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Properties

Label 16.0.13279393802...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{15}\cdot 61^{11}$
Root discriminant $76.33$
Ramified primes $5, 61$
Class number $32$ (GRH)
Class group $[2, 4, 4]$ (GRH)
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8746001, -56633126, 132820860, -94819515, 12850245, 6262112, 2136023, -2059220, -238150, 326245, -25202, -17923, 3025, 320, -90, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 90*x^14 + 320*x^13 + 3025*x^12 - 17923*x^11 - 25202*x^10 + 326245*x^9 - 238150*x^8 - 2059220*x^7 + 2136023*x^6 + 6262112*x^5 + 12850245*x^4 - 94819515*x^3 + 132820860*x^2 - 56633126*x + 8746001)
 
gp: K = bnfinit(x^16 - x^15 - 90*x^14 + 320*x^13 + 3025*x^12 - 17923*x^11 - 25202*x^10 + 326245*x^9 - 238150*x^8 - 2059220*x^7 + 2136023*x^6 + 6262112*x^5 + 12850245*x^4 - 94819515*x^3 + 132820860*x^2 - 56633126*x + 8746001, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 90 x^{14} + 320 x^{13} + 3025 x^{12} - 17923 x^{11} - 25202 x^{10} + 326245 x^{9} - 238150 x^{8} - 2059220 x^{7} + 2136023 x^{6} + 6262112 x^{5} + 12850245 x^{4} - 94819515 x^{3} + 132820860 x^{2} - 56633126 x + 8746001 \)

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:  \(1327939380231806599761962890625=5^{15}\cdot 61^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.33$
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:  $5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{18461256676124093471089794456067244888104813048669339790191} a^{15} - \frac{2579020867341562145532339959662657622425578989892874310306}{18461256676124093471089794456067244888104813048669339790191} a^{14} + \frac{136798555930648870320160688364325418600506111079533722764}{18461256676124093471089794456067244888104813048669339790191} a^{13} + \frac{4703947724626245428396437305914625952070238434977347232818}{18461256676124093471089794456067244888104813048669339790191} a^{12} - \frac{452310109256544114277549966878201575899951116740560266657}{1678296061465826679189981314187931353464073913515394526381} a^{11} - \frac{187728580113906840822939863981596147446549452481406875025}{18461256676124093471089794456067244888104813048669339790191} a^{10} + \frac{6758701942528080546553194723896439704953705656782076405363}{18461256676124093471089794456067244888104813048669339790191} a^{9} + \frac{1975382224110629668951454931314369182143354498391787076644}{18461256676124093471089794456067244888104813048669339790191} a^{8} - \frac{342464243213583314658999751881656156755038251888765810976}{1678296061465826679189981314187931353464073913515394526381} a^{7} - \frac{2454795161994536894233457993108436536363738194912510047864}{18461256676124093471089794456067244888104813048669339790191} a^{6} + \frac{5212632845845788824372459493415768798534563562171789250203}{18461256676124093471089794456067244888104813048669339790191} a^{5} - \frac{2413303762201937388395086884655954345962394542894689691733}{18461256676124093471089794456067244888104813048669339790191} a^{4} - \frac{3531804849466903451466174692006506657833804574818159649902}{18461256676124093471089794456067244888104813048669339790191} a^{3} - \frac{3253169902941419687898894718662677651573419305139509883897}{18461256676124093471089794456067244888104813048669339790191} a^{2} + \frac{146969605305185042107083263194776014686822337659802757044}{18461256676124093471089794456067244888104813048669339790191} a - \frac{191582623761627260465791696514158003896257715332362259568}{1678296061465826679189981314187931353464073913515394526381}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}$, which has order $32$ (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:  \( -\frac{6832038750229535728889107726398075870782381087}{66755342327903690354002677466605598562669500557471641} a^{15} - \frac{1970003167397350047578009020110735875042123034}{66755342327903690354002677466605598562669500557471641} a^{14} + \frac{575915173837395021036646121964310954977411825117}{66755342327903690354002677466605598562669500557471641} a^{13} - \frac{1569775698239469354784494326156656942626504098029}{66755342327903690354002677466605598562669500557471641} a^{12} - \frac{19844380513317041225019892945914057843352388181754}{66755342327903690354002677466605598562669500557471641} a^{11} + \frac{98345197337398225572280835891038305891379905476237}{66755342327903690354002677466605598562669500557471641} a^{10} + \frac{185803212801451699164612700004633677239561727223466}{66755342327903690354002677466605598562669500557471641} a^{9} - \frac{1847621191896956762142201229317894939940724929634889}{66755342327903690354002677466605598562669500557471641} a^{8} + \frac{1166178714443679852880618990480716634216676386339419}{66755342327903690354002677466605598562669500557471641} a^{7} + \frac{11895979393927663515889357929026824448535858868633613}{66755342327903690354002677466605598562669500557471641} a^{6} - \frac{13240772424709259924424184502852654839106325830643071}{66755342327903690354002677466605598562669500557471641} a^{5} - \frac{38140269603217091425642147088808868339535853192005093}{66755342327903690354002677466605598562669500557471641} a^{4} - \frac{69656114019024335749835230581905810717717194616211769}{66755342327903690354002677466605598562669500557471641} a^{3} + \frac{584387613085658044569604470176558660090567541380255634}{66755342327903690354002677466605598562669500557471641} a^{2} - \frac{748029743989332265220028706038701655284377782275278620}{66755342327903690354002677466605598562669500557471641} a + \frac{161748363996543825967211991620788476927793687572004214}{66755342327903690354002677466605598562669500557471641} \) (order $10$)
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:  \( 57781591.4741 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.17732890625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
5Data not computed
61Data not computed