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Properties

Label 16.8.32634053000...0625.4
Degree $16$
Signature $[8, 4]$
Discriminant $5^{10}\cdot 109^{14}$
Root discriminant $165.81$
Ramified primes $5, 109$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times OD_{16}).C_2$ (as 16T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![187726144, -1620521312, -1155728032, 12770440, 129529688, 23045434, 1825173, -45344, -123966, -42946, -10549, -2398, -982, 0, 13, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 13*x^14 - 982*x^12 - 2398*x^11 - 10549*x^10 - 42946*x^9 - 123966*x^8 - 45344*x^7 + 1825173*x^6 + 23045434*x^5 + 129529688*x^4 + 12770440*x^3 - 1155728032*x^2 - 1620521312*x + 187726144)
 
gp: K = bnfinit(x^16 + 13*x^14 - 982*x^12 - 2398*x^11 - 10549*x^10 - 42946*x^9 - 123966*x^8 - 45344*x^7 + 1825173*x^6 + 23045434*x^5 + 129529688*x^4 + 12770440*x^3 - 1155728032*x^2 - 1620521312*x + 187726144, 1)
 

Normalized defining polynomial

\( x^{16} + 13 x^{14} - 982 x^{12} - 2398 x^{11} - 10549 x^{10} - 42946 x^{9} - 123966 x^{8} - 45344 x^{7} + 1825173 x^{6} + 23045434 x^{5} + 129529688 x^{4} + 12770440 x^{3} - 1155728032 x^{2} - 1620521312 x + 187726144 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(326340530003540716649666265244140625=5^{10}\cdot 109^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $165.81$
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:  $5, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{12} a^{11} + \frac{1}{12} a^{10} - \frac{1}{12} a^{7} - \frac{1}{4} a^{6} - \frac{1}{12} a^{5} + \frac{1}{6} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{24} a^{12} - \frac{1}{24} a^{10} + \frac{1}{12} a^{8} - \frac{1}{12} a^{7} + \frac{5}{24} a^{6} + \frac{1}{4} a^{5} + \frac{1}{6} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} - \frac{5}{12} a + \frac{1}{6}$, $\frac{1}{48} a^{13} + \frac{1}{48} a^{11} - \frac{1}{12} a^{10} + \frac{1}{24} a^{9} - \frac{1}{24} a^{8} + \frac{1}{16} a^{7} + \frac{1}{8} a^{6} - \frac{5}{24} a^{5} - \frac{1}{6} a^{4} - \frac{5}{16} a^{3} - \frac{11}{24} a^{2} + \frac{1}{12} a + \frac{1}{3}$, $\frac{1}{4032} a^{14} + \frac{5}{2016} a^{13} + \frac{61}{4032} a^{12} + \frac{5}{224} a^{11} - \frac{115}{2016} a^{10} - \frac{17}{224} a^{9} + \frac{367}{4032} a^{8} - \frac{5}{112} a^{7} + \frac{151}{2016} a^{6} + \frac{25}{336} a^{5} + \frac{85}{4032} a^{4} + \frac{479}{1008} a^{3} + \frac{25}{336} a^{2} + \frac{11}{36} a + \frac{73}{252}$, $\frac{1}{55572354224174372206674096760129202968405370366625906048} a^{15} - \frac{1209137338386292257738246290431993822732872337707751}{13893088556043593051668524190032300742101342591656476512} a^{14} + \frac{35515477941458048622985130734670917535496500760228311}{7938907746310624600953442394304171852629338623803700864} a^{13} - \frac{33528503651795652667518246216116458090891046390675423}{1984726936577656150238360598576042963157334655950925216} a^{12} + \frac{955871530575190173955845515175556732320074072038156919}{27786177112087186103337048380064601484202685183312953024} a^{11} + \frac{2982100202570931486166112588187196133418158801943535433}{27786177112087186103337048380064601484202685183312953024} a^{10} - \frac{1587906041465571206778225210950251313595883964871450437}{55572354224174372206674096760129202968405370366625906048} a^{9} + \frac{1289240228969184483299678505196221073060389705499387409}{27786177112087186103337048380064601484202685183312953024} a^{8} - \frac{365286441687117550598623945639993152655696597156186331}{3969453873155312300476721197152085926314669311901850432} a^{7} + \frac{438634601162371418914245380019411996665555638122976075}{6946544278021796525834262095016150371050671295828238256} a^{6} - \frac{9739111484170077950354125292483985952714027327938997331}{55572354224174372206674096760129202968405370366625906048} a^{5} + \frac{1228963855808795999353472786674605879476665993077189479}{3087353012454131789259672042229400164911409464812550336} a^{4} + \frac{205415021468797149734609012986934482079369973439518671}{1984726936577656150238360598576042963157334655950925216} a^{3} + \frac{179804321381443783036698073042478183389169239737412219}{6946544278021796525834262095016150371050671295828238256} a^{2} + \frac{61657907534634070213699909862069188954807323954432653}{385919126556766473657459005278675020613926183101568792} a - \frac{3960145312123662192354158478131787963388314542455343}{13256763889354573522584469646977386204295174228679844}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 1597586691690 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).C_2$ (as 16T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times OD_{16}).C_2$
Character table for $(C_2\times OD_{16}).C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{545}) \), \(\Q(\sqrt{109}) \), \(\Q(\sqrt{5}) \), 4.4.32375725.1 x2, 4.4.6475145.1 x2, \(\Q(\sqrt{5}, \sqrt{109})\), 8.8.1048187569275625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$109$109.8.7.2$x^{8} - 3924$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
109.8.7.2$x^{8} - 3924$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$