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Properties

Label 16.0.65790008098...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 89^{4}$
Root discriminant $41.08$
Ramified primes $2, 5, 89$
Class number $104$ (GRH)
Class group $[2, 52]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T217)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![657121, 317500, 304498, 294708, 245011, 69508, 16098, -5232, -1015, -356, 912, -532, 213, -92, 22, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 22*x^14 - 92*x^13 + 213*x^12 - 532*x^11 + 912*x^10 - 356*x^9 - 1015*x^8 - 5232*x^7 + 16098*x^6 + 69508*x^5 + 245011*x^4 + 294708*x^3 + 304498*x^2 + 317500*x + 657121)
 
gp: K = bnfinit(x^16 - 4*x^15 + 22*x^14 - 92*x^13 + 213*x^12 - 532*x^11 + 912*x^10 - 356*x^9 - 1015*x^8 - 5232*x^7 + 16098*x^6 + 69508*x^5 + 245011*x^4 + 294708*x^3 + 304498*x^2 + 317500*x + 657121, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 22 x^{14} - 92 x^{13} + 213 x^{12} - 532 x^{11} + 912 x^{10} - 356 x^{9} - 1015 x^{8} - 5232 x^{7} + 16098 x^{6} + 69508 x^{5} + 245011 x^{4} + 294708 x^{3} + 304498 x^{2} + 317500 x + 657121 \)

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:  \(65790008098816000000000000=2^{32}\cdot 5^{12}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.08$
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:  $2, 5, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3}$, $\frac{1}{20120931} a^{14} + \frac{313453}{20120931} a^{13} + \frac{1062025}{20120931} a^{12} - \frac{1895048}{20120931} a^{11} + \frac{600976}{2235659} a^{10} + \frac{91634}{6706977} a^{9} + \frac{2945449}{6706977} a^{8} - \frac{4386227}{20120931} a^{7} + \frac{3234046}{20120931} a^{6} - \frac{1741198}{6706977} a^{5} - \frac{9962366}{20120931} a^{4} + \frac{1034906}{2235659} a^{3} + \frac{3122312}{20120931} a^{2} - \frac{7162787}{20120931} a - \frac{9539014}{20120931}$, $\frac{1}{33118790400791226399593777414338893177} a^{15} + \frac{554859134312769190014454629836}{33118790400791226399593777414338893177} a^{14} - \frac{2375757779083582711315320467394471127}{33118790400791226399593777414338893177} a^{13} + \frac{3305381599255175385846541626785678468}{33118790400791226399593777414338893177} a^{12} - \frac{3154507165577483653636416210767142911}{33118790400791226399593777414338893177} a^{11} + \frac{1759971962493800245863983884090219385}{11039596800263742133197925804779631059} a^{10} + \frac{20708057310450823020660065538633821}{11039596800263742133197925804779631059} a^{9} + \frac{2402451185608719402135319760617174993}{33118790400791226399593777414338893177} a^{8} - \frac{12526223689680255440811848291433424399}{33118790400791226399593777414338893177} a^{7} + \frac{7737316955922546950890927441394147089}{33118790400791226399593777414338893177} a^{6} + \frac{1679033459788097571688285449708141655}{33118790400791226399593777414338893177} a^{5} - \frac{1549627702036100217197553054057620369}{33118790400791226399593777414338893177} a^{4} + \frac{9697801198944207497239950883070386325}{33118790400791226399593777414338893177} a^{3} - \frac{5389439423537877330602125995519571976}{11039596800263742133197925804779631059} a^{2} + \frac{2894055424915752457781489666713277258}{11039596800263742133197925804779631059} a - \frac{15817380382550452455976766236067478523}{33118790400791226399593777414338893177}$

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

Class group and class number

$C_{2}\times C_{52}$, which has order $104$ (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:  \( 18891.2708489 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T217):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.4.91136000000.1, 8.4.227840000.1, 8.0.8111104000000.28

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
5Data not computed
$89$89.2.1.2$x^{2} + 267$$2$$1$$1$$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$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{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}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$