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Properties

Label 16.0.89791815397...0000.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{8}\cdot 13^{8}$
Root discriminant $55.86$
Ramified primes $2, 3, 5, 13$
Class number $1536$ (GRH)
Class group $[2, 4, 4, 48]$ (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![129600, 0, -112320, 0, 121716, 0, 25788, 0, 3649, 0, -336, 0, 14, 0, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 + 14*x^12 - 336*x^10 + 3649*x^8 + 25788*x^6 + 121716*x^4 - 112320*x^2 + 129600)
 
gp: K = bnfinit(x^16 - 12*x^14 + 14*x^12 - 336*x^10 + 3649*x^8 + 25788*x^6 + 121716*x^4 - 112320*x^2 + 129600, 1)
 

Normalized defining polynomial

\( x^{16} - 12 x^{14} + 14 x^{12} - 336 x^{10} + 3649 x^{8} + 25788 x^{6} + 121716 x^{4} - 112320 x^{2} + 129600 \)

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:  \(8979181539709000089600000000=2^{32}\cdot 3^{8}\cdot 5^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.86$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1560=2^{3}\cdot 3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1560}(1,·)$, $\chi_{1560}(131,·)$, $\chi_{1560}(389,·)$, $\chi_{1560}(311,·)$, $\chi_{1560}(391,·)$, $\chi_{1560}(1481,·)$, $\chi_{1560}(779,·)$, $\chi_{1560}(1039,·)$, $\chi_{1560}(209,·)$, $\chi_{1560}(469,·)$, $\chi_{1560}(599,·)$, $\chi_{1560}(859,·)$, $\chi_{1560}(181,·)$, $\chi_{1560}(649,·)$, $\chi_{1560}(571,·)$, $\chi_{1560}(1301,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{6} + \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{3}{8} a^{5} - \frac{1}{2} a$, $\frac{1}{360} a^{10} + \frac{1}{24} a^{8} + \frac{23}{360} a^{6} + \frac{1}{24} a^{4} + \frac{83}{180} a^{2} + \frac{1}{3}$, $\frac{1}{2160} a^{11} - \frac{5}{144} a^{9} + \frac{203}{2160} a^{7} - \frac{29}{144} a^{5} + \frac{353}{1080} a^{3} + \frac{2}{9} a$, $\frac{1}{19440} a^{12} - \frac{1}{2160} a^{10} + \frac{653}{19440} a^{8} + \frac{1}{6480} a^{6} + \frac{289}{4860} a^{4} - \frac{757}{1620} a^{2} - \frac{1}{27}$, $\frac{1}{19440} a^{13} - \frac{11}{9720} a^{9} + \frac{61}{648} a^{7} - \frac{2759}{19440} a^{5} - \frac{91}{648} a^{3} + \frac{5}{27} a$, $\frac{1}{1549890508320} a^{14} - \frac{2367007}{258315084720} a^{12} - \frac{883511561}{774945254160} a^{10} - \frac{69062479}{1793854755} a^{8} + \frac{325035861781}{1549890508320} a^{6} + \frac{960359747}{3743696880} a^{4} + \frac{650107}{2310924} a^{2} + \frac{45057043}{358770951}$, $\frac{1}{46496715249600} a^{15} - \frac{135245137}{7749452541600} a^{13} - \frac{1601053463}{23248357624800} a^{11} - \frac{19706778821}{1291575423600} a^{9} - \frac{650342763671}{46496715249600} a^{7} + \frac{5909588647}{112310906400} a^{5} - \frac{4513670393}{9359242200} a^{3} - \frac{778787363}{10763128530} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{48}$, which has order $1536$ (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:  \( \frac{1657}{206569440} a^{14} - \frac{13}{141680} a^{12} + \frac{4183}{103284720} a^{10} - \frac{18313}{8607060} a^{8} + \frac{4993357}{206569440} a^{6} + \frac{355877}{1496880} a^{4} + \frac{25001}{24948} a^{2} - \frac{5567}{15939} \) (order $4$)
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:  \( 2239795.2502793795 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{390}) \), \(\Q(\sqrt{-390}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-65}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{39}) \), \(\Q(\sqrt{-39}) \), \(\Q(i, \sqrt{390})\), \(\Q(i, \sqrt{65})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{65})\), \(\Q(\sqrt{-6}, \sqrt{-65})\), \(\Q(\sqrt{-6}, \sqrt{65})\), \(\Q(\sqrt{6}, \sqrt{-65})\), \(\Q(i, \sqrt{26})\), \(\Q(i, \sqrt{15})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{39})\), \(\Q(\sqrt{15}, \sqrt{26})\), \(\Q(\sqrt{-15}, \sqrt{-26})\), \(\Q(\sqrt{10}, \sqrt{39})\), \(\Q(\sqrt{-10}, \sqrt{-39})\), \(\Q(\sqrt{-15}, \sqrt{26})\), \(\Q(\sqrt{15}, \sqrt{-26})\), \(\Q(\sqrt{10}, \sqrt{-39})\), \(\Q(\sqrt{-10}, \sqrt{39})\), \(\Q(\sqrt{10}, \sqrt{26})\), \(\Q(\sqrt{-10}, \sqrt{-26})\), \(\Q(\sqrt{15}, \sqrt{39})\), \(\Q(\sqrt{-15}, \sqrt{-39})\), \(\Q(\sqrt{-10}, \sqrt{26})\), \(\Q(\sqrt{10}, \sqrt{-26})\), \(\Q(\sqrt{15}, \sqrt{-39})\), \(\Q(\sqrt{-15}, \sqrt{39})\), \(\Q(\sqrt{6}, \sqrt{26})\), \(\Q(\sqrt{6}, \sqrt{-26})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(\sqrt{-6}, \sqrt{26})\), \(\Q(\sqrt{-6}, \sqrt{-26})\), \(\Q(\sqrt{-6}, \sqrt{-10})\), \(\Q(\sqrt{-6}, \sqrt{10})\), 8.0.94758543360000.170, 8.0.94758543360000.160, 8.0.94758543360000.137, 8.0.1169858560000.1, 8.0.370150560000.7, 8.0.151613669376.2, 8.0.3317760000.5, 8.8.94758543360000.5, 8.0.5922408960000.6, 8.0.94758543360000.22, 8.0.94758543360000.16, 8.0.5922408960000.3, 8.0.94758543360000.62, 8.0.94758543360000.123, 8.0.94758543360000.161

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
$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$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$