↑

Properties

Label 16.0.71594983564...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 11^{2}\cdot 101^{6}\cdot 1511^{2}$
Root discriminant $63.60$
Ramified primes $5, 11, 101, 1511$
Class number $2480$ (GRH)
Class group $[2, 2, 620]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T707)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4954805, -576065, 3251335, -919720, 1300506, -437821, 348382, -128672, 62333, -23117, 8548, -2332, 698, -152, 32, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 32*x^14 - 152*x^13 + 698*x^12 - 2332*x^11 + 8548*x^10 - 23117*x^9 + 62333*x^8 - 128672*x^7 + 348382*x^6 - 437821*x^5 + 1300506*x^4 - 919720*x^3 + 3251335*x^2 - 576065*x + 4954805)
 
gp: K = bnfinit(x^16 - 6*x^15 + 32*x^14 - 152*x^13 + 698*x^12 - 2332*x^11 + 8548*x^10 - 23117*x^9 + 62333*x^8 - 128672*x^7 + 348382*x^6 - 437821*x^5 + 1300506*x^4 - 919720*x^3 + 3251335*x^2 - 576065*x + 4954805, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 32 x^{14} - 152 x^{13} + 698 x^{12} - 2332 x^{11} + 8548 x^{10} - 23117 x^{9} + 62333 x^{8} - 128672 x^{7} + 348382 x^{6} - 437821 x^{5} + 1300506 x^{4} - 919720 x^{3} + 3251335 x^{2} - 576065 x + 4954805 \)

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:  \(71594983564208250058837890625=5^{12}\cdot 11^{2}\cdot 101^{6}\cdot 1511^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.60$
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:  $5, 11, 101, 1511$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{10} + \frac{1}{5} a^{8} + \frac{1}{5} a^{6} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{11} + \frac{1}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{5}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{4}$, $\frac{1}{30443068069952669396504836471091769679351225} a^{15} + \frac{1983915274083328533007918724989662318294313}{30443068069952669396504836471091769679351225} a^{14} + \frac{1785483134235338903586569156897138930164644}{30443068069952669396504836471091769679351225} a^{13} - \frac{1968086287172453127600661840068024298888216}{30443068069952669396504836471091769679351225} a^{12} + \frac{6629038016559080187005590972552572811207689}{30443068069952669396504836471091769679351225} a^{11} + \frac{11052436560065127098858264526383904886927834}{30443068069952669396504836471091769679351225} a^{10} - \frac{11165000990121857583157937777846723150029261}{30443068069952669396504836471091769679351225} a^{9} + \frac{963150459241727016234904006064695135105499}{30443068069952669396504836471091769679351225} a^{8} - \frac{3134556268636634949057007529060603172838291}{30443068069952669396504836471091769679351225} a^{7} + \frac{12589338112122164997877047583026464652009074}{30443068069952669396504836471091769679351225} a^{6} + \frac{11362738811085546692220126486735390353285733}{30443068069952669396504836471091769679351225} a^{5} - \frac{14914456785307281995406318068366409141205219}{30443068069952669396504836471091769679351225} a^{4} - \frac{308307485409376796661747475947466827737974}{1217722722798106775860193458843670787174049} a^{3} - \frac{1679024545800700378264348422415224421665339}{6088613613990533879300967294218353935870245} a^{2} - \frac{1175494755230946460923117034664409358658074}{6088613613990533879300967294218353935870245} a - \frac{2747898178212464248103952914944127815177709}{6088613613990533879300967294218353935870245}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{620}$, which has order $2480$ (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:  \( 38120.6275869 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T707):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.8.16098453125.1, 8.0.53514477878125.2, 8.0.529846315625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$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}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$101$101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
1511Data not computed