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Properties

Label 16.8.39728624488...6209.2
Degree $16$
Signature $[8, 4]$
Discriminant $71^{10}\cdot 73^{14}$
Root discriminant $612.98$
Ramified primes $71, 73$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-15665152, -65689600, 53723776, 55014144, 208451456, -258626096, -65989784, 85441192, 3448164, 2484917, -861906, -6448, -2951, -487, 124, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 124*x^14 - 487*x^13 - 2951*x^12 - 6448*x^11 - 861906*x^10 + 2484917*x^9 + 3448164*x^8 + 85441192*x^7 - 65989784*x^6 - 258626096*x^5 + 208451456*x^4 + 55014144*x^3 + 53723776*x^2 - 65689600*x - 15665152)
 
gp: K = bnfinit(x^16 - 6*x^15 + 124*x^14 - 487*x^13 - 2951*x^12 - 6448*x^11 - 861906*x^10 + 2484917*x^9 + 3448164*x^8 + 85441192*x^7 - 65989784*x^6 - 258626096*x^5 + 208451456*x^4 + 55014144*x^3 + 53723776*x^2 - 65689600*x - 15665152, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 124 x^{14} - 487 x^{13} - 2951 x^{12} - 6448 x^{11} - 861906 x^{10} + 2484917 x^{9} + 3448164 x^{8} + 85441192 x^{7} - 65989784 x^{6} - 258626096 x^{5} + 208451456 x^{4} + 55014144 x^{3} + 53723776 x^{2} - 65689600 x - 15665152 \)

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:  \(397286244885795491769071181517306466749106209=71^{10}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $612.98$
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:  $71, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} + \frac{1}{16} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{9} - \frac{1}{32} a^{8} - \frac{1}{8} a^{7} - \frac{3}{32} a^{6} + \frac{1}{32} a^{5} - \frac{5}{32} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{1}{64} a^{9} - \frac{1}{16} a^{8} - \frac{3}{64} a^{7} + \frac{1}{64} a^{6} - \frac{5}{64} a^{5} - \frac{3}{16} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2}$, $\frac{1}{384} a^{12} - \frac{1}{192} a^{11} - \frac{1}{192} a^{10} - \frac{3}{128} a^{9} + \frac{1}{128} a^{8} + \frac{5}{96} a^{7} + \frac{5}{48} a^{6} - \frac{41}{384} a^{5} - \frac{17}{192} a^{4} + \frac{7}{24} a^{3} - \frac{7}{16} a^{2} + \frac{1}{24} a - \frac{1}{6}$, $\frac{1}{1536} a^{13} - \frac{7}{1536} a^{10} + \frac{13}{512} a^{9} - \frac{5}{768} a^{8} + \frac{19}{768} a^{7} - \frac{21}{512} a^{6} + \frac{29}{768} a^{5} + \frac{17}{96} a^{4} + \frac{25}{192} a^{3} + \frac{19}{96} a^{2} - \frac{5}{24} a - \frac{1}{3}$, $\frac{1}{6144} a^{14} - \frac{1}{3072} a^{13} - \frac{31}{6144} a^{11} - \frac{67}{6144} a^{10} + \frac{5}{384} a^{9} - \frac{43}{3072} a^{8} + \frac{125}{6144} a^{7} + \frac{37}{384} a^{6} - \frac{85}{512} a^{5} - \frac{157}{768} a^{4} + \frac{21}{64} a^{3} - \frac{83}{192} a^{2} + \frac{11}{24} a + \frac{5}{12}$, $\frac{1}{15439811372179060414739829884573290221382987657124511744} a^{15} - \frac{149754771483715464576905696581899689019632302521449}{3859952843044765103684957471143322555345746914281127936} a^{14} + \frac{100868623497388294597093271376036255157899110522959}{1286650947681588367894985823714440851781915638093709312} a^{13} + \frac{6253449415358229247322023423503835659837042836254123}{5146603790726353471579943294857763407127662552374837248} a^{12} - \frac{22545265878667982282029822047169330330117295838651319}{5146603790726353471579943294857763407127662552374837248} a^{11} + \frac{8910680666318443841621119957301457842410975385434481}{2573301895363176735789971647428881703563831276187418624} a^{10} + \frac{146095276516263490996046316009620413552400607205643357}{7719905686089530207369914942286645110691493828562255872} a^{9} - \frac{689257696175281721245667319396426183312895837660410391}{15439811372179060414739829884573290221382987657124511744} a^{8} + \frac{307517425759326096929315779434919258318162946974413081}{2573301895363176735789971647428881703563831276187418624} a^{7} - \frac{304627956946835813957569012178705792575375967605719035}{3859952843044765103684957471143322555345746914281127936} a^{6} - \frac{181368662625340115294261979680089556125554008942878927}{964988210761191275921239367785830638836436728570281984} a^{5} + \frac{5417910483763639233568725810391630292367372734318983}{23536297823443689656615594336239771678937481184641024} a^{4} + \frac{78762354003638992899252512418010717210590701452901561}{160831368460198545986873227964305106472739454761713664} a^{3} + \frac{17609094257609455434251532474583131472889481463159559}{241247052690297818980309841946457659709109182142570496} a^{2} + \frac{748007356661603429246890010090343256743464374591351}{30155881586287227372538730243307207463638647767821312} a - \frac{6271198886874517887945587346831490358611957473855055}{15077940793143613686269365121653603731819323883910656}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 3999364699030000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.8.280732967047165372057.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
71Data not computed
$73$73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$