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Properties

Label 16.8.35120559063...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{28}\cdot 5^{14}\cdot 11^{8}$
Root discriminant $45.61$
Ramified primes $2, 5, 11$
Class number $2$ (GRH)
Class group $[2]$ (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![-12475, -18900, 63600, -1150, -60590, 79670, -76440, 43100, -14349, 5826, -3128, 888, -60, -18, 12, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 12*x^14 - 18*x^13 - 60*x^12 + 888*x^11 - 3128*x^10 + 5826*x^9 - 14349*x^8 + 43100*x^7 - 76440*x^6 + 79670*x^5 - 60590*x^4 - 1150*x^3 + 63600*x^2 - 18900*x - 12475)
 
gp: K = bnfinit(x^16 - 6*x^15 + 12*x^14 - 18*x^13 - 60*x^12 + 888*x^11 - 3128*x^10 + 5826*x^9 - 14349*x^8 + 43100*x^7 - 76440*x^6 + 79670*x^5 - 60590*x^4 - 1150*x^3 + 63600*x^2 - 18900*x - 12475, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 12 x^{14} - 18 x^{13} - 60 x^{12} + 888 x^{11} - 3128 x^{10} + 5826 x^{9} - 14349 x^{8} + 43100 x^{7} - 76440 x^{6} + 79670 x^{5} - 60590 x^{4} - 1150 x^{3} + 63600 x^{2} - 18900 x - 12475 \)

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:  \(351205590630400000000000000=2^{28}\cdot 5^{14}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.61$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{165} a^{14} + \frac{14}{165} a^{13} + \frac{1}{33} a^{12} + \frac{8}{55} a^{11} + \frac{41}{165} a^{10} + \frac{34}{165} a^{9} - \frac{58}{165} a^{8} - \frac{2}{33} a^{7} - \frac{16}{55} a^{6} - \frac{2}{165} a^{5} - \frac{37}{165} a^{4} + \frac{8}{33} a^{3} + \frac{13}{33} a^{2} + \frac{1}{3} a + \frac{2}{33}$, $\frac{1}{192742670742768802769405174402685} a^{15} + \frac{138061056185358916430560077868}{192742670742768802769405174402685} a^{14} + \frac{177823990084104088226861121329}{12849511382851253517960344960179} a^{13} - \frac{4337269568530683787716366685013}{192742670742768802769405174402685} a^{12} - \frac{69510465769655540300965698880987}{192742670742768802769405174402685} a^{11} - \frac{2580172222674298689516197899151}{38548534148553760553881034880537} a^{10} + \frac{85630039692319244525594306098507}{192742670742768802769405174402685} a^{9} + \frac{3370590518631770576845214909717}{12849511382851253517960344960179} a^{8} - \frac{52943488185810454924354686834464}{192742670742768802769405174402685} a^{7} + \frac{51484242228990186166215710601844}{192742670742768802769405174402685} a^{6} + \frac{18440813320961510105839934004391}{192742670742768802769405174402685} a^{5} + \frac{6196732332699789823825078512634}{17522060976615345706309561309335} a^{4} - \frac{16821304819240639033608125433298}{38548534148553760553881034880537} a^{3} + \frac{12439784224565674640503444148938}{38548534148553760553881034880537} a^{2} + \frac{3532155199018704260496951685258}{12849511382851253517960344960179} a - \frac{3574271821213708833553637045174}{38548534148553760553881034880537}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 16823087.2023 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 4.4.22000.1, 4.4.4400.1, 8.4.4685120000000.13, 8.4.4685120000000.7, 8.8.7744000000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$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}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.8.6.2$x^{8} - 781 x^{4} + 290521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$