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Properties

Label 16.0.11939508427...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 941^{4}$
Root discriminant $87.56$
Ramified primes $2, 5, 29, 941$
Class number $151840$ (GRH)
Class group $[4, 37960]$ (GRH)
Galois group 16T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2667232231, 112463354, 1229052881, 148225950, 263102522, 21797522, 34322677, 851928, 2933108, -45672, 162976, -5200, 5670, -170, 113, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 113*x^14 - 170*x^13 + 5670*x^12 - 5200*x^11 + 162976*x^10 - 45672*x^9 + 2933108*x^8 + 851928*x^7 + 34322677*x^6 + 21797522*x^5 + 263102522*x^4 + 148225950*x^3 + 1229052881*x^2 + 112463354*x + 2667232231)
 
gp: K = bnfinit(x^16 - 2*x^15 + 113*x^14 - 170*x^13 + 5670*x^12 - 5200*x^11 + 162976*x^10 - 45672*x^9 + 2933108*x^8 + 851928*x^7 + 34322677*x^6 + 21797522*x^5 + 263102522*x^4 + 148225950*x^3 + 1229052881*x^2 + 112463354*x + 2667232231, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 113 x^{14} - 170 x^{13} + 5670 x^{12} - 5200 x^{11} + 162976 x^{10} - 45672 x^{9} + 2933108 x^{8} + 851928 x^{7} + 34322677 x^{6} + 21797522 x^{5} + 263102522 x^{4} + 148225950 x^{3} + 1229052881 x^{2} + 112463354 x + 2667232231 \)

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:  \(11939508427262544380953600000000=2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 941^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.56$
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:  $2, 5, 29, 941$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{19} a^{14} - \frac{2}{19} a^{12} - \frac{3}{19} a^{11} + \frac{4}{19} a^{10} - \frac{2}{19} a^{9} + \frac{5}{19} a^{8} - \frac{3}{19} a^{7} - \frac{9}{19} a^{6} - \frac{9}{19} a^{5} + \frac{4}{19} a^{4} - \frac{2}{19} a^{3} - \frac{5}{19} a^{2} + \frac{7}{19} a - \frac{5}{19}$, $\frac{1}{376315379149896161242685961991476646252007407880740882090099} a^{15} - \frac{9122802886349027696991833980824526116432708901373110971390}{376315379149896161242685961991476646252007407880740882090099} a^{14} + \frac{5894100822110192898608953143937499366060518614142728640263}{376315379149896161242685961991476646252007407880740882090099} a^{13} - \frac{6661862808779772070257886296795644453802906784513218734757}{16361538223908528749681998347455506358782930777423516612613} a^{12} - \frac{2942961702309430081775070539150814770042247990178541666679}{376315379149896161242685961991476646252007407880740882090099} a^{11} + \frac{93307369383014244696532463516141500656443326204329132123081}{376315379149896161242685961991476646252007407880740882090099} a^{10} - \frac{29374484431905118569090035313299866963442938717254070940706}{376315379149896161242685961991476646252007407880740882090099} a^{9} - \frac{107461433952944625997327378890169329078711892647706138651538}{376315379149896161242685961991476646252007407880740882090099} a^{8} + \frac{143444743642384829748891553608314765310022531710963021534941}{376315379149896161242685961991476646252007407880740882090099} a^{7} - \frac{4088330548501032173717647160839526050315670802649007673271}{19806072586836640065404524315340876118526705677933730636321} a^{6} - \frac{58734913983129255650088531274934606009803799067930404693106}{376315379149896161242685961991476646252007407880740882090099} a^{5} - \frac{1064857800133789080103636888379320350538631496320678939874}{9178423881704784420553316146133576737853839216603436148539} a^{4} - \frac{75161418279059461068368076275091910521396206728741175569865}{376315379149896161242685961991476646252007407880740882090099} a^{3} - \frac{77689571265913893024523245505832726029436772752902185943142}{376315379149896161242685961991476646252007407880740882090099} a^{2} + \frac{96187259181697673737870367435398357180483705345162153347668}{376315379149896161242685961991476646252007407880740882090099} a + \frac{89062701378412663299608902018161717764024908250723987596538}{376315379149896161242685961991476646252007407880740882090099}$

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

Class group and class number

$C_{4}\times C_{37960}$, which has order $151840$ (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:  \( 4024.00673312 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 44 conjugacy class representatives for t16n1025
Character table for t16n1025 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.0.3455359377440000.1, 8.0.3672007840000.1, 8.8.494613125.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
941Data not computed