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Properties

Label 16.0.63456228123...0000.2
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 $64$ (GRH)
Class group $[2, 2, 2, 8]$ (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![62001, 0, -48336, 0, 29938, 0, -7284, 0, 2163, 0, -68, 0, 102, 0, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 12*x^14 + 102*x^12 - 68*x^10 + 2163*x^8 - 7284*x^6 + 29938*x^4 - 48336*x^2 + 62001)
 
gp: K = bnfinit(x^16 + 12*x^14 + 102*x^12 - 68*x^10 + 2163*x^8 - 7284*x^6 + 29938*x^4 - 48336*x^2 + 62001, 1)
 

Normalized defining polynomial

\( x^{16} + 12 x^{14} + 102 x^{12} - 68 x^{10} + 2163 x^{8} - 7284 x^{6} + 29938 x^{4} - 48336 x^{2} + 62001 \)

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$$\chi_{840}(1,·)$, $\chi_{840}(391,·)$, $\chi_{840}(139,·)$, $\chi_{840}(589,·)$, $\chi_{840}(211,·)$, $\chi_{840}(799,·)$, $\chi_{840}(839,·)$, $\chi_{840}(41,·)$, $\chi_{840}(71,·)$, $\chi_{840}(449,·)$, $\chi_{840}(659,·)$, $\chi_{840}(629,·)$, $\chi_{840}(251,·)$, $\chi_{840}(769,·)$, $\chi_{840}(701,·)$, $\chi_{840}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{24} a^{8} - \frac{1}{12} a^{6} - \frac{1}{2} a^{5} + \frac{1}{24} a^{4} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{72} a^{9} + \frac{1}{12} a^{7} - \frac{1}{6} a^{6} - \frac{5}{24} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{1}{6} a^{2} - \frac{5}{24} a$, $\frac{1}{936} a^{10} - \frac{1}{156} a^{8} - \frac{1}{6} a^{7} + \frac{17}{104} a^{6} + \frac{1}{3} a^{5} - \frac{67}{234} a^{4} - \frac{1}{6} a^{3} - \frac{101}{312} a^{2} + \frac{5}{26}$, $\frac{1}{936} a^{11} - \frac{1}{156} a^{9} + \frac{17}{104} a^{7} - \frac{67}{234} a^{5} - \frac{101}{312} a^{3} + \frac{5}{26} a - \frac{1}{2}$, $\frac{1}{8424} a^{12} + \frac{1}{4212} a^{10} + \frac{35}{2808} a^{8} + \frac{83}{2106} a^{6} - \frac{1199}{8424} a^{4} + \frac{151}{702} a^{2} - \frac{1}{2} a - \frac{19}{117}$, $\frac{1}{8424} a^{13} + \frac{1}{4212} a^{11} - \frac{1}{702} a^{9} - \frac{185}{4212} a^{7} - \frac{1}{6} a^{6} + \frac{139}{2106} a^{5} + \frac{1}{3} a^{4} - \frac{161}{702} a^{3} + \frac{1}{3} a^{2} + \frac{43}{936} a$, $\frac{1}{109840536} a^{14} + \frac{173}{109840536} a^{12} + \frac{18011}{36613512} a^{10} + \frac{561793}{54920268} a^{8} - \frac{8583239}{109840536} a^{6} - \frac{1}{2} a^{5} + \frac{3773213}{18306756} a^{4} - \frac{1}{2} a^{3} + \frac{2098139}{6102252} a^{2} - \frac{233105}{1356056}$, $\frac{1}{9116764488} a^{15} + \frac{71801}{4558382244} a^{13} - \frac{1648495}{4558382244} a^{11} + \frac{56317673}{9116764488} a^{9} - \frac{312457711}{2279191122} a^{7} - \frac{1}{6} a^{6} + \frac{2015559155}{9116764488} a^{5} - \frac{1}{6} a^{4} + \frac{491054939}{3038921496} a^{3} - \frac{1}{6} a^{2} + \frac{19286015}{1012973832} a - \frac{1}{2}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{8}$, which has order $64$ (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:  \( 829665.224905 \) (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

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{70}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{210}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-2}, \sqrt{-35})\), \(\Q(\sqrt{-6}, \sqrt{-35})\), \(\Q(\sqrt{3}, \sqrt{-35})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-105})\), \(\Q(\sqrt{-6}, \sqrt{70})\), \(\Q(\sqrt{3}, \sqrt{70})\), \(\Q(\sqrt{-5}, \sqrt{7})\), \(\Q(\sqrt{10}, \sqrt{-14})\), \(\Q(\sqrt{30}, \sqrt{-35})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{-2}, \sqrt{7})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{21})\), \(\Q(\sqrt{-2}, \sqrt{-15})\), \(\Q(\sqrt{7}, \sqrt{10})\), \(\Q(\sqrt{-5}, \sqrt{-14})\), \(\Q(\sqrt{-15}, \sqrt{-42})\), \(\Q(\sqrt{21}, \sqrt{30})\), \(\Q(\sqrt{-6}, \sqrt{7})\), \(\Q(\sqrt{-5}, \sqrt{-6})\), \(\Q(\sqrt{-6}, \sqrt{-14})\), \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{7}, \sqrt{30})\), \(\Q(\sqrt{-5}, \sqrt{-42})\), \(\Q(\sqrt{-14}, \sqrt{-15})\), \(\Q(\sqrt{10}, \sqrt{21})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{-14})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\sqrt{7}, \sqrt{-15})\), \(\Q(\sqrt{-5}, \sqrt{21})\), \(\Q(\sqrt{-14}, \sqrt{30})\), \(\Q(\sqrt{10}, \sqrt{-42})\), 8.0.7965941760000.35, 8.0.98344960000.4, 8.0.497871360000.8, 8.0.7965941760000.36, 8.0.497871360000.5, 8.0.31116960000.5, 8.0.7965941760000.7, 8.0.12745506816.3, 8.0.3317760000.1, 8.0.7965941760000.10, 8.0.7965941760000.12, 8.0.7965941760000.29, 8.0.7965941760000.5, 8.8.7965941760000.1, 8.0.7965941760000.42

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$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{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$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$