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Properties

Label 16.0.23879602510...056.28
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 13^{12}\cdot 17^{12}$
Root discriminant $385.59$
Ramified primes $2, 13, 17$
Class number $498401280$ (GRH)
Class group $[2, 2, 2, 4, 4, 8, 312, 1560]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9770775678976, 0, 4885387839488, 0, 811007540096, 0, 55955375424, 0, 1869438116, 0, 31648968, 0, 259454, 0, 884, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 884*x^14 + 259454*x^12 + 31648968*x^10 + 1869438116*x^8 + 55955375424*x^6 + 811007540096*x^4 + 4885387839488*x^2 + 9770775678976)
 
gp: K = bnfinit(x^16 + 884*x^14 + 259454*x^12 + 31648968*x^10 + 1869438116*x^8 + 55955375424*x^6 + 811007540096*x^4 + 4885387839488*x^2 + 9770775678976, 1)
 

Normalized defining polynomial

\( x^{16} + 884 x^{14} + 259454 x^{12} + 31648968 x^{10} + 1869438116 x^{8} + 55955375424 x^{6} + 811007540096 x^{4} + 4885387839488 x^{2} + 9770775678976 \)

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:  \(238796025103060449638782189092971852333056=2^{44}\cdot 13^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $385.59$
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:  $2, 13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3536=2^{4}\cdot 13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{3536}(1,·)$, $\chi_{3536}(3141,·)$, $\chi_{3536}(1665,·)$, $\chi_{3536}(2313,·)$, $\chi_{3536}(2189,·)$, $\chi_{3536}(21,·)$, $\chi_{3536}(1373,·)$, $\chi_{3536}(837,·)$, $\chi_{3536}(545,·)$, $\chi_{3536}(421,·)$, $\chi_{3536}(2209,·)$, $\chi_{3536}(2605,·)$, $\chi_{3536}(1769,·)$, $\chi_{3536}(441,·)$, $\chi_{3536}(1789,·)$, $\chi_{3536}(3433,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{442} a^{4}$, $\frac{1}{442} a^{5}$, $\frac{1}{1326} a^{6} - \frac{1}{1326} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{1326} a^{7} - \frac{1}{1326} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{2930460} a^{8} - \frac{1}{6630} a^{6} - \frac{1}{3315} a^{4} - \frac{4}{15} a^{2} + \frac{2}{5}$, $\frac{1}{5860920} a^{9} + \frac{1}{3315} a^{7} - \frac{7}{13260} a^{5} - \frac{7}{15} a^{3} + \frac{1}{30} a$, $\frac{1}{11721840} a^{10} - \frac{3}{8840} a^{6} - \frac{1}{2210} a^{4} - \frac{7}{60} a^{2} + \frac{1}{5}$, $\frac{1}{23443680} a^{11} + \frac{11}{53040} a^{7} - \frac{2}{3315} a^{5} - \frac{47}{120} a^{3} + \frac{13}{30} a$, $\frac{1}{2176042377600} a^{12} - \frac{61}{2461586400} a^{10} - \frac{167}{1641057600} a^{8} - \frac{953}{5569200} a^{6} + \frac{2027}{3712800} a^{4} + \frac{247}{6300} a^{2} - \frac{731}{1575}$, $\frac{1}{4352084755200} a^{13} - \frac{61}{4923172800} a^{11} - \frac{167}{3282115200} a^{9} - \frac{953}{11138400} a^{7} + \frac{2027}{7425600} a^{5} - \frac{6053}{12600} a^{3} + \frac{422}{1575} a$, $\frac{1}{443912645030400} a^{14} + \frac{2687}{77255942400} a^{10} + \frac{137}{1054965600} a^{8} - \frac{482903}{2272233600} a^{6} + \frac{1243}{6683040} a^{4} - \frac{211}{5950} a^{2} + \frac{232}{4725}$, $\frac{1}{887825290060800} a^{15} + \frac{2687}{154511884800} a^{11} + \frac{137}{2109931200} a^{9} - \frac{482903}{4544467200} a^{7} - \frac{13877}{13366080} a^{5} + \frac{5739}{11900} a^{3} - \frac{4493}{9450} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{8}\times C_{312}\times C_{1560}$, which has order $498401280$ (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:  \( 897048.5749561078 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{34}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{442}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{221}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{2}, \sqrt{17})\), \(\Q(\sqrt{13}, \sqrt{34})\), \(\Q(\sqrt{26}, \sqrt{34})\), \(\Q(\sqrt{2}, \sqrt{221})\), \(\Q(\sqrt{2}, \sqrt{13})\), \(\Q(\sqrt{17}, \sqrt{26})\), \(\Q(\sqrt{13}, \sqrt{17})\), 4.0.22105827328.8, 4.0.22105827328.4, 4.0.22105827328.2, 4.0.22105827328.7, 8.8.9770775678976.8, 8.0.488667601855351619584.4, 8.0.488667601855351619584.2, 8.0.488667601855351619584.3, 8.0.488667601855351619584.5, 8.0.488667601855351619584.1, 8.0.488667601855351619584.6

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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.22.5$x^{8} + 10 x^{4} + 16 x + 36$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.8.22.5$x^{8} + 10 x^{4} + 16 x + 36$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$13$13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$17$17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$