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Properties

Label 16.0.14620212588...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 11^{10}\cdot 31^{4}$
Root discriminant $43.18$
Ramified primes $5, 11, 31$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T1161

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14771, -11125, 4339, -14930, 20497, -8605, -528, 3705, -1880, -200, -132, 125, 97, -25, 11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 11*x^14 - 25*x^13 + 97*x^12 + 125*x^11 - 132*x^10 - 200*x^9 - 1880*x^8 + 3705*x^7 - 528*x^6 - 8605*x^5 + 20497*x^4 - 14930*x^3 + 4339*x^2 - 11125*x + 14771)
 
gp: K = bnfinit(x^16 + 11*x^14 - 25*x^13 + 97*x^12 + 125*x^11 - 132*x^10 - 200*x^9 - 1880*x^8 + 3705*x^7 - 528*x^6 - 8605*x^5 + 20497*x^4 - 14930*x^3 + 4339*x^2 - 11125*x + 14771, 1)
 

Normalized defining polynomial

\( x^{16} + 11 x^{14} - 25 x^{13} + 97 x^{12} + 125 x^{11} - 132 x^{10} - 200 x^{9} - 1880 x^{8} + 3705 x^{7} - 528 x^{6} - 8605 x^{5} + 20497 x^{4} - 14930 x^{3} + 4339 x^{2} - 11125 x + 14771 \)

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:  \(146202125884644293212890625=5^{14}\cdot 11^{10}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.18$
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:  $5, 11, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11241934469315911407129587487335908399} a^{15} + \frac{3119508081305956248159302590992113104}{11241934469315911407129587487335908399} a^{14} + \frac{911715152164143773580617507120199966}{11241934469315911407129587487335908399} a^{13} + \frac{3121353774387317982721779306114365907}{11241934469315911407129587487335908399} a^{12} + \frac{2314984768268720927009891343576887154}{11241934469315911407129587487335908399} a^{11} + \frac{236715932465037558704743047692546797}{11241934469315911407129587487335908399} a^{10} - \frac{2793295619756716550471954708315786670}{11241934469315911407129587487335908399} a^{9} - \frac{4025139441388346192550504797593234666}{11241934469315911407129587487335908399} a^{8} - \frac{162225893114822217922044451866556223}{591680761542942705638399341438732021} a^{7} + \frac{86062998621425379524280752095668800}{591680761542942705638399341438732021} a^{6} - \frac{80470330553273405179426624052022337}{387652912735031427832054740942617531} a^{5} - \frac{2420085348964511748215422532438116310}{11241934469315911407129587487335908399} a^{4} - \frac{2363622480952777310667519924038756487}{11241934469315911407129587487335908399} a^{3} + \frac{5076323631871037659927948887809428027}{11241934469315911407129587487335908399} a^{2} - \frac{4942092920236489538599357407364372936}{11241934469315911407129587487335908399} a - \frac{2967065400904239506204684891994191106}{11241934469315911407129587487335908399}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 152559.675032 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1161:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1161
Character table for t16n1161 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.15125.1, 8.4.78009078125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
$11$11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
31Data not computed