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Properties

Label 16.0.63456228123...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}$
Root discriminant $40.99$
Ramified primes $2, 3, 5, 7$
Class number $32$ (GRH)
Class group $[2, 2, 2, 4]$ (GRH)
Galois group $C_2^4$ (as 16T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14641, 0, -20570, 0, 23213, 0, -10410, 0, 3788, 0, 810, 0, 53, 0, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 10*x^14 + 53*x^12 + 810*x^10 + 3788*x^8 - 10410*x^6 + 23213*x^4 - 20570*x^2 + 14641)
 
gp: K = bnfinit(x^16 + 10*x^14 + 53*x^12 + 810*x^10 + 3788*x^8 - 10410*x^6 + 23213*x^4 - 20570*x^2 + 14641, 1)
 

Normalized defining polynomial

\( x^{16} + 10 x^{14} + 53 x^{12} + 810 x^{10} + 3788 x^{8} - 10410 x^{6} + 23213 x^{4} - 20570 x^{2} + 14641 \)

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:  \(63456228123711897600000000=2^{32}\cdot 3^{8}\cdot 5^{8}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.99$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(840=2^{3}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{840}(799,·)$, $\chi_{840}(1,·)$, $\chi_{840}(391,·)$, $\chi_{840}(209,·)$, $\chi_{840}(659,·)$, $\chi_{840}(281,·)$, $\chi_{840}(349,·)$, $\chi_{840}(671,·)$, $\chi_{840}(251,·)$, $\chi_{840}(421,·)$, $\chi_{840}(811,·)$, $\chi_{840}(239,·)$, $\chi_{840}(629,·)$, $\chi_{840}(379,·)$, $\chi_{840}(769,·)$, $\chi_{840}(701,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{52} a^{10} - \frac{1}{4} a^{9} - \frac{1}{26} a^{8} - \frac{1}{4} a^{7} - \frac{9}{52} a^{6} - \frac{1}{2} a^{5} + \frac{5}{52} a^{4} + \frac{1}{4} a^{3} + \frac{4}{13} a^{2} - \frac{1}{4} a - \frac{19}{52}$, $\frac{1}{52} a^{11} + \frac{11}{52} a^{9} - \frac{1}{4} a^{8} + \frac{1}{13} a^{7} - \frac{1}{4} a^{6} - \frac{21}{52} a^{5} - \frac{1}{2} a^{4} + \frac{3}{52} a^{3} + \frac{1}{4} a^{2} - \frac{3}{26} a - \frac{1}{4}$, $\frac{1}{1092} a^{12} - \frac{1}{156} a^{10} - \frac{1}{4} a^{9} - \frac{2}{13} a^{8} - \frac{1}{4} a^{7} + \frac{271}{1092} a^{6} - \frac{1}{2} a^{5} - \frac{19}{52} a^{4} + \frac{1}{4} a^{3} + \frac{31}{78} a^{2} - \frac{1}{4} a + \frac{79}{273}$, $\frac{1}{24024} a^{13} - \frac{1}{2184} a^{12} + \frac{7}{1716} a^{11} - \frac{1}{156} a^{10} - \frac{87}{572} a^{9} + \frac{5}{52} a^{8} + \frac{241}{6006} a^{7} + \frac{58}{273} a^{6} - \frac{101}{572} a^{5} + \frac{7}{52} a^{4} - \frac{5}{1716} a^{3} + \frac{23}{156} a^{2} + \frac{11971}{24024} a - \frac{463}{2184}$, $\frac{1}{213577372008} a^{14} - \frac{6862673}{71192457336} a^{12} - \frac{3903413}{15255526572} a^{10} - \frac{1}{4} a^{9} - \frac{12945767815}{106788686004} a^{8} - \frac{1}{4} a^{7} - \frac{11558396519}{53394343002} a^{6} - \frac{1}{2} a^{5} - \frac{7011968375}{15255526572} a^{4} + \frac{1}{4} a^{3} + \frac{5586226573}{23730819112} a^{2} - \frac{1}{4} a + \frac{168142567}{1765102248}$, $\frac{1}{4698702184176} a^{15} - \frac{1}{427154744016} a^{14} - \frac{6862673}{1566234061392} a^{13} + \frac{6862673}{142384914672} a^{12} + \frac{1318238093}{167810792292} a^{11} - \frac{144736049}{15255526572} a^{10} + \frac{3483260801}{2349351092088} a^{9} + \frac{17053024969}{213577372008} a^{8} - \frac{29277678769}{2349351092088} a^{7} + \frac{41599450231}{213577372008} a^{6} - \frac{16388726191}{41952698073} a^{5} + \frac{1386272705}{7627763286} a^{4} + \frac{83167750593}{522078020464} a^{3} + \frac{10842802043}{47461638224} a^{2} - \frac{8283981659}{38832249456} a - \frac{1288303609}{3530204496}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $32$ (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:  \( -\frac{1895125}{35596228668} a^{14} - \frac{1652254}{2966352389} a^{12} - \frac{3974120}{1271293881} a^{10} - \frac{807700529}{17798114334} a^{8} - \frac{2019642340}{8899057167} a^{6} + \frac{509928592}{1271293881} a^{4} - \frac{16233449475}{11865409556} a^{2} + \frac{89025941}{73545927} \) (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:  \( 968872.10852 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4$ (as 16T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_2^4$
Character table for $C_2^4$

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{210}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{2}, \sqrt{-35})\), \(\Q(\sqrt{-6}, \sqrt{-35})\), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\sqrt{-6}, \sqrt{-70})\), \(\Q(\sqrt{-3}, \sqrt{-70})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{105})\), \(\Q(\sqrt{15}, \sqrt{-21})\), \(\Q(\sqrt{30}, \sqrt{-35})\), \(\Q(\sqrt{-10}, \sqrt{14})\), \(\Q(\sqrt{-5}, \sqrt{7})\), \(\Q(\sqrt{15}, \sqrt{-42})\), \(\Q(\sqrt{-21}, \sqrt{30})\), \(\Q(\sqrt{7}, \sqrt{-10})\), \(\Q(\sqrt{-5}, \sqrt{14})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{2}, \sqrt{-21})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(\sqrt{-6}, \sqrt{-10})\), \(\Q(\sqrt{-6}, \sqrt{14})\), \(\Q(\sqrt{-6}, \sqrt{7})\), \(\Q(\sqrt{-5}, \sqrt{-6})\), \(\Q(\sqrt{14}, \sqrt{15})\), \(\Q(\sqrt{-10}, \sqrt{-21})\), \(\Q(\sqrt{-5}, \sqrt{-42})\), \(\Q(\sqrt{7}, \sqrt{30})\), \(\Q(\sqrt{7}, \sqrt{15})\), \(\Q(\sqrt{-5}, \sqrt{-21})\), \(\Q(\sqrt{-10}, \sqrt{-42})\), \(\Q(\sqrt{14}, \sqrt{30})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{7})\), \(\Q(\sqrt{-3}, \sqrt{14})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), 8.0.497871360000.14, 8.0.7965941760000.16, 8.0.98344960000.3, 8.0.7965941760000.51, 8.0.7965941760000.36, 8.0.31116960000.2, 8.0.497871360000.12, 8.0.7965941760000.20, 8.0.7965941760000.25, 8.0.7965941760000.61, 8.0.7965941760000.48, 8.0.3317760000.3, 8.0.12745506816.4, 8.8.7965941760000.4, 8.0.7965941760000.33

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 R R R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$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}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$