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Properties

Label 16.12.5136202967...0000.1
Degree $16$
Signature $[12, 2]$
Discriminant $2^{28}\cdot 5^{10}\cdot 241^{6}$
Root discriminant $71.93$
Ramified primes $2, 5, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T869

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2416, 9216, -20304, -54496, 45396, 50352, 25448, -28648, -41518, 1964, 9972, 176, -719, 18, -1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - x^14 + 18*x^13 - 719*x^12 + 176*x^11 + 9972*x^10 + 1964*x^9 - 41518*x^8 - 28648*x^7 + 25448*x^6 + 50352*x^5 + 45396*x^4 - 54496*x^3 - 20304*x^2 + 9216*x + 2416)
 
gp: K = bnfinit(x^16 - 2*x^15 - x^14 + 18*x^13 - 719*x^12 + 176*x^11 + 9972*x^10 + 1964*x^9 - 41518*x^8 - 28648*x^7 + 25448*x^6 + 50352*x^5 + 45396*x^4 - 54496*x^3 - 20304*x^2 + 9216*x + 2416, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - x^{14} + 18 x^{13} - 719 x^{12} + 176 x^{11} + 9972 x^{10} + 1964 x^{9} - 41518 x^{8} - 28648 x^{7} + 25448 x^{6} + 50352 x^{5} + 45396 x^{4} - 54496 x^{3} - 20304 x^{2} + 9216 x + 2416 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(513620296716624855040000000000=2^{28}\cdot 5^{10}\cdot 241^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.93$
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:  $2, 5, 241$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{180} a^{12} + \frac{2}{45} a^{11} - \frac{1}{36} a^{10} - \frac{1}{45} a^{9} + \frac{7}{180} a^{8} - \frac{7}{18} a^{7} + \frac{3}{10} a^{6} - \frac{1}{9} a^{5} + \frac{22}{45} a^{4} + \frac{13}{45} a^{3} - \frac{1}{9} a^{2} - \frac{14}{45} a + \frac{16}{45}$, $\frac{1}{540} a^{13} + \frac{1}{540} a^{12} - \frac{1}{540} a^{11} + \frac{31}{540} a^{10} + \frac{7}{108} a^{9} - \frac{29}{540} a^{8} + \frac{16}{135} a^{7} - \frac{47}{135} a^{6} - \frac{16}{45} a^{5} + \frac{31}{90} a^{4} + \frac{8}{45} a^{3} + \frac{22}{45} a^{2} + \frac{8}{45} a - \frac{37}{135}$, $\frac{1}{842400} a^{14} - \frac{1}{1200} a^{13} - \frac{803}{842400} a^{12} + \frac{961}{421200} a^{11} + \frac{70087}{842400} a^{10} - \frac{7271}{105300} a^{9} + \frac{5333}{140400} a^{8} + \frac{7}{24} a^{7} - \frac{145757}{421200} a^{6} + \frac{257}{900} a^{5} + \frac{4091}{23400} a^{4} + \frac{5801}{17550} a^{3} - \frac{14363}{70200} a^{2} + \frac{1601}{4050} a - \frac{33349}{105300}$, $\frac{1}{4145810606510573808475200} a^{15} - \frac{875187969979639}{19931781762070066386900} a^{14} - \frac{1492240866123821016863}{4145810606510573808475200} a^{13} - \frac{1379476775447667542461}{518226325813821726059400} a^{12} + \frac{2678272194750627242239}{78222841632274977518400} a^{11} - \frac{101839840491811776628459}{2072905303255286904237600} a^{10} + \frac{166452585312265104045059}{2072905303255286904237600} a^{9} + \frac{146651981554644340277}{2657570901609342184920} a^{8} + \frac{6019681946101038676811}{39111420816137488759200} a^{7} - \frac{37364604320488477624829}{79727127048280265547600} a^{6} + \frac{26016488691858939219331}{115161405736404828013200} a^{5} - \frac{62306482208126187833003}{172742108604607242019800} a^{4} + \frac{119221000088220675044597}{345484217209214484039600} a^{3} + \frac{2694587006479153765987}{39863563524140132773800} a^{2} + \frac{252184055778476462862451}{518226325813821726059400} a + \frac{776035493349881453473}{3986356352414013277380}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 4913128294.03 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T869:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n869
Character table for t16n869 is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), 4.4.385600.1, 4.4.385600.2, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.148687360000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
241Data not computed