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Properties

Label 16.0.26701300602...3125.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{9}$
Root discriminant $44.84$
Ramified primes $5, 101$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group 16T1281

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8249296, -10598192, 15383004, -12197136, 6626619, -2495735, 678517, -78822, -27087, 24548, -9063, 2110, -226, -51, 29, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 29*x^14 - 51*x^13 - 226*x^12 + 2110*x^11 - 9063*x^10 + 24548*x^9 - 27087*x^8 - 78822*x^7 + 678517*x^6 - 2495735*x^5 + 6626619*x^4 - 12197136*x^3 + 15383004*x^2 - 10598192*x + 8249296)
 
gp: K = bnfinit(x^16 - 7*x^15 + 29*x^14 - 51*x^13 - 226*x^12 + 2110*x^11 - 9063*x^10 + 24548*x^9 - 27087*x^8 - 78822*x^7 + 678517*x^6 - 2495735*x^5 + 6626619*x^4 - 12197136*x^3 + 15383004*x^2 - 10598192*x + 8249296, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 29 x^{14} - 51 x^{13} - 226 x^{12} + 2110 x^{11} - 9063 x^{10} + 24548 x^{9} - 27087 x^{8} - 78822 x^{7} + 678517 x^{6} - 2495735 x^{5} + 6626619 x^{4} - 12197136 x^{3} + 15383004 x^{2} - 10598192 x + 8249296 \)

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:  \(267013006026455298095703125=5^{12}\cdot 101^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.84$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{6} + \frac{1}{5} a^{4} + \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{5} + \frac{1}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{5} a^{10} - \frac{1}{5}$, $\frac{1}{25} a^{11} - \frac{2}{25} a^{10} - \frac{1}{25} a^{9} - \frac{1}{25} a^{8} + \frac{4}{25} a^{7} - \frac{1}{25} a^{6} + \frac{9}{25} a^{5} - \frac{1}{25} a^{4} - \frac{11}{25} a^{3} - \frac{1}{25} a^{2} - \frac{7}{25} a + \frac{1}{25}$, $\frac{1}{25} a^{12} + \frac{2}{25} a^{9} + \frac{2}{25} a^{8} + \frac{12}{25} a^{7} + \frac{7}{25} a^{6} - \frac{3}{25} a^{5} + \frac{12}{25} a^{4} + \frac{7}{25} a^{3} - \frac{9}{25} a^{2} - \frac{8}{25} a - \frac{3}{25}$, $\frac{1}{150} a^{13} - \frac{1}{150} a^{12} - \frac{1}{150} a^{11} - \frac{1}{150} a^{10} + \frac{1}{25} a^{9} - \frac{2}{75} a^{8} - \frac{29}{150} a^{7} + \frac{13}{75} a^{6} + \frac{61}{150} a^{5} + \frac{28}{75} a^{4} - \frac{1}{6} a^{3} - \frac{21}{50} a^{2} + \frac{67}{150} a - \frac{4}{75}$, $\frac{1}{27000} a^{14} - \frac{23}{9000} a^{13} + \frac{151}{27000} a^{12} - \frac{209}{27000} a^{11} + \frac{97}{3375} a^{10} - \frac{179}{13500} a^{9} + \frac{359}{9000} a^{8} + \frac{1177}{4500} a^{7} + \frac{2899}{9000} a^{6} + \frac{11}{2250} a^{5} - \frac{9359}{27000} a^{4} - \frac{3829}{27000} a^{3} - \frac{12539}{27000} a^{2} + \frac{403}{13500} a + \frac{589}{6750}$, $\frac{1}{1285961683278276995286209829809668038000} a^{15} - \frac{17835144035080091046263795143299989}{1285961683278276995286209829809668038000} a^{14} - \frac{2790979310908615680311352425748375569}{1285961683278276995286209829809668038000} a^{13} + \frac{6552762893371937745632052990065036591}{1285961683278276995286209829809668038000} a^{12} - \frac{444898512840344254347376612776244103}{26790868401630770735129371454368084125} a^{11} + \frac{23630307063649855625325818603582349161}{642980841639138497643104914904834019000} a^{10} + \frac{26008088284387472601945263832155826077}{1285961683278276995286209829809668038000} a^{9} + \frac{5312276310884240278847104100736243377}{214326947213046165881034971634944673000} a^{8} - \frac{30507810501223891714577167993024203167}{142884631475364110587356647756629782000} a^{7} - \frac{2543654218440258731614618037650180763}{35721157868841027646839161939157445500} a^{6} + \frac{370918571597338769544047112914787891241}{1285961683278276995286209829809668038000} a^{5} + \frac{4024134143929250154521856068027213209}{12989511952305828235214240705148162000} a^{4} + \frac{42227268241017800022005018388475883509}{142884631475364110587356647756629782000} a^{3} - \frac{73669256388031932164355869225189954699}{214326947213046165881034971634944673000} a^{2} + \frac{10868484373197656006974200646363167341}{35721157868841027646839161939157445500} a + \frac{368715787132134188673593071878879814}{1461320094634405676461602079329168225}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, 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:  \( -\frac{194961824280181887043049071}{29952756231297067414022076952686000} a^{15} + \frac{112023367034287647116221691}{665616805139934831422712821170800} a^{14} - \frac{900840679365688591843856681}{5990551246259413482804415390537200} a^{13} - \frac{16517746310809648250614392857}{5990551246259413482804415390537200} a^{12} + \frac{17245960982218291148575134641}{1497637811564853370701103847634300} a^{11} - \frac{756281081476524417670156170883}{14976378115648533707011038476343000} a^{10} + \frac{10215102216210804834771365573}{133123361027986966284542564234160} a^{9} + \frac{19451941576491216144356935771}{39937008308396089885362769270248} a^{8} - \frac{165776253788450792200435019105}{79874016616792179770725538540496} a^{7} + \frac{345108023308801862482034568881}{49921260385495112356703461587810} a^{6} - \frac{373989132037306871106879498775279}{29952756231297067414022076952686000} a^{5} - \frac{6796836874300191013008707344669}{5990551246259413482804415390537200} a^{4} + \frac{424563670661120496348515531231341}{5990551246259413482804415390537200} a^{3} - \frac{1205603692882525846264545403762919}{2995275623129706741402207695268600} a^{2} + \frac{1102999032864583255443149039048149}{1497637811564853370701103847634300} a + \frac{227675945795020712187783252836392}{624015754818688904458793269847625} \) (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:  \( 3925602.28351 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1281:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1281
Character table for t16n1281 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.159390625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$101$101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.2$x^{4} - 101 x^{2} + 30603$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
101.4.3.3$x^{4} + 202$$4$$1$$3$$C_4$$[\ ]_{4}$