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Properties

Label 16.0.10504372386...000.29
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 13^{12}$
Root discriminant $317.20$
Ramified primes $2, 3, 5, 13$
Class number $243507200$ (GRH)
Class group $[2, 2, 2, 4, 4, 40, 47560]$ (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![11156640625, 0, 17850625000, 0, 7634575000, 0, 1340170000, 0, 108831775, 0, 4123600, 0, 72280, 0, 520, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 520*x^14 + 72280*x^12 + 4123600*x^10 + 108831775*x^8 + 1340170000*x^6 + 7634575000*x^4 + 17850625000*x^2 + 11156640625)
 
gp: K = bnfinit(x^16 + 520*x^14 + 72280*x^12 + 4123600*x^10 + 108831775*x^8 + 1340170000*x^6 + 7634575000*x^4 + 17850625000*x^2 + 11156640625, 1)
 

Normalized defining polynomial

\( x^{16} + 520 x^{14} + 72280 x^{12} + 4123600 x^{10} + 108831775 x^{8} + 1340170000 x^{6} + 7634575000 x^{4} + 17850625000 x^{2} + 11156640625 \)

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:  \(10504372386022553663859326976000000000000=2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $317.20$
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:  \(3120=2^{4}\cdot 3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{3120}(1,·)$, $\chi_{3120}(961,·)$, $\chi_{3120}(649,·)$, $\chi_{3120}(1867,·)$, $\chi_{3120}(1997,·)$, $\chi_{3120}(2159,·)$, $\chi_{3120}(1123,·)$, $\chi_{3120}(1253,·)$, $\chi_{3120}(2471,·)$, $\chi_{3120}(3119,·)$, $\chi_{3120}(2803,·)$, $\chi_{3120}(2933,·)$, $\chi_{3120}(311,·)$, $\chi_{3120}(2809,·)$, $\chi_{3120}(187,·)$, $\chi_{3120}(317,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{195} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{195} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{585} a^{6} + \frac{4}{9}$, $\frac{1}{585} a^{7} + \frac{4}{9} a$, $\frac{1}{38025} a^{8} + \frac{2}{9} a^{2}$, $\frac{1}{38025} a^{9} + \frac{2}{9} a^{3}$, $\frac{1}{570375} a^{10} - \frac{1}{114075} a^{8} + \frac{2}{8775} a^{6} - \frac{1}{1755} a^{4} + \frac{44}{135} a^{2} - \frac{2}{27}$, $\frac{1}{2851875} a^{11} - \frac{1}{570375} a^{9} - \frac{1}{3375} a^{7} + \frac{8}{8775} a^{5} + \frac{134}{675} a^{3} - \frac{10}{27} a$, $\frac{1}{6117271875} a^{12} + \frac{16}{31370625} a^{10} + \frac{4}{31370625} a^{8} - \frac{269}{1447875} a^{6} + \frac{1153}{482625} a^{4} - \frac{569}{1485} a^{2} + \frac{335}{891}$, $\frac{1}{6117271875} a^{13} + \frac{1}{6274125} a^{11} + \frac{59}{31370625} a^{9} + \frac{32}{289575} a^{7} + \frac{713}{482625} a^{5} + \frac{3106}{7425} a^{3} - \frac{226}{891} a$, $\frac{1}{15874320515625} a^{14} + \frac{2}{634972820625} a^{12} + \frac{2878}{27135590625} a^{10} + \frac{85442}{9768812625} a^{8} - \frac{98797}{289018125} a^{6} - \frac{62134}{83494125} a^{4} - \frac{136894}{2312145} a^{2} - \frac{210244}{462429}$, $\frac{1}{79371602578125} a^{15} + \frac{529}{15874320515625} a^{13} + \frac{1286}{10436765625} a^{11} + \frac{433438}{244220315625} a^{9} - \frac{14827666}{18786178125} a^{7} - \frac{5288}{7590375} a^{5} + \frac{2060033}{11560725} a^{3} + \frac{33829}{462429} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{40}\times C_{47560}$, which has order $243507200$ (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:  \( 3096919.8109767493 \) (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

\(\Q(\sqrt{195}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{78}) \), \(\Q(\sqrt{130}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{10}, \sqrt{78})\), \(\Q(\sqrt{6}, \sqrt{130})\), \(\Q(\sqrt{13}, \sqrt{15})\), \(\Q(\sqrt{10}, \sqrt{13})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{15}, \sqrt{78})\), \(\Q(\sqrt{6}, \sqrt{13})\), 4.0.562432000.3, 4.0.5061888000.8, 4.0.562432000.6, 4.0.5061888000.5, 8.8.94758543360000.10, 8.0.102490840498176000000.5, 8.0.102490840498176000000.4, 8.0.316329754624000000.4, 8.0.25622710124544000000.16, 8.0.102490840498176000000.6, 8.0.102490840498176000000.7

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 ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{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}$
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$