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Properties

Label 16.0.44016602739...0625.5
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 101^{12}$
Root discriminant $71.24$
Ramified primes $5, 101$
Class number $320$ (GRH)
Class group $[4, 80]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5041, -25915, 59273, -87122, 102866, -103912, 86087, -60065, 37640, -20367, 8916, -3169, 867, -166, 30, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 30*x^14 - 166*x^13 + 867*x^12 - 3169*x^11 + 8916*x^10 - 20367*x^9 + 37640*x^8 - 60065*x^7 + 86087*x^6 - 103912*x^5 + 102866*x^4 - 87122*x^3 + 59273*x^2 - 25915*x + 5041)
 
gp: K = bnfinit(x^16 - 4*x^15 + 30*x^14 - 166*x^13 + 867*x^12 - 3169*x^11 + 8916*x^10 - 20367*x^9 + 37640*x^8 - 60065*x^7 + 86087*x^6 - 103912*x^5 + 102866*x^4 - 87122*x^3 + 59273*x^2 - 25915*x + 5041, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 30 x^{14} - 166 x^{13} + 867 x^{12} - 3169 x^{11} + 8916 x^{10} - 20367 x^{9} + 37640 x^{8} - 60065 x^{7} + 86087 x^{6} - 103912 x^{5} + 102866 x^{4} - 87122 x^{3} + 59273 x^{2} - 25915 x + 5041 \)

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:  \(440166027395300672133281640625=5^{8}\cdot 101^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.24$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2627} a^{14} + \frac{38}{2627} a^{13} - \frac{930}{2627} a^{12} + \frac{250}{2627} a^{11} + \frac{504}{2627} a^{10} - \frac{1029}{2627} a^{9} - \frac{1145}{2627} a^{8} + \frac{342}{2627} a^{7} - \frac{394}{2627} a^{6} + \frac{209}{2627} a^{5} + \frac{1216}{2627} a^{4} - \frac{1223}{2627} a^{3} + \frac{1232}{2627} a^{2} + \frac{34}{71} a - \frac{1}{37}$, $\frac{1}{13291504979890997951631923} a^{15} + \frac{2103867860167029126390}{13291504979890997951631923} a^{14} - \frac{6150373148161781796647579}{13291504979890997951631923} a^{13} + \frac{76636424064654394060626}{187204295491422506361013} a^{12} + \frac{3964017256928216885299245}{13291504979890997951631923} a^{11} - \frac{4190275746915413967165988}{13291504979890997951631923} a^{10} - \frac{1233397003282342650004228}{13291504979890997951631923} a^{9} + \frac{6551895843595722921917829}{13291504979890997951631923} a^{8} - \frac{5900458981356033345229729}{13291504979890997951631923} a^{7} - \frac{830363605130079755747001}{13291504979890997951631923} a^{6} + \frac{3716841531172191154139560}{13291504979890997951631923} a^{5} + \frac{6470278265978191100094996}{13291504979890997951631923} a^{4} + \frac{2533510018617409155563591}{13291504979890997951631923} a^{3} + \frac{4758023629962212889906909}{13291504979890997951631923} a^{2} + \frac{2593524330044696124785420}{13291504979890997951631923} a - \frac{19767422795531386748540}{187204295491422506361013}$

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

Class group and class number

$C_{4}\times C_{80}$, which has order $320$ (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:  \( 822826.064416 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{101}) \), 4.0.25757525.2, 4.4.51005.1, 4.0.5151505.1, 8.4.6568812813125.4, 8.4.262752512525.1, 8.0.663450094125625.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
5.4.2.1$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}$
$101$101.8.6.1$x^{8} - 707 x^{4} + 826281$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
101.8.6.1$x^{8} - 707 x^{4} + 826281$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$