↑

Properties

Label 16.0.17958563260...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 5^{12}\cdot 7^{12}$
Root discriminant $32.80$
Ramified primes $3, 5, 7$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $D_8$ (as 16T7)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![126736, 783912, 1857092, 2191586, 1486133, 650220, 203726, 47763, 12238, 5376, 1189, 58, 78, 7, -2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 2*x^14 + 7*x^13 + 78*x^12 + 58*x^11 + 1189*x^10 + 5376*x^9 + 12238*x^8 + 47763*x^7 + 203726*x^6 + 650220*x^5 + 1486133*x^4 + 2191586*x^3 + 1857092*x^2 + 783912*x + 126736)
 
gp: K = bnfinit(x^16 - 2*x^15 - 2*x^14 + 7*x^13 + 78*x^12 + 58*x^11 + 1189*x^10 + 5376*x^9 + 12238*x^8 + 47763*x^7 + 203726*x^6 + 650220*x^5 + 1486133*x^4 + 2191586*x^3 + 1857092*x^2 + 783912*x + 126736, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 2 x^{14} + 7 x^{13} + 78 x^{12} + 58 x^{11} + 1189 x^{10} + 5376 x^{9} + 12238 x^{8} + 47763 x^{7} + 203726 x^{6} + 650220 x^{5} + 1486133 x^{4} + 2191586 x^{3} + 1857092 x^{2} + 783912 x + 126736 \)

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:  \(1795856326022129150390625=3^{12}\cdot 5^{12}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.80$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{3} a^{11} - \frac{1}{6} a^{10} + \frac{1}{6} a^{7} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{2} + \frac{1}{6} a$, $\frac{1}{12} a^{14} - \frac{1}{6} a^{12} - \frac{1}{12} a^{11} - \frac{1}{2} a^{9} + \frac{1}{12} a^{8} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} - \frac{5}{12} a^{5} - \frac{1}{3} a^{3} + \frac{1}{12} a^{2}$, $\frac{1}{3547512197182954396582867965714641496519096} a^{15} + \frac{839951128405993371147480626288664339071}{22452608842930091117613088390598996813412} a^{14} - \frac{47640722059489966134878877495291754892219}{591252032863825732763811327619106916086516} a^{13} - \frac{156632085890642710215686279418973220768091}{1182504065727651465527622655238213832173032} a^{12} + \frac{883560477366088511420353600315092859655441}{1773756098591477198291433982857320748259548} a^{11} + \frac{422123273543488632259076322432084118040897}{1773756098591477198291433982857320748259548} a^{10} + \frac{111165818267916071306834088023588503196149}{3547512197182954396582867965714641496519096} a^{9} + \frac{262778872816037560923191347249783748242099}{886878049295738599145716991428660374129774} a^{8} - \frac{64713293162889127765501549594951145128923}{591252032863825732763811327619106916086516} a^{7} - \frac{1049704783890366481809088408267589250615229}{3547512197182954396582867965714641496519096} a^{6} + \frac{26292362805020772454680211651989146771483}{591252032863825732763811327619106916086516} a^{5} - \frac{55279078922529075683606152698359999610917}{886878049295738599145716991428660374129774} a^{4} - \frac{63148254881874959881212316949062728879305}{1182504065727651465527622655238213832173032} a^{3} - \frac{87449107666020476829152826702030113577977}{1773756098591477198291433982857320748259548} a^{2} - \frac{299972907241698817690820672726092246048801}{886878049295738599145716991428660374129774} a + \frac{313585228953152052281505151037805899332}{1660820317033218350460144178705356505861}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, 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:  \( \frac{6100711299731842774139}{45181082526140886223445544} a^{15} - \frac{107562345021881698805}{285956218519879026730668} a^{14} + \frac{379516813607605402025}{7530180421023481037240924} a^{13} + \frac{36522649202226018599485}{45181082526140886223445544} a^{12} + \frac{226327852273036162947745}{22590541263070443111722772} a^{11} - \frac{330611994203806231245}{7530180421023481037240924} a^{10} + \frac{7340328745359291285567775}{45181082526140886223445544} a^{9} + \frac{3366624954666858097741090}{5647635315767610777930693} a^{8} + \frac{9164614295178724532149905}{7530180421023481037240924} a^{7} + \frac{251411701475677728162571985}{45181082526140886223445544} a^{6} + \frac{175349825550361851902795067}{7530180421023481037240924} a^{5} + \frac{132473313605464377369222870}{1882545105255870259310231} a^{4} + \frac{6719958096492159625825796455}{45181082526140886223445544} a^{3} + \frac{4255672070119127493986839415}{22590541263070443111722772} a^{2} + \frac{226391915985742289672281910}{1882545105255870259310231} a + \frac{1796882055842400276475006}{63456576581658548066637} \) (order $6$)
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:  \( 103783.231276 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times Q_8$ (as 16T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $D_8$
Character table for $D_8$

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{21})\), 8.0.121550625.1, 8.0.1340095640625.1, 8.8.1340095640625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$