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Properties

Label 16.0.25775033599...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{15}\cdot 61^{5}$
Root discriminant $16.34$
Ramified primes $5, 61$
Class number $2$
Class group $[2]$
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 24, 175, 375, 360, 357, 398, 20, 0, 20, -57, -8, 45, 5, -10, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 10*x^14 + 5*x^13 + 45*x^12 - 8*x^11 - 57*x^10 + 20*x^9 + 20*x^7 + 398*x^6 + 357*x^5 + 360*x^4 + 375*x^3 + 175*x^2 + 24*x + 1)
 
gp: K = bnfinit(x^16 - x^15 - 10*x^14 + 5*x^13 + 45*x^12 - 8*x^11 - 57*x^10 + 20*x^9 + 20*x^7 + 398*x^6 + 357*x^5 + 360*x^4 + 375*x^3 + 175*x^2 + 24*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 10 x^{14} + 5 x^{13} + 45 x^{12} - 8 x^{11} - 57 x^{10} + 20 x^{9} + 20 x^{7} + 398 x^{6} + 357 x^{5} + 360 x^{4} + 375 x^{3} + 175 x^{2} + 24 x + 1 \)

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:  \(25775033599853515625=5^{15}\cdot 61^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.34$
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:  $5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1831137652609575539} a^{15} - \frac{727678615321459900}{1831137652609575539} a^{14} - \frac{669596247228142647}{1831137652609575539} a^{13} + \frac{426226666879168031}{1831137652609575539} a^{12} - \frac{855585924149264691}{1831137652609575539} a^{11} - \frac{885368273596281936}{1831137652609575539} a^{10} - \frac{149651601543984516}{1831137652609575539} a^{9} + \frac{859934124698071668}{1831137652609575539} a^{8} - \frac{337020325283341007}{1831137652609575539} a^{7} - \frac{462682761521921499}{1831137652609575539} a^{6} + \frac{406298134464916576}{1831137652609575539} a^{5} + \frac{440034146320605377}{1831137652609575539} a^{4} + \frac{581006206276120218}{1831137652609575539} a^{3} + \frac{77413577228227970}{1831137652609575539} a^{2} + \frac{778631047584069821}{1831137652609575539} a - \frac{645060815323954518}{1831137652609575539}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{5523291253295359}{5532137923291769} a^{15} + \frac{6174072215399641}{5532137923291769} a^{14} + \frac{54446012257468171}{5532137923291769} a^{13} - \frac{33908891288381568}{5532137923291769} a^{12} - \frac{244031669419491649}{5532137923291769} a^{11} + \frac{72012641964992758}{5532137923291769} a^{10} + \frac{304048590376967691}{5532137923291769} a^{9} - \frac{142861673355958840}{5532137923291769} a^{8} + \frac{19624763197137523}{5532137923291769} a^{7} - \frac{118491971936640404}{5532137923291769} a^{6} - \frac{2184049508609182335}{5532137923291769} a^{5} - \frac{1713469722702580419}{5532137923291769} a^{4} - \frac{1807188883187045413}{5532137923291769} a^{3} - \frac{1857024294207404847}{5532137923291769} a^{2} - \frac{750621425069824649}{5532137923291769} a - \frac{51009580557978530}{5532137923291769} \) (order $10$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3374.49345422 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.4765625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
5Data not computed
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.4.2.2$x^{4} - 61 x^{2} + 7442$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.4.3.2$x^{4} - 244$$4$$1$$3$$C_4$$[\ ]_{4}$