762400921} a^{6} + \frac{159912214039626528894090586171398269}{4115425511730528812042106297762400921} a^{5} - \frac{1361960457209123431460404768246273515}{4115425511730528812042106297762400921} a^{4} - \frac{1635081571447916388662100381418577596}{4115425511730528812042106297762400921} a^{3} - \frac{890521496054182719051139396013723103}{4115425511730528812042106297762400921} a^{2} + \frac{1817230326985518748900378611326664426}{4115425511730528812042106297762400921} a - \frac{1191218163801506931302731184232201805}{4115425511730528812042106297762400921}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{190}$, which has order $760$ (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:  \( 7114.13535725 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n797 are not computed
Character table for t16n797 is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), 4.4.8000.1, \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
241Data not computed