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Properties

Label 16.0.81925410828...5625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 41^{10}$
Root discriminant $41.65$
Ramified primes $5, 41$
Class number $8$ (GRH)
Class group $[2, 4]$ (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![22705, -1225, 37865, 3160, 39510, -9715, 21925, -7725, 5916, -1787, 1033, 6, -35, 6, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 + 6*x^13 - 35*x^12 + 6*x^11 + 1033*x^10 - 1787*x^9 + 5916*x^8 - 7725*x^7 + 21925*x^6 - 9715*x^5 + 39510*x^4 + 3160*x^3 + 37865*x^2 - 1225*x + 22705)
 
gp: K = bnfinit(x^16 - 2*x^15 + 3*x^14 + 6*x^13 - 35*x^12 + 6*x^11 + 1033*x^10 - 1787*x^9 + 5916*x^8 - 7725*x^7 + 21925*x^6 - 9715*x^5 + 39510*x^4 + 3160*x^3 + 37865*x^2 - 1225*x + 22705, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 3 x^{14} + 6 x^{13} - 35 x^{12} + 6 x^{11} + 1033 x^{10} - 1787 x^{9} + 5916 x^{8} - 7725 x^{7} + 21925 x^{6} - 9715 x^{5} + 39510 x^{4} + 3160 x^{3} + 37865 x^{2} - 1225 x + 22705 \)

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:  \(81925410828566900634765625=5^{14}\cdot 41^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.65$
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:  $5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{99} a^{14} - \frac{7}{99} a^{13} - \frac{2}{99} a^{12} - \frac{1}{99} a^{11} - \frac{49}{99} a^{10} - \frac{2}{33} a^{9} - \frac{46}{99} a^{8} - \frac{10}{33} a^{7} - \frac{14}{99} a^{6} - \frac{20}{99} a^{5} + \frac{13}{99} a^{4} + \frac{29}{99} a^{3} + \frac{37}{99} a^{2} + \frac{1}{3} a - \frac{14}{99}$, $\frac{1}{318928846761648264070468462402254303} a^{15} + \frac{114990070589797910056843892133185}{106309615587216088023489487467418101} a^{14} - \frac{14609064118758910136677553059129588}{106309615587216088023489487467418101} a^{13} + \frac{5363936840311382294792648144355608}{35436538529072029341163162489139367} a^{12} - \frac{52264475911719856223659811201640443}{318928846761648264070468462402254303} a^{11} - \frac{57881454402427971320740153982460331}{318928846761648264070468462402254303} a^{10} + \frac{70570312749897339630534155556802445}{318928846761648264070468462402254303} a^{9} + \frac{84000087687414106479453245206880087}{318928846761648264070468462402254303} a^{8} + \frac{46457188592570962844897369604059965}{318928846761648264070468462402254303} a^{7} + \frac{2710321204355310995181237044293178}{318928846761648264070468462402254303} a^{6} - \frac{114674170315572685414701795849475927}{318928846761648264070468462402254303} a^{5} + \frac{446674026244585036907418521817302}{3221503502642911758287560226285397} a^{4} - \frac{12476959559131042701824966828033720}{35436538529072029341163162489139367} a^{3} - \frac{147532293289824227949760397735605631}{318928846761648264070468462402254303} a^{2} - \frac{47114699216853312256424374182613211}{318928846761648264070468462402254303} a + \frac{63457909789919821237444633328201}{1334430321178444619541709047708177}$

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

Class group and class number

$C_{2}\times C_{4}$, 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:  \( -1 \) (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:  \( 367138.88247 \) (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

\(\Q(\sqrt{5}) \), 4.0.210125.1, 4.4.5125.1, 4.0.1025.1, 8.4.9051265703125.2, 8.4.5384453125.1, 8.0.44152515625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$41$41.8.6.2$x^{8} + 943 x^{4} + 242064$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$