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Properties

Label 16.0.26347107536...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{4}\cdot 5^{12}\cdot 59^{4}$
Root discriminant $68.99$
Ramified primes $2, 3, 5, 59$
Class number $96$ (GRH)
Class group $[2, 48]$ (GRH)
Galois group $C_2^4.C_2^4$ (as 16T459)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2722361, 1211320, 1811136, 934120, 887830, 255520, 137584, 1280, 15534, 1040, 3464, 40, 230, -40, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 16*x^14 - 40*x^13 + 230*x^12 + 40*x^11 + 3464*x^10 + 1040*x^9 + 15534*x^8 + 1280*x^7 + 137584*x^6 + 255520*x^5 + 887830*x^4 + 934120*x^3 + 1811136*x^2 + 1211320*x + 2722361)
 
gp: K = bnfinit(x^16 + 16*x^14 - 40*x^13 + 230*x^12 + 40*x^11 + 3464*x^10 + 1040*x^9 + 15534*x^8 + 1280*x^7 + 137584*x^6 + 255520*x^5 + 887830*x^4 + 934120*x^3 + 1811136*x^2 + 1211320*x + 2722361, 1)
 

Normalized defining polynomial

\( x^{16} + 16 x^{14} - 40 x^{13} + 230 x^{12} + 40 x^{11} + 3464 x^{10} + 1040 x^{9} + 15534 x^{8} + 1280 x^{7} + 137584 x^{6} + 255520 x^{5} + 887830 x^{4} + 934120 x^{3} + 1811136 x^{2} + 1211320 x + 2722361 \)

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:  \(263471075369680896000000000000=2^{40}\cdot 3^{4}\cdot 5^{12}\cdot 59^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.99$
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:  $2, 3, 5, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{42} a^{14} - \frac{1}{42} a^{13} + \frac{3}{14} a^{12} - \frac{1}{7} a^{11} - \frac{2}{21} a^{10} + \frac{4}{21} a^{9} + \frac{1}{21} a^{8} - \frac{1}{7} a^{7} + \frac{19}{42} a^{6} + \frac{1}{6} a^{5} + \frac{1}{42} a^{4} + \frac{2}{7} a^{3} - \frac{1}{3} a - \frac{8}{21}$, $\frac{1}{5321930552805908619957038444048437331886} a^{15} + \frac{32042859954893423913357765662979696593}{5321930552805908619957038444048437331886} a^{14} + \frac{35498490732685233185204651608468941371}{5321930552805908619957038444048437331886} a^{13} + \frac{30867970089238108005148371675514263803}{136459757764254067178385601129447111074} a^{12} - \frac{153231773077669766755842697111691075725}{760275793257986945708148349149776761698} a^{11} - \frac{646995471102925529806263096172835750849}{2660965276402954309978519222024218665943} a^{10} + \frac{122722477281197852298760210257733978619}{591325616978434291106337604894270814654} a^{9} + \frac{18898035044690197145668184424662771927}{5321930552805908619957038444048437331886} a^{8} + \frac{1097823352192545624894936568684169173411}{5321930552805908619957038444048437331886} a^{7} + \frac{43728226091977928807160249383186641249}{171675179122771245805065756259627010706} a^{6} + \frac{285299948169165827370198120545295705319}{591325616978434291106337604894270814654} a^{5} - \frac{2576416547029683117960015281132225225567}{5321930552805908619957038444048437331886} a^{4} + \frac{43224956514467273606606403572358283393}{136459757764254067178385601129447111074} a^{3} - \frac{37846891382922946105835979691047431443}{380137896628993472854074174574888380849} a^{2} - \frac{1932315215086764424867895393005681910683}{5321930552805908619957038444048437331886} a + \frac{1279269431134741330639170504930272951807}{5321930552805908619957038444048437331886}$

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

Class group and class number

$C_{2}\times C_{48}$, which has order $96$ (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:  \( 2295366.07419 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^4$ (as 16T459):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^4.C_2^4$
Character table for $C_2^4.C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.0.472000.1, 4.0.118000.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.3564544000000.19

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

Sibling fields

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$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$$[\ ]_{4}^{2}$
$59$59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$