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Properties

Label 16.0.23205298903...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{10}\cdot 29^{8}\cdot 41^{6}$
Root discriminant $59.27$
Ramified primes $5, 29, 41$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T516)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![71723961, -44690913, 26851698, -17457960, 4464779, -2947174, 1081137, -362698, 157584, -14239, 23063, 414, 1545, 11, 61, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 61*x^14 + 11*x^13 + 1545*x^12 + 414*x^11 + 23063*x^10 - 14239*x^9 + 157584*x^8 - 362698*x^7 + 1081137*x^6 - 2947174*x^5 + 4464779*x^4 - 17457960*x^3 + 26851698*x^2 - 44690913*x + 71723961)
 
gp: K = bnfinit(x^16 - x^15 + 61*x^14 + 11*x^13 + 1545*x^12 + 414*x^11 + 23063*x^10 - 14239*x^9 + 157584*x^8 - 362698*x^7 + 1081137*x^6 - 2947174*x^5 + 4464779*x^4 - 17457960*x^3 + 26851698*x^2 - 44690913*x + 71723961, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 61 x^{14} + 11 x^{13} + 1545 x^{12} + 414 x^{11} + 23063 x^{10} - 14239 x^{9} + 157584 x^{8} - 362698 x^{7} + 1081137 x^{6} - 2947174 x^{5} + 4464779 x^{4} - 17457960 x^{3} + 26851698 x^{2} - 44690913 x + 71723961 \)

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:  \(23205298903819174488291015625=5^{10}\cdot 29^{8}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.27$
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:  $5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{8} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{3762} a^{14} + \frac{25}{342} a^{13} - \frac{65}{3762} a^{12} - \frac{104}{1881} a^{11} + \frac{27}{418} a^{10} - \frac{131}{1254} a^{9} + \frac{679}{1881} a^{8} - \frac{398}{1881} a^{7} + \frac{109}{418} a^{6} - \frac{245}{1881} a^{5} + \frac{593}{1254} a^{4} - \frac{338}{1881} a^{3} + \frac{676}{1881} a^{2} + \frac{97}{627} a - \frac{104}{209}$, $\frac{1}{516775596213080445095130233133842816308678115764146} a^{15} - \frac{33149586676760520661864563772532711149065166351}{516775596213080445095130233133842816308678115764146} a^{14} - \frac{2967194910030977147663106256858620897081807304279}{258387798106540222547565116566921408154339057882073} a^{13} + \frac{7587449135798882182932928153990782411536704378594}{258387798106540222547565116566921408154339057882073} a^{12} - \frac{882296305209996892013213353221565496866532128577}{172258532071026815031710077711280938769559371921382} a^{11} + \frac{230980583175414075905820345320360881329642938636}{28709755345171135838618346285213489794926561986897} a^{10} - \frac{3975322358437663453766109554521861672600135831769}{46979599655734585917739112103076619664425283251286} a^{9} - \frac{63030447471139550014496463573366634451898131285408}{258387798106540222547565116566921408154339057882073} a^{8} - \frac{151210713656689665864592698830107929932300718892}{28709755345171135838618346285213489794926561986897} a^{7} + \frac{86572456007013609867720993066690311062256818922203}{258387798106540222547565116566921408154339057882073} a^{6} - \frac{4624639994819688042290303762404216734208765501074}{9569918448390378612872782095071163264975520662299} a^{5} + \frac{6024820446334253150765931286960578411754392865387}{46979599655734585917739112103076619664425283251286} a^{4} - \frac{1844640864555010745786437933859638380942954748589}{46979599655734585917739112103076619664425283251286} a^{3} + \frac{81582189932398334464714072508238450284244229988497}{172258532071026815031710077711280938769559371921382} a^{2} - \frac{9953302220834418316466664742781762621497052288546}{28709755345171135838618346285213489794926561986897} a + \frac{3642882176865831975671342412944589503895848131}{10169945216142804051937069176483701663098321639}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 8981036.97091 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T516):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.1025.1, 4.0.29725.2, 4.4.725.1, 8.0.883575625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$