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Properties

Label 16.0.40216693307...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 11329^{4}$
Root discriminant $34.50$
Ramified primes $5, 11329$
Class number $22$
Class group $[22]$
Galois group 16T1496

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32141, -28815, 34438, -14272, 16238, -7487, 4840, -2325, 1251, -678, 425, -181, 120, -30, 17, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 17*x^14 - 30*x^13 + 120*x^12 - 181*x^11 + 425*x^10 - 678*x^9 + 1251*x^8 - 2325*x^7 + 4840*x^6 - 7487*x^5 + 16238*x^4 - 14272*x^3 + 34438*x^2 - 28815*x + 32141)
 
gp: K = bnfinit(x^16 - 2*x^15 + 17*x^14 - 30*x^13 + 120*x^12 - 181*x^11 + 425*x^10 - 678*x^9 + 1251*x^8 - 2325*x^7 + 4840*x^6 - 7487*x^5 + 16238*x^4 - 14272*x^3 + 34438*x^2 - 28815*x + 32141, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 17 x^{14} - 30 x^{13} + 120 x^{12} - 181 x^{11} + 425 x^{10} - 678 x^{9} + 1251 x^{8} - 2325 x^{7} + 4840 x^{6} - 7487 x^{5} + 16238 x^{4} - 14272 x^{3} + 34438 x^{2} - 28815 x + 32141 \)

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:  \(4021669330769062744140625=5^{12}\cdot 11329^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.50$
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:  $5, 11329$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{119} a^{14} - \frac{5}{17} a^{13} + \frac{40}{119} a^{12} - \frac{48}{119} a^{11} - \frac{22}{119} a^{10} + \frac{22}{119} a^{9} - \frac{30}{119} a^{8} + \frac{41}{119} a^{7} - \frac{57}{119} a^{6} + \frac{30}{119} a^{5} - \frac{3}{7} a^{4} - \frac{18}{119} a^{3} - \frac{7}{17} a^{2} - \frac{2}{17} a + \frac{47}{119}$, $\frac{1}{850319467327205480831164589231} a^{15} - \frac{44781997962911159438941952}{850319467327205480831164589231} a^{14} - \frac{148976130972458681012550011501}{850319467327205480831164589231} a^{13} + \frac{285937061327059667363954173595}{850319467327205480831164589231} a^{12} - \frac{244813437064290274432747630698}{850319467327205480831164589231} a^{11} - \frac{423887365145307353448169035077}{850319467327205480831164589231} a^{10} - \frac{287807970840273803539432272632}{850319467327205480831164589231} a^{9} + \frac{360272669066856777999733094790}{850319467327205480831164589231} a^{8} - \frac{60430415772245049073269060137}{850319467327205480831164589231} a^{7} - \frac{20479431755183248581308710367}{50018792195717969460656740543} a^{6} - \frac{347543962907036022705623712890}{850319467327205480831164589231} a^{5} - \frac{5399951491459463133420800119}{121474209618172211547309227033} a^{4} - \frac{237116432472938332043233806840}{850319467327205480831164589231} a^{3} - \frac{28537581544459287865767636413}{121474209618172211547309227033} a^{2} - \frac{384876596956403028720174483204}{850319467327205480831164589231} a - \frac{106661039606693796765991385141}{850319467327205480831164589231}$

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

Class group and class number

$C_{22}$, which has order $22$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 19287.1768698 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1496:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 40 conjugacy class representatives for t16n1496
Character table for t16n1496 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.1416125.1, 8.4.177015625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
11329Data not computed