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Properties

Label 16.0.35074927889...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{8}\cdot 13^{8}$
Root discriminant $39.50$
Ramified primes $2, 3, 5, 13$
Class number $24$ (GRH)
Class group $[2, 12]$ (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![38416, 0, -8624, 0, 53372, 0, -48180, 0, 21761, 0, -5280, 0, 722, 0, -44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 44*x^14 + 722*x^12 - 5280*x^10 + 21761*x^8 - 48180*x^6 + 53372*x^4 - 8624*x^2 + 38416)
 
gp: K = bnfinit(x^16 - 44*x^14 + 722*x^12 - 5280*x^10 + 21761*x^8 - 48180*x^6 + 53372*x^4 - 8624*x^2 + 38416, 1)
 

Normalized defining polynomial

\( x^{16} - 44 x^{14} + 722 x^{12} - 5280 x^{10} + 21761 x^{8} - 48180 x^{6} + 53372 x^{4} - 8624 x^{2} + 38416 \)

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:  \(35074927889488281600000000=2^{24}\cdot 3^{8}\cdot 5^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.50$
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$$\chi_{1560}(1,·)$, $\chi_{1560}(1091,·)$, $\chi_{1560}(961,·)$, $\chi_{1560}(521,·)$, $\chi_{1560}(779,·)$, $\chi_{1560}(209,·)$, $\chi_{1560}(259,·)$, $\chi_{1560}(859,·)$, $\chi_{1560}(1249,·)$, $\chi_{1560}(1379,·)$, $\chi_{1560}(131,·)$, $\chi_{1560}(1169,·)$, $\chi_{1560}(649,·)$, $\chi_{1560}(1171,·)$, $\chi_{1560}(1481,·)$, $\chi_{1560}(571,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{14} a^{7} - \frac{1}{2} a^{3} - \frac{3}{7} a$, $\frac{1}{56} a^{8} - \frac{1}{4} a^{6} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{3}{28} a^{2} - \frac{1}{2}$, $\frac{1}{56} a^{9} - \frac{1}{28} a^{7} + \frac{1}{8} a^{5} + \frac{11}{28} a^{3} - \frac{1}{2} a^{2} + \frac{3}{14} a$, $\frac{1}{112} a^{10} - \frac{1}{112} a^{8} - \frac{1}{28} a^{7} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{27}{112} a^{4} - \frac{25}{56} a^{2} + \frac{3}{14} a - \frac{1}{4}$, $\frac{1}{112} a^{11} - \frac{1}{112} a^{9} + \frac{1}{112} a^{7} - \frac{1}{4} a^{6} - \frac{27}{112} a^{5} - \frac{1}{4} a^{4} + \frac{3}{56} a^{3} + \frac{9}{28} a$, $\frac{1}{11760} a^{12} - \frac{23}{5880} a^{10} - \frac{1}{980} a^{8} + \frac{125}{588} a^{6} - \frac{979}{3920} a^{4} - \frac{2603}{5880} a^{2} - \frac{1}{2} a + \frac{1}{60}$, $\frac{1}{164640} a^{13} + \frac{73}{20580} a^{11} + \frac{13}{1715} a^{9} + \frac{103}{16464} a^{7} - \frac{8119}{54880} a^{5} - \frac{1}{4} a^{4} - \frac{12263}{82320} a^{3} - \frac{1}{4} a^{2} - \frac{59}{840} a$, $\frac{1}{3079591200} a^{14} + \frac{1411}{34217680} a^{12} - \frac{46889}{96237225} a^{10} - \frac{6977491}{1539795600} a^{8} - \frac{121684217}{3079591200} a^{6} - \frac{1}{4} a^{5} + \frac{5403287}{192474450} a^{4} - \frac{1}{4} a^{3} - \frac{444091}{3666180} a^{2} - \frac{1}{2} a - \frac{332341}{1122300}$, $\frac{1}{6159182400} a^{15} + \frac{41}{17108840} a^{13} + \frac{7723141}{3079591200} a^{11} + \frac{9614107}{1539795600} a^{9} - \frac{44806667}{6159182400} a^{7} - \frac{1}{4} a^{6} + \frac{244445813}{1539795600} a^{5} - \frac{1}{4} a^{4} - \frac{328256}{6415815} a^{3} + \frac{6876473}{15712200} a - \frac{1}{2}$

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

Class group and class number

$C_{2}\times C_{12}$, which has order $24$ (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:  \( \frac{1}{30450} a^{14} - \frac{113}{85260} a^{12} + \frac{7853}{426300} a^{10} - \frac{9337}{106575} a^{8} + \frac{55687}{426300} a^{6} + \frac{124963}{426300} a^{4} - \frac{19513}{21315} a^{2} + \frac{964}{2175} \) (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:  \( 546536.633433 \) (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

\(\Q(\sqrt{390}) \), \(\Q(\sqrt{-195}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-130}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{78}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{-2}, \sqrt{-195})\), \(\Q(\sqrt{-3}, \sqrt{-130})\), \(\Q(\sqrt{6}, \sqrt{65})\), \(\Q(\sqrt{-3}, \sqrt{65})\), \(\Q(\sqrt{6}, \sqrt{-130})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{65})\), \(\Q(\sqrt{-10}, \sqrt{-39})\), \(\Q(\sqrt{5}, \sqrt{78})\), \(\Q(\sqrt{13}, \sqrt{30})\), \(\Q(\sqrt{-15}, \sqrt{-26})\), \(\Q(\sqrt{5}, \sqrt{-39})\), \(\Q(\sqrt{-10}, \sqrt{78})\), \(\Q(\sqrt{13}, \sqrt{-15})\), \(\Q(\sqrt{-26}, \sqrt{30})\), \(\Q(\sqrt{-2}, \sqrt{-39})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{13})\), \(\Q(\sqrt{-2}, \sqrt{-15})\), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-26})\), \(\Q(\sqrt{30}, \sqrt{-39})\), \(\Q(\sqrt{-10}, \sqrt{13})\), \(\Q(\sqrt{5}, \sqrt{-26})\), \(\Q(\sqrt{-15}, \sqrt{78})\), \(\Q(\sqrt{-15}, \sqrt{-39})\), \(\Q(\sqrt{-10}, \sqrt{-26})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{30}, \sqrt{65})\), \(\Q(\sqrt{6}, \sqrt{-26})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{13})\), 8.0.5922408960000.18, 8.0.5922408960000.8, 8.0.5922408960000.9, 8.0.5922408960000.13, 8.0.5922408960000.21, 8.0.5922408960000.6, 8.8.5922408960000.1, 8.0.1445900625.1, 8.0.5922408960000.1, 8.0.5922408960000.14, 8.0.5922408960000.7, 8.0.9475854336.1, 8.0.207360000.2, 8.0.5922408960000.12, 8.0.73116160000.1

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$2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[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}$
$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}$