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Properties

Label 16.0.17229451455...9761.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{8}\cdot 89^{12}$
Root discriminant $119.47$
Ramified primes $17, 89$
Class number $1177813$ (GRH)
Class group $[301, 3913]$ (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7929856, -3638272, 18312704, -10615248, 6836220, -4677356, 3929497, -1745730, 562581, -147216, 43602, -16220, 3150, -404, 53, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 53*x^14 - 404*x^13 + 3150*x^12 - 16220*x^11 + 43602*x^10 - 147216*x^9 + 562581*x^8 - 1745730*x^7 + 3929497*x^6 - 4677356*x^5 + 6836220*x^4 - 10615248*x^3 + 18312704*x^2 - 3638272*x + 7929856)
 
gp: K = bnfinit(x^16 - 2*x^15 + 53*x^14 - 404*x^13 + 3150*x^12 - 16220*x^11 + 43602*x^10 - 147216*x^9 + 562581*x^8 - 1745730*x^7 + 3929497*x^6 - 4677356*x^5 + 6836220*x^4 - 10615248*x^3 + 18312704*x^2 - 3638272*x + 7929856, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 53 x^{14} - 404 x^{13} + 3150 x^{12} - 16220 x^{11} + 43602 x^{10} - 147216 x^{9} + 562581 x^{8} - 1745730 x^{7} + 3929497 x^{6} - 4677356 x^{5} + 6836220 x^{4} - 10615248 x^{3} + 18312704 x^{2} - 3638272 x + 7929856 \)

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:  \(1722945145525970304337872614249761=17^{8}\cdot 89^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $119.47$
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:  $17, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{7} + \frac{3}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{128} a^{8} - \frac{1}{32} a^{7} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} + \frac{1}{128} a^{4} - \frac{1}{32} a^{3} - \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{128} a^{9} + \frac{1}{64} a^{7} + \frac{1}{128} a^{5} - \frac{1}{32} a^{3}$, $\frac{1}{256} a^{10} - \frac{1}{256} a^{9} + \frac{3}{128} a^{7} - \frac{3}{256} a^{6} + \frac{15}{256} a^{5} - \frac{3}{128} a^{4} + \frac{3}{64} a^{3} + \frac{1}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{1024} a^{11} - \frac{3}{1024} a^{9} - \frac{1}{256} a^{8} + \frac{7}{1024} a^{7} + \frac{3}{128} a^{6} - \frac{41}{1024} a^{5} + \frac{15}{256} a^{4} + \frac{41}{256} a^{3} - \frac{5}{64} a^{2} - \frac{1}{8} a$, $\frac{1}{40960} a^{12} + \frac{19}{40960} a^{11} - \frac{15}{8192} a^{10} + \frac{107}{40960} a^{9} + \frac{107}{40960} a^{8} - \frac{531}{40960} a^{7} + \frac{43}{8192} a^{6} + \frac{857}{40960} a^{5} - \frac{161}{5120} a^{4} + \frac{1711}{10240} a^{3} + \frac{13}{512} a^{2} - \frac{121}{320} a - \frac{2}{5}$, $\frac{1}{163840} a^{13} - \frac{9}{40960} a^{11} - \frac{17}{40960} a^{10} - \frac{283}{81920} a^{9} - \frac{1}{40960} a^{8} + \frac{259}{10240} a^{7} + \frac{1033}{40960} a^{6} + \frac{6989}{163840} a^{5} - \frac{1931}{40960} a^{4} - \frac{2729}{40960} a^{3} - \frac{283}{10240} a^{2} - \frac{509}{1280} a - \frac{7}{20}$, $\frac{1}{45875200} a^{14} - \frac{13}{5734400} a^{13} - \frac{37}{11468800} a^{12} + \frac{507}{11468800} a^{11} + \frac{37213}{22937600} a^{10} - \frac{3383}{1638400} a^{9} - \frac{803}{358400} a^{8} + \frac{210117}{11468800} a^{7} + \frac{1206301}{45875200} a^{6} + \frac{709839}{11468800} a^{5} - \frac{536881}{11468800} a^{4} - \frac{415357}{2867200} a^{3} - \frac{297}{8960} a^{2} + \frac{1007}{8960} a - \frac{131}{700}$, $\frac{1}{18621504140461727639876403200} a^{15} - \frac{104396809777711582881}{18621504140461727639876403200} a^{14} - \frac{621386483064807666007}{931075207023086381993820160} a^{13} - \frac{823180350349557512809}{83131714912775569820876800} a^{12} - \frac{643678926397435701514229}{1862150414046172763987640320} a^{11} - \frac{14931618962046156842261463}{9310752070230863819938201600} a^{10} + \frac{8920417665094609097874521}{4655376035115431909969100800} a^{9} - \frac{11984059468779165806550411}{4655376035115431909969100800} a^{8} - \frac{1112189250889130782103677}{532042975441763646853611520} a^{7} - \frac{266035376406990632274915321}{18621504140461727639876403200} a^{6} + \frac{379700717512601561248771}{26451000199519499488460800} a^{5} + \frac{264255070596631674716016789}{4655376035115431909969100800} a^{4} + \frac{218355234331492834410173789}{1163844008778857977492275200} a^{3} - \frac{730010909737985996183193}{7274025054867862359326720} a^{2} - \frac{7487652413972725402669599}{18185062637169655898316800} a - \frac{8190975807895912728523}{25831054882343261219200}$

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

Class group and class number

$C_{301}\times C_{3913}$, which has order $1177813$ (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:  \( 325756301.284 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{89}) \), \(\Q(\sqrt{1513}) \), \(\Q(\sqrt{17}, \sqrt{89})\), 4.0.11984473.1 x2, 4.0.203736041.1 x2, 4.0.25721.1 x2, 4.0.134657.1 x2, 4.4.704969.1, 4.4.203736041.1, 8.0.41508374402353681.3, 8.0.5240294710561.2, 8.8.41508374402353681.1, 8.0.143627593087729.1 x2, 8.0.41508374402353681.1 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.8.6.1$x^{8} - 4361 x^{4} + 10265616$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
89.8.6.1$x^{8} - 4361 x^{4} + 10265616$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$