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Properties

Label 16.0.26268600651...000.91
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 17^{12}$
Root discriminant $387.90$
Ramified primes $2, 3, 5, 17$
Class number $127296000$ (GRH)
Class group $[2, 2, 2, 2, 10, 30, 26520]$ (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![2642656640625, 0, 1127533500000, 0, 184097479500, 0, 14709522000, 0, 619088575, 0, 13640800, 0, 146540, 0, 680, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 680*x^14 + 146540*x^12 + 13640800*x^10 + 619088575*x^8 + 14709522000*x^6 + 184097479500*x^4 + 1127533500000*x^2 + 2642656640625)
 
gp: K = bnfinit(x^16 + 680*x^14 + 146540*x^12 + 13640800*x^10 + 619088575*x^8 + 14709522000*x^6 + 184097479500*x^4 + 1127533500000*x^2 + 2642656640625, 1)
 

Normalized defining polynomial

\( x^{16} + 680 x^{14} + 146540 x^{12} + 13640800 x^{10} + 619088575 x^{8} + 14709522000 x^{6} + 184097479500 x^{4} + 1127533500000 x^{2} + 2642656640625 \)

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:  \(262686006513622818702917369856000000000000=2^{48}\cdot 3^{8}\cdot 5^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $387.90$
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:  $2, 3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4080=2^{4}\cdot 3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{4080}(1,·)$, $\chi_{4080}(3013,·)$, $\chi_{4080}(3911,·)$, $\chi_{4080}(1801,·)$, $\chi_{4080}(13,·)$, $\chi_{4080}(3277,·)$, $\chi_{4080}(2449,·)$, $\chi_{4080}(4067,·)$, $\chi_{4080}(2197,·)$, $\chi_{4080}(1883,·)$, $\chi_{4080}(1631,·)$, $\chi_{4080}(803,·)$, $\chi_{4080}(2279,·)$, $\chi_{4080}(169,·)$, $\chi_{4080}(1067,·)$, $\chi_{4080}(4079,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{85} a^{4}$, $\frac{1}{85} a^{5}$, $\frac{1}{425} a^{6} - \frac{2}{5} a^{2}$, $\frac{1}{425} a^{7} - \frac{2}{5} a^{3}$, $\frac{1}{7225} a^{8}$, $\frac{1}{36125} a^{9} + \frac{1}{425} a^{5} - \frac{1}{5} a$, $\frac{1}{541875} a^{10} - \frac{1}{21675} a^{8} - \frac{4}{6375} a^{6} + \frac{1}{255} a^{4} - \frac{26}{75} a^{2}$, $\frac{1}{541875} a^{11} + \frac{1}{108375} a^{9} - \frac{4}{6375} a^{7} - \frac{4}{1275} a^{5} - \frac{26}{75} a^{3} - \frac{2}{5} a$, $\frac{1}{1298874375} a^{12} + \frac{52}{76404375} a^{10} + \frac{296}{15280875} a^{8} - \frac{103}{898875} a^{6} + \frac{319}{179775} a^{4} - \frac{193}{1175} a^{2} - \frac{7}{47}$, $\frac{1}{3896623125} a^{13} - \frac{89}{229213125} a^{11} + \frac{31}{9168525} a^{9} + \frac{2576}{2696625} a^{7} - \frac{1232}{539325} a^{5} + \frac{2758}{10575} a^{3} + \frac{59}{705} a$, $\frac{1}{717165517561959375} a^{14} - \frac{1057052}{3335653570055625} a^{12} + \frac{24318106}{38177562819375} a^{10} - \frac{32470251256}{1687448276616375} a^{8} + \frac{91398702289}{99261663330375} a^{6} - \frac{1260678278}{315116391525} a^{4} - \frac{1723665758}{5190152331} a^{2} - \frac{740701914}{1730050777}$, $\frac{1}{10757482763429390625} a^{15} - \frac{3625163}{50034803550834375} a^{13} - \frac{2423762212}{9735278518940625} a^{11} - \frac{143009453029}{25311724149245625} a^{9} + \frac{1161563941432}{1488924949955625} a^{7} - \frac{23995758806}{14180237618625} a^{5} - \frac{9668535322}{1946307124125} a^{3} - \frac{3044429554}{8650253885} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{10}\times C_{30}\times C_{26520}$, which has order $127296000$ (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:  \( 3083829.668996395 \) (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

\(\Q(\sqrt{51}) \), \(\Q(\sqrt{255}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{170}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{5}, \sqrt{51})\), \(\Q(\sqrt{30}, \sqrt{51})\), \(\Q(\sqrt{6}, \sqrt{34})\), \(\Q(\sqrt{6}, \sqrt{170})\), \(\Q(\sqrt{30}, \sqrt{34})\), \(\Q(\sqrt{5}, \sqrt{34})\), \(\Q(\sqrt{5}, \sqrt{6})\), 4.0.11319552000.2, 4.0.1257728000.5, 4.0.1257728000.8, 4.0.11319552000.7, 8.8.277102632960000.13, 8.0.512529029922816000000.1, 8.0.512529029922816000000.4, 8.0.512529029922816000000.13, 8.0.512529029922816000000.7, 8.0.128132257480704000000.37, 8.0.1581879721984000000.5

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$17$17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$