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Properties

Label 16.8.73904015491...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 5^{10}\cdot 29^{6}\cdot 89^{6}$
Root discriminant $73.59$
Ramified primes $2, 5, 29, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T790

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![368009, 5204053, -14277659, 7809328, -264332, 88205, 245065, -303172, 64261, -12791, -9449, 178, -614, 42, -2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 2*x^14 + 42*x^13 - 614*x^12 + 178*x^11 - 9449*x^10 - 12791*x^9 + 64261*x^8 - 303172*x^7 + 245065*x^6 + 88205*x^5 - 264332*x^4 + 7809328*x^3 - 14277659*x^2 + 5204053*x + 368009)
 
gp: K = bnfinit(x^16 - 2*x^15 - 2*x^14 + 42*x^13 - 614*x^12 + 178*x^11 - 9449*x^10 - 12791*x^9 + 64261*x^8 - 303172*x^7 + 245065*x^6 + 88205*x^5 - 264332*x^4 + 7809328*x^3 - 14277659*x^2 + 5204053*x + 368009, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 2 x^{14} + 42 x^{13} - 614 x^{12} + 178 x^{11} - 9449 x^{10} - 12791 x^{9} + 64261 x^{8} - 303172 x^{7} + 245065 x^{6} + 88205 x^{5} - 264332 x^{4} + 7809328 x^{3} - 14277659 x^{2} + 5204053 x + 368009 \)

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:  \(739040154910723253702500000000=2^{8}\cdot 5^{10}\cdot 29^{6}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.59$
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:  $2, 5, 29, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{22} a^{13} - \frac{5}{22} a^{12} + \frac{5}{22} a^{11} - \frac{9}{22} a^{10} + \frac{3}{11} a^{9} + \frac{3}{11} a^{8} + \frac{1}{11} a^{7} + \frac{9}{22} a^{6} + \frac{3}{11} a^{5} + \frac{5}{11} a^{4} - \frac{7}{22} a^{3} + \frac{9}{22} a^{2} + \frac{2}{11} a + \frac{7}{22}$, $\frac{1}{22} a^{14} + \frac{1}{11} a^{12} - \frac{3}{11} a^{11} + \frac{5}{22} a^{10} - \frac{4}{11} a^{9} + \frac{5}{11} a^{8} - \frac{3}{22} a^{7} + \frac{7}{22} a^{6} - \frac{2}{11} a^{5} - \frac{1}{22} a^{4} - \frac{2}{11} a^{3} + \frac{5}{22} a^{2} + \frac{5}{22} a - \frac{9}{22}$, $\frac{1}{540575862728118285202476366082146445882087597717903726} a^{15} + \frac{214228814603772834508100241649707074093403442306454}{270287931364059142601238183041073222941043798858951863} a^{14} + \frac{1424246010258537242876268020936880640764152298006604}{270287931364059142601238183041073222941043798858951863} a^{13} - \frac{15238204909132319024947883435957391870114838388376147}{270287931364059142601238183041073222941043798858951863} a^{12} - \frac{18782034294417969818176360597547211973530678655191665}{49143260248010753200225124189286040534735236156173066} a^{11} - \frac{115848176951117943075109820683728364444281983160705678}{270287931364059142601238183041073222941043798858951863} a^{10} + \frac{103965153130764811558123989903815514790484363362181472}{270287931364059142601238183041073222941043798858951863} a^{9} + \frac{61793193092907103534764445029422940962332392497598429}{540575862728118285202476366082146445882087597717903726} a^{8} - \frac{155824951456471825290591805567752409228227169272507061}{540575862728118285202476366082146445882087597717903726} a^{7} + \frac{76435981188305660654985315429392837856493983234992395}{270287931364059142601238183041073222941043798858951863} a^{6} - \frac{125585231411689464320989305508390879784786820679619061}{540575862728118285202476366082146445882087597717903726} a^{5} + \frac{37286419984985093371226219829325759731474419945723550}{270287931364059142601238183041073222941043798858951863} a^{4} + \frac{190388336179264488758299444772985037073318463719260669}{540575862728118285202476366082146445882087597717903726} a^{3} - \frac{161699225730439010109787035173041804863497303384453971}{540575862728118285202476366082146445882087597717903726} a^{2} - \frac{236199470708154491094575137693177526472400083289963925}{540575862728118285202476366082146445882087597717903726} a + \frac{35278681775694458822685991536575541453310380016741968}{270287931364059142601238183041073222941043798858951863}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 909645027.352 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T790:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n790 are not computed
Character table for t16n790 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.64525.2, 4.4.2225.1, 4.4.725.1, 8.8.4163475625.2

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$89$89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.8.6.2$x^{8} + 979 x^{4} + 285156$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$