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Properties

Label 16.8.70440191930...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{38}\cdot 5^{8}\cdot 7^{8}\cdot 241^{8}$
Root discriminant $476.42$
Ramified primes $2, 5, 7, 241$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T1228

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3636911, -22885728, 32333530, 212805460, 248959625, 101652956, 863210, -11926252, -5361528, -1591384, -229166, -10564, -3887, -564, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 564*x^13 - 3887*x^12 - 10564*x^11 - 229166*x^10 - 1591384*x^9 - 5361528*x^8 - 11926252*x^7 + 863210*x^6 + 101652956*x^5 + 248959625*x^4 + 212805460*x^3 + 32333530*x^2 - 22885728*x - 3636911)
 
gp: K = bnfinit(x^16 - 4*x^15 + 10*x^14 - 564*x^13 - 3887*x^12 - 10564*x^11 - 229166*x^10 - 1591384*x^9 - 5361528*x^8 - 11926252*x^7 + 863210*x^6 + 101652956*x^5 + 248959625*x^4 + 212805460*x^3 + 32333530*x^2 - 22885728*x - 3636911, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 10 x^{14} - 564 x^{13} - 3887 x^{12} - 10564 x^{11} - 229166 x^{10} - 1591384 x^{9} - 5361528 x^{8} - 11926252 x^{7} + 863210 x^{6} + 101652956 x^{5} + 248959625 x^{4} + 212805460 x^{3} + 32333530 x^{2} - 22885728 x - 3636911 \)

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:  \(7044019193056619128660199158998630400000000=2^{38}\cdot 5^{8}\cdot 7^{8}\cdot 241^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $476.42$
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:  $2, 5, 7, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{8}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{12} - \frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{4} a^{9} - \frac{1}{6} a^{7} - \frac{5}{12} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{12} a^{2} - \frac{1}{8} a + \frac{5}{24}$, $\frac{1}{72} a^{14} + \frac{1}{72} a^{13} + \frac{1}{36} a^{12} + \frac{1}{36} a^{10} - \frac{1}{12} a^{9} - \frac{5}{36} a^{8} + \frac{1}{12} a^{7} - \frac{7}{36} a^{6} - \frac{7}{36} a^{5} - \frac{1}{6} a^{4} - \frac{4}{9} a^{3} - \frac{17}{72} a^{2} + \frac{35}{72} a - \frac{7}{36}$, $\frac{1}{73876037076385028701578285602999557386477366730567929744} a^{15} - \frac{2759216831923145004840208072162563538387771473939449}{2736149521347593655614010577888872495795458027058071472} a^{14} + \frac{93851097304608610128356035603097187098601952533408955}{73876037076385028701578285602999557386477366730567929744} a^{13} + \frac{3596283064349823408070730385421846605736917613542612031}{73876037076385028701578285602999557386477366730567929744} a^{12} + \frac{3608133656016708944605938800782715800389432193101575125}{36938018538192514350789142801499778693238683365283964872} a^{11} + \frac{31643735374283233403203629737405802147416444187429739}{900927281419329618311930312231701919347284960128877192} a^{10} + \frac{1392501814104030533081925115316280619218958698281672831}{18469009269096257175394571400749889346619341682641982436} a^{9} - \frac{1779905386739429381619930086931887667522650244926847319}{9234504634548128587697285700374944673309670841320991218} a^{8} + \frac{98861452398549115898680302634496319677306274146100281}{450463640709664809155965156115850959673642480064438596} a^{7} - \frac{499326807846501082349392910067256360489821922495880131}{2052112141010695241710507933416654371846593520293553604} a^{6} - \frac{13617668304381444320059089120622886848584552972609669485}{36938018538192514350789142801499778693238683365283964872} a^{5} + \frac{3670275381709504277285643459437642344509733718825367101}{36938018538192514350789142801499778693238683365283964872} a^{4} + \frac{212138978360033536635731618047537320577334919142670017}{600618187612886412207953541487801279564856640085918128} a^{3} - \frac{23735834583168355315186806874565204120402274242307900749}{73876037076385028701578285602999557386477366730567929744} a^{2} + \frac{33056371946574530800910687369253938056469678755291622513}{73876037076385028701578285602999557386477366730567929744} a + \frac{24465378912894060879746326097973737703437537152657566389}{73876037076385028701578285602999557386477366730567929744}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 794266084703000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1228:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1228 are not computed
Character table for t16n1228 is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{16870}) \), \(\Q(\sqrt{1687}) \), 4.4.385600.2, 4.4.75577600.2, \(\Q(\sqrt{10}, \sqrt{1687})\), 8.8.331757139925442560000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.22.97$x^{8} + 4 x^{7} + 6 x^{4} + 4 x^{2} + 10$$8$$1$$22$$D_4\times C_2$$[2, 3, 7/2]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$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}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7Data not computed
241Data not computed