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Properties

Label 16.0.26968313608...625.18
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{10}$
Root discriminant $59.83$
Ramified primes $5, 101$
Class number $64$ (GRH)
Class group $[2, 4, 8]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![497311, 144471, -73468, -832761, 1402694, -1489799, 1006203, -440459, 147063, -39236, 6778, -1128, 450, -42, 16, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 16*x^14 - 42*x^13 + 450*x^12 - 1128*x^11 + 6778*x^10 - 39236*x^9 + 147063*x^8 - 440459*x^7 + 1006203*x^6 - 1489799*x^5 + 1402694*x^4 - 832761*x^3 - 73468*x^2 + 144471*x + 497311)
 
gp: K = bnfinit(x^16 - 3*x^15 + 16*x^14 - 42*x^13 + 450*x^12 - 1128*x^11 + 6778*x^10 - 39236*x^9 + 147063*x^8 - 440459*x^7 + 1006203*x^6 - 1489799*x^5 + 1402694*x^4 - 832761*x^3 - 73468*x^2 + 144471*x + 497311, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 16 x^{14} - 42 x^{13} + 450 x^{12} - 1128 x^{11} + 6778 x^{10} - 39236 x^{9} + 147063 x^{8} - 440459 x^{7} + 1006203 x^{6} - 1489799 x^{5} + 1402694 x^{4} - 832761 x^{3} - 73468 x^{2} + 144471 x + 497311 \)

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:  \(26968313608671985107666015625=5^{12}\cdot 101^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{275} a^{14} - \frac{2}{275} a^{13} - \frac{4}{55} a^{12} + \frac{6}{275} a^{11} - \frac{19}{275} a^{10} + \frac{79}{275} a^{9} + \frac{78}{275} a^{8} - \frac{4}{25} a^{7} - \frac{63}{275} a^{6} - \frac{126}{275} a^{5} + \frac{124}{275} a^{4} - \frac{116}{275} a^{3} - \frac{68}{275} a^{2} - \frac{7}{55} a + \frac{34}{275}$, $\frac{1}{175738894765556472299126989240129327844938891225} a^{15} + \frac{207554777311735160195063917254031964237171168}{175738894765556472299126989240129327844938891225} a^{14} + \frac{4436192809386338108647163765206819720343582}{194186624050338643424449711867546218613192145} a^{13} + \frac{7248233852591218020062630339294446494605282326}{175738894765556472299126989240129327844938891225} a^{12} - \frac{56708436131758204746514461052835401025173697739}{175738894765556472299126989240129327844938891225} a^{11} + \frac{84609931132566093754845801660104942634296287124}{175738894765556472299126989240129327844938891225} a^{10} - \frac{29626416090923395367108258708030220691865163172}{175738894765556472299126989240129327844938891225} a^{9} + \frac{3473444705257706691585282891360553254322769596}{15976263160505133845375180840011757076812626475} a^{8} - \frac{21392252036316948974730048371095117607784174063}{175738894765556472299126989240129327844938891225} a^{7} + \frac{14961024577755314187142228945799932575194234374}{175738894765556472299126989240129327844938891225} a^{6} + \frac{43877038910235231860050416695687888918403725949}{175738894765556472299126989240129327844938891225} a^{5} + \frac{79236918599809398965713677993816587662396462679}{175738894765556472299126989240129327844938891225} a^{4} - \frac{20643814311065892503695507286316891360451989188}{175738894765556472299126989240129327844938891225} a^{3} - \frac{2734933118304118201859984201559946165116134043}{35147778953111294459825397848025865568987778245} a^{2} - \frac{59624169201482380472630001238088007811308296836}{175738894765556472299126989240129327844938891225} a - \frac{569080982076312062634611707019313215743305926}{3195252632101026769075036168002351415362525295}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{8}$, which has order $64$ (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:  \( 981317.138997 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.1275125.2, 4.4.2525.1, 4.0.12625.1, 8.4.16098453125.1, 8.4.6568812813125.5, 8.0.1625943765625.4

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
101Data not computed